# A Novel State-Centric Necessary Condition for Time-Optimal Control of
Controllable Linear Systems Based on Augmented Switching Laws

Yunan Wang, Chuxiong Hu, Yujie Lin, Zeyang Li, Shize Lin, and Suqin He
Corresponding author: Chuxiong Hu (e-mail: cxhu@tsinghua.edu.cn). The authors
are all with the Department of Mechanical Engineering, Tsinghua University,
Beijing 100084, China.

###### Abstract

Most existing necessary conditions for optimal control based on adjoining
methods require both state information and costate information, yet the lack
of costates for a given feasible trajectory in practice impedes the
determination of optimality. This paper establishes a novel theoretical
framework for time-optimal control of controllable linear systems, proposing
the augmented switching law that represents the input control and the
feasibility in a compact form. Given a feasible trajectory, the disturbed
trajectory under the constraints of augmented switching law is guaranteed to
be feasible, resulting in a novel state-centric necessary condition without
dependence on costate information. A first order necessary condition is
proposed that the Jacobian matrix of the augmented switching law is not full
row rank, which also results in an approach to optimizing a given feasible
trajectory further. The proposed necessary condition is applied to the chain-
of-integrators systems with full box constraints, contributing to some
conclusions challenging to reason by traditional costate-based necessary
conditions.

###### Index Terms:

Optimal control, linear systems, variational methods, switched systems,
necessary condition.

##  I Introduction

Time-optimal control for controllable linear systems achieves universal
applications in aerospace [1], manufacturing [2, 3], robotic control [4, 5],
and autonomous driving [6]. Numerous works have been conducted on the
behaviors of the optimal controls, resulting in some well-known necessary or
sufficient conditions [7]. However, while referring to existing conclusions of
optimal control for problems in general forms, it remains difficult to fully
solve a high order problem in practice, especially when state inequality
constraints are introduced into the system [8]. Specifically, when given a
planned feasible trajectory, it is challenging to determine whether the
trajectory is optimal based on state information since most existing necessary
conditions require costate information.

In the domain of optimal control theory, numerous necessary conditions are
developed based on the variational method [9] and Pontryagin’s maximum
principle (PMP) [10]. The former one provides a necessary condition on the
extreme of a functional in an open feasible set, while PMP has the potential
to deal with inequality constraints on states or controls [11]. Among them,
the state-control inequality constraints pose a challenge in solution since
they involve the connection of different arcs. Chang [12], Dreyfus [13], and
other scholars developed the direct adjoining method for the connection of
unconstrained arcs and constrained arcs, i.e., the junction conditions of
costates. Jacobson et al. [14] pointed out that constrained arcs are not
allowed when the state constraint is of odd order $p>1$, and instead, the
unconstrained arc is tangent to the constrained boundary at a single point.
Makowski and Neustadt [15] generalized the PMP-based necessary condition to
the mixed constraints of state and control. Bryson et al. [16] developed the
necessary condition based on the indirect adjoining method, further
supplemented by Kreindler [17]. Hartl et al. [7] systematically reviewed
existing PMP-based necessary conditions and summarized their general forms.
The necessary conditions are usually applied to rule out some forms of non-
optimal controls in practice. In some simple cases, the optimal control can be
fully solved based on some necessary conditions [18, 19], while in other
cases, the necessary conditions are applied to guide the design of suboptimal
trajectories [8]. However, the above necessary conditions require information
on the states and the costates simultaneously, while the costates are not
directly observable in practice. Given a planned feasible trajectory, it is
challenging to determine whether the trajectory is optimal based on the pure
information on states, let alone the further optimization based on the given
trajectory.

In this paper, a necessary condition is called to be state-centric if it only
requires information on states and controls, without dependence on costate
information. There have existed some state-centric necessary so far, most of
which are based on Bellman’s principle of optimality [20]. Bellman’s principle
of optimality states that a sub-arc of an optimal trajectory is also optimal
for the induced sub-problem, where the costates are not introduced. The
Hamilton-Jacobi-Bellman (HJB) equation [21] represents a state-centric
necessary in the form of the partial differential equation, which is widely
used in optimal control [22] including reinforcement learning [23]. Evans and
James [24] investigated the HJB equation for time-optimal control. Wolenski
and Zhuang [25] showed that the minimal time function is the unique proximal
solution to the HJB equation. However, most numerical methods based on the HJB
equation require high computational cost [26], where the continuous time
domain is discretized into a grid. A trade-off exists between the
computational cost and the computational accuracy in practice. It remains
challenging to develop a state-centric necessary condition for time-optimal
control in a compact form with limited computational cost.

This paper sets out to establish a theoretical framework for time-optimal
control of controllable linear systems and develop a novel state-centric
necessary condition. The optimal control problem is formulated and
investigated based on Hamiltonian in Section II. Section III analyzes the
structure of arcs and the switching law of the optimal trajectory, where the
connections between adjacent arcs are fully discussed. The keypoints for
feasibility are introduced in Section IV, where the augmented switching law is
proposed to represent the control and the feasibility in a compact form. Based
on the uniqueness of the optimal control and the feasibility of the disturbed
trajectory, Section V develops a novel state-centric necessary condition for
time-optimal control. The proposed necessary condition is further applied to
the chain-of-integrators systems with full box constraints in Section VI as
examples. The contributions of this paper are as follows:

  1. 1.

This paper establishes an innovative theoretical framework for time-optimal
control of controllable linear systems, proposing the augmented switching law
which fully represents the input control and the feasibility of the Bang-
Singular-Bang trajectory in a compact form. The established framework focuses
on the control of each unconstrained arc and constrained arc, the constraints
induced by the connections of adjoining arcs, the feasibility of the disturbed
trajectory near the constrained boundary, and the optimality of a given
feasible trajectory. In the proposed theory, the control can be fully
represented by the augmented switching law and the motion time of each arc,
where the constraints directly induced by the augmented switching law can
guarantee the feasibility of the trajectory near the constrained boundary in a
sufficient and necessary sense. The developed framework provides a novel
variational approach as well as a local optimization method with feasibility
assurance, i.e., disturbing the motion time of each arc under fixed augmented
switching law, resulting in disturbed trajectories with feasibility
guarantees.

  2. 2.

This paper proposes a novel state-centric necessary condition for optimal
control of controllable linear systems, which only requires the state and
control information without dependence on costate information. Given a Bang-
Singular-Bang feasible trajectory which is claimed to be optimal, all
disturbed feasible trajectories should result in a terminal time which is
strictly longer than that of the original trajectory. In this way, a first
order necessary condition is proposed that the Jacobian matrix of the
augmented switching law should not be full row rank. The higher order
necessary conditions are also expected to be defined based on the strict local
optimality of the trajectory. Based on the proposed necessary condition, a
novel optimization approach with a given Bang-Singular-Bang feasible
trajectory as initial value is proposed, where the optimized trajectory always
has the same augmented switching law as the initial trajectory in a Bang-
Singular-Bang form and the feasibility is always guaranteed during iteration.

  3. 3.

The proposed state-centric necessary condition is applied to high-order chain-
of-integrators systems with full box constraints as examples, contributing to
some conclusions that are challenging to prove by traditional costate-based
necessary conditions, which demonstrates the effectiveness of the proposed
necessary condition. Note that time-optimal control for chain-of-integrators
systems with full box constraints represents a challenging and open problem in
optimal control theory and kinematics, yet to be resolved. The number of arcs
is proved to be less than the order of the system plus the sum of the order of
arcs’ constraints under some conditions, while the traditional methods have to
analyze the costates case-by-case. The proposed state-centric necessary
condition successfully proves the recursive equations of motion time if the
chattering phenomenon occurs and is induced by the 2nd order state
constraints, which is challenging based on costate analysis due to the high
degree-of-freedom of costates in high order problems. As a result, it is
hopeful to determine the existence of chattering induced by the 2nd order
state constraints in chain-of-integrators systems in future works.

##  II Problem Formulation and Assumptions

This section formulates the time-optimal control problem for controllable
linear systems and introduces some necessary assumptions. The optimal control
problem is defined as follows:

$\displaystyle\min_{u}\quad$ $\displaystyle
J=\int_{0}^{{t_{\mathrm{f}}}}\mathrm{d}t={t_{\mathrm{f}}},$ (1a)
$\displaystyle{\mathrm{s.t.}}\quad$
$\displaystyle\dot{{\boldsymbol{x}}}\left(t\right)={\boldsymbol{A}}x\left(t\right)+{\boldsymbol{b}}u\left(t\right),\,\forall
t\in\left[0,t_{\mathrm{f}}\right],$ (1b)
$\displaystyle{\boldsymbol{C}}{\boldsymbol{x}}\left(t\right)+{\boldsymbol{d}}\leq{\boldsymbol{0}},\,\forall
t\in\left[0,t_{\mathrm{f}}\right],$ (1c)
$\displaystyle{\boldsymbol{x}}\left(0\right)={\boldsymbol{x}}_{0},\,{\boldsymbol{x}}\left({t_{\mathrm{f}}}\right)={\boldsymbol{x}}_{t_{\mathrm{f}}},$
(1d) $\displaystyle\left|u\left(t\right)\right|\leq{u_{\mathrm{m}}},\,\forall
t\in\left[0,t_{\mathrm{f}}\right].$ (1e)

Among them, ${t_{\mathrm{f}}}$ is free.
${\boldsymbol{x}}=\left(x_{k}\right)_{k=1}^{n}\in{\mathbb{R}}^{n}$ is the
system state vector. The input control $u$ is subject to a box constraint
(1e). Denote that
${\boldsymbol{A}}=\left({\boldsymbol{a}}_{k}\right)_{k=1}^{n}$,
${\boldsymbol{b}}=\left(b_{k}\right)_{k=1}^{n}$,
${\boldsymbol{C}}=\left({\boldsymbol{c}}_{p}\right)_{p=1}^{P}$, and
${\boldsymbol{d}}=\left(d_{p}\right)_{p=1}^{P}$. Among them, the notation
$\left(\bullet\right)$ means $\left[\bullet\right]^{\top}$. Assume that
$\forall 1\leq p\leq P$, ${\boldsymbol{c}}_{p}\not={\boldsymbol{0}}$.

Assume that the system (1b) is controllable, i.e.,

${\mathrm{rank}}\left[{\boldsymbol{b}},{\boldsymbol{A}}{\boldsymbol{b}},{\boldsymbol{A}}^{2}{\boldsymbol{b}},\dots,{\boldsymbol{A}}^{n-1}{\boldsymbol{b}}\right]=n.$
(2)

######  Remark.

Generally, an equality constraint
${\boldsymbol{F}}{\boldsymbol{x}}\left(t\right)+{\boldsymbol{g}}={\boldsymbol{0}}$
can be introduced into problem (1a). However, it is not considered in this
paper since the equality constraint can be equivalently eliminated by a linear
transformation.

In order to solve (1a), the Hamiltonian is constructed as

$\displaystyle{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_{0},{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right)$
(3) $\displaystyle=$
$\displaystyle\lambda_{0}+{\boldsymbol{\lambda}}^{\top}\left({\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{b}}u\right)+{\boldsymbol{\eta}}^{\top}\left({\boldsymbol{C}}{\boldsymbol{x}}+{\boldsymbol{d}}\right).$

Among them, $\lambda_{0}\geq 0$ is a constant.
${\boldsymbol{\lambda}}=\left(\lambda_{k}\right)_{k=1}^{n}$ is the costate
vector. $\lambda_{0}$ and ${\boldsymbol{\lambda}}$ satisfy
$\left(\lambda_{0},{\boldsymbol{\lambda}}\left(t\right)\right)\not=0$. The
Euler-Lagrange equations [27] holds that

$\dot{\boldsymbol{\lambda}}=-\frac{\partial{\mathcal{H}}}{\partial{\boldsymbol{x}}}=-{\boldsymbol{A}}^{\top}{\boldsymbol{\lambda}}-{\boldsymbol{C}}^{\top}{\boldsymbol{\eta}}.$
(4)

In (3), ${\boldsymbol{\eta}}=\left(\eta_{p}\right)_{p=1}^{P}$ is the
multiplier vector induced by the state inequality constraint (1c), satisfying

$\eta_{p}\geq
0,\,\eta_{k}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)=0,\,\forall
1\leq p\leq P.$ (5)

Therefore, $\forall t\in\left[0,t_{\mathrm{f}}\right]$,
$\eta_{p}\left(t\right)>0$ only if
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}=0$.

Pontryagin’s maximum principle (PMP) [7] states that the optimal control
$u\left(t\right)$ minimizes the Hamiltonian ${\mathcal{H}}$, i.e.,

$u\left(t\right)\in\mathop{\arg\min}\limits_{\left|U\right|\leq{u_{\mathrm{m}}}}{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),U,\lambda_{0},{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right).$
(6)

By (3), ${\mathcal{H}}$ is affine in $u$; hence,

$u\left(t\right)=\begin{dcases}{u_{\mathrm{m}}},&{\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\left(t\right)<0,\\\
*,&{\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\left(t\right)=0,\\\
-{u_{\mathrm{m}}},&{\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\left(t\right)>0.\end{dcases}$
(7)

$u\left(t\right)\in\left[-{u_{\mathrm{m}}},{u_{\mathrm{m}}}\right]$ is
undetermined during ${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\equiv 0$,
which is called a singular arc. The notation “$\equiv$” means that the
equality holds for a continuous period. (7) is the well-known Bang-Singular-
Bang law [8].

Note that the objective function $J=\int_{0}^{{t_{\mathrm{f}}}}\mathrm{d}t$ is
in a Lagrangian form; hence, the continuity of the system is guaranteed by the
following equality, i.e.,

$\forall
t\in\left[0,t_{\mathrm{f}}\right],\,{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_{0},{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right)\equiv
0.$ (8)

For convenience of discussion, the following assumptions are introduced:

######  Assumption 1.

Assume that problem (1a) is feasible, and the optimal control exists.
Furthermore, assume that the chattering phenomenon does not occur in the
optimal profile.

######  Remark.

In Assumption 1, the chattering phenomenon [28] means that the optimal control
switches infinitely many times in a finite period. At a one-sided neighborhood
of a chattering limit time point, an infinite number of constrained arcs are
joined at the boundary of inequality state constraints, where chattering limit
time points are usually isolated [28]. If a chattering phenomenon occurs,
then, by Bellman’s principle of optimality [20], a sub-arc of the optimal
trajectory is also optimal in the sub-problem. Hence, a sub-arc that contains
no chattering limit time points can be considered in this paper.

In the following of this paper, Assumption 1 is considered to hold throughout
unless otherwise specified.

##  III Arc Representation of the Optimal Trajectory

An arc of a trajectory is defined as a continuous segment of the optimal
trajectory conditioned by the same input control. Two kinds of arcs exist,
i.e., unconstrained arcs and constrained arcs, which are discussed in Section
III-A and Section III-B, respectively. Through discussion on the two arcs, the
system behavior representing a single arc is defined in Section III-C.
Finally, if the chattering phenomenon does not occur, an optimal trajectory
can be represented by a switching law, i.e., a sequence of system behaviors,
as discussed in Section III-D.

###  III-A Unconstrained Arcs in Problem (1a)

In an unconstrained arc,
${\boldsymbol{C}}{\boldsymbol{x}}+{\boldsymbol{d}}<{\boldsymbol{0}}$ holds
strictly almost everywhere, and the control does not switch. The following
proposition states that $u\equiv\pm{u_{\mathrm{m}}}$ in an unconstrained arc.

######  Proposition 1.

In an unconstrained arc, ${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\not=0$
almost everywhere, and
$u\equiv-{u_{\mathrm{m}}}{\mathrm{sgn}}\left({\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\right)$.

###### Proof.

Assume that a singular condition holds in the unconstrained arc, i.e.,
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\equiv 0$ for a period. By (5),
${\boldsymbol{\eta}}\equiv{\boldsymbol{0}}$ in an unconstrained arc. Hence,
(4) implies that
$\dot{\boldsymbol{\lambda}}=-{\boldsymbol{A}}^{\top}{\boldsymbol{\lambda}}$.
Considering the $k$-th order derivative of
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\equiv 0$, it has

$\forall
k\in{\mathbb{N}},\,{\boldsymbol{\lambda}}^{\top}{\boldsymbol{A}}^{k}{\boldsymbol{b}}\equiv
0.$ (9)

By (2), ${\boldsymbol{\lambda}}\equiv 0$. Then, (8) implies that
$\lambda_{0}=0$, which contradicts
$\left(\lambda_{0},{\boldsymbol{\lambda}}\right)\not=0$. Therefore,
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\not\equiv 0$.∎

######  Remark.

By
$\dot{\boldsymbol{\lambda}}=-{\boldsymbol{A}}^{\top}{\boldsymbol{\lambda}}$,
either ${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}\equiv 0$ for a period or
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}$ crosses 0 for finite times,
while the former case is ruled out in an unconstrained arc by Proposition 1.
The control switches between $u\equiv{u_{\mathrm{m}}}$ and
$u\equiv-{u_{\mathrm{m}}}$ when
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}$ crosses 0, where two
unconstrained arcs are connected.

###  III-B Constrained Arcs in Problem (1a)

Compared to unconstrained arcs, constrained arcs exhibit more complex
behavior. In a constrained arc, $\exists 1\leq p\leq P$, s.t.
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$. The following
proposition characterizes an unconstrained arc.

######  Proposition 2.

If ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$ in a
constrained arc, then $\exists 1\leq r_{p}\leq n$, s.t. $\forall 1\leq
r<r_{p}$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r-1}{\boldsymbol{b}}=0$, and
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\not=0$.
During the constrained arc,

$u=-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}{\boldsymbol{x}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}},\,\dot{\boldsymbol{x}}=\left({\boldsymbol{A}}-\frac{{\boldsymbol{b}}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}\right){\boldsymbol{x}},$
(10)

and

$\begin{dcases}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0,\\\
\forall 1\leq
r<r_{p},\,{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r}{\boldsymbol{x}}\equiv
0.\end{dcases}$ (11)

Conversely, if the initial value ${\boldsymbol{x}}\left(t_{0}\right)$
satisfies

$\begin{dcases}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{0}\right)+d_{p}=0,\\\
\forall 1\leq
r<r_{p},\,{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r}{\boldsymbol{x}}\left(t_{0}\right)=0,\end{dcases}$
(12)

then under the driving of (10),
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$ holds.

###### Proof.

Since ${\boldsymbol{c}}_{p}\not={\boldsymbol{0}}$, (2) implies the existence
of the above $r_{p}$. Considering the $k$-th order derivative of
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$, (11) holds, and
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}\left({\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{b}}u\right)=0$.
Hence, (10) holds.

Assume that (10) holds and ${\boldsymbol{x}}\left(t_{0}\right)$ satisfies
(12). Note that
$\frac{\mathrm{d}^{r_{p}}}{\mathrm{d}t^{r_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}\left({\boldsymbol{A}}-\frac{{\boldsymbol{b}}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}\right){\boldsymbol{x}}=0$;
hence, the initial condition implies that (11) holds. ∎

In the above proposition, $r_{p}$ is called the order of the equality state
constraint ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}=0$. Similarly,
the analytical expression of a constrained arc can be obtained by Lemma 1.

Cases exist where multiple inequality constraints are active in a constrained
arc. For this case, the active constraints should be consistent. Specifically,
denote that
$\mathcal{P}=\left\\{p\in\left[P\right]:{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv
0\right\\}$, where $\left[P\right]\triangleq\left\\{1,2,\dots,P\right\\}$.
Then, $\forall p_{1},p_{2}\in\mathcal{P}$,
$\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p_{1}}}{\boldsymbol{x}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p_{1}}-1}{\boldsymbol{b}}}=\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p_{2}}}{\boldsymbol{x}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p_{2}}-1}{\boldsymbol{b}}}$
holds in the arc due to (10), and $\forall p\in\mathcal{P}$, (11) holds. The
following example illustrates the above analysis:

######  Example 1.

Consider an $n$-th order chain-of-integrators system, i.e.,

$\begin{dcases}\dot{x}_{k}\left(t\right)=x_{k-1}\left(t\right),&t\in\left[0,{t_{\mathrm{f}}}\right],\,\forall
1<k\leq n,\\\
\dot{x}_{1}\left(t\right)=u\left(t\right),&t\in\left[0,{t_{\mathrm{f}}}\right].\end{dcases}$
(13)

Two inequality constraints are introduced as $x_{1}+x_{3}\leq 1$ and
$x_{2}\leq 0$. Assume that both $x_{1}+x_{3}\equiv 1$ and $x_{2}\equiv 0$ hold
in a constrained arc. Then, $u\equiv-x_{2}=0$; hence, $x_{3}\equiv 1$,
$x_{2}\equiv 0$, and $x_{1}\equiv 0$. Therefore, $x_{1}+x_{3}\equiv 1$ and
$x_{2}\equiv 0$ are consistent. Similarly, $x_{1}+x_{3}\equiv 1$ and
$x_{2}\equiv 1$ are not consistent.

###  III-C System Behavior

Following the terminology introduced in our previous work [8], the system
behavior is defined as follows:

######  Definition 1.

A system behavior in problem (1a) is an arc with constant system dynamics,
denoted as
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$.
During the arc,
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}+\widehat{{\boldsymbol{b}}}$.
$\widehat{{\boldsymbol{F}}}{\boldsymbol{x}}+\widehat{{\boldsymbol{g}}}\equiv{\boldsymbol{0}}$
provides fully equality state constraints, where $\widehat{{\boldsymbol{F}}}$
has full row rank. $\mathcal{P}$ is the set of active inequality state
constraints.

In Definition 1, an unconstrained arc can be denoted as
${\mathcal{S}}=\left({\boldsymbol{A}},u_{0}{\boldsymbol{b}},\sim,\sim,\varnothing\right)$,
where $\sim$ means that no inequality constraints are introduced, and
$u_{0}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$ is a constant.

For a constrained arc, $\mathcal{P}\not=\varnothing$. (10) implies that
$\forall p\in\mathcal{P}$,
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{p}{\boldsymbol{x}}+\widehat{{\boldsymbol{b}}}_{p}$,
where

$\widehat{{\boldsymbol{A}}}_{p}={\boldsymbol{A}}-\frac{{\boldsymbol{b}}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}},\,\widehat{{\boldsymbol{b}}}_{p}={\boldsymbol{0}}.$
(14)

Although it allows that $\exists p_{1},p_{2}\in\mathcal{P}$, s.t.
$\widehat{{\boldsymbol{A}}}_{p_{1}}\not=\widehat{{\boldsymbol{A}}}_{p_{2}}$,
it is guaranteed by the consistency of active constraints that
$\widehat{{\boldsymbol{A}}}_{p_{1}}{\boldsymbol{x}}\equiv\widehat{{\boldsymbol{A}}}_{p_{2}}{\boldsymbol{x}}$
during the arc. For convenience of discussion, let
$\widehat{{\boldsymbol{A}}}$ be $\widehat{{\boldsymbol{A}}}_{p}$ with the
minimum $p\in\mathcal{P}$, i.e.,
$\widehat{{\boldsymbol{A}}}\triangleq\widehat{{\boldsymbol{A}}}_{\min\left(\mathcal{P}\right)}$.
Evidently,
$\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}}\right)$ is well-
defined in Definition 1.

The active inequality state constraints state that $\forall p\in\mathcal{P}$,
(11) holds. The above equalities induce
$\left(\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}}\right)$ where
$\widehat{{\boldsymbol{F}}}$ has full row rank. For well-posedness,
$\left(\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}}\right)$ are
determined by the equality constraints with minimum indices.

######  Example 2.

Consider a chain-of-integrators system (13) with full box state constraints,
i.e., $\forall 1\leq k\leq n$, $t\in\left[0,{t_{\mathrm{f}}}\right]$,
$-x_{\mathrm{m}k}\leq x_{k}\left(t\right)\leq x_{\mathrm{m}k}$. The
$\left(2k-1\right)$-th constraint is $-x_{k}\left(t\right)\leq
x_{\mathrm{m}k}$, and the $\left(2k\right)$-th constraint is
$x_{k}\left(t\right)\leq x_{\mathrm{m}k}$. A constrained arc $x_{k}\equiv
x_{\mathrm{m}k}$ is of $k$-th order, where $\forall 1\leq j<k$, $x_{j}\equiv
0$, and $u\equiv 0$. Then, $x_{k}\equiv x_{\mathrm{m}k}$ can be represented as
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},{\boldsymbol{0}},{\boldsymbol{I}}_{k\times
n},x_{\mathrm{m}k}{\boldsymbol{e}}_{k},\left\\{2k\right\\}\right)$. Among
them, $\widehat{{\boldsymbol{A}}}=\left(\hat{a}_{ij}\right)_{i,j=1}^{n}$,
where $\hat{a}_{ij}=1$ if $k<i=j+1\leq n$, and otherwise, $\hat{a}_{ij}=0$.

In a system behavior
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$,
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}+\widehat{{\boldsymbol{b}}}$
holds. Therefore, the analytical expression of
${\boldsymbol{x}}\left(t\right)$ in ${\mathcal{S}}$ can be given by the
following lemma.

######  Lemma 1.

The initial value problem (IVP) where
$\dot{\boldsymbol{x}}\left(t\right)={\boldsymbol{A}}{\boldsymbol{x}}\left(t\right)+{\boldsymbol{b}}\left(t\right)$,
$t\in{\mathbb{R}}$, ${\boldsymbol{x}}\left(t_{0}\right)={\boldsymbol{x}}_{0}$
has a unique solution:

$\forall
t\in{\mathbb{R}},\,{\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}\left(t-t_{0}\right)}{\boldsymbol{x}}_{0}+\int_{t_{0}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}\left(t-\tau\right)}{\boldsymbol{b}}\left(\tau\right)\mathrm{d}\tau.$
(15)

###### Proof.

$\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{e}}^{-{\boldsymbol{A}}t}{\boldsymbol{x}}\left(t\right)\right)={\mathrm{e}}^{-{\boldsymbol{A}}t}\left(\dot{\boldsymbol{x}}\left(t\right)-{\boldsymbol{A}}{\boldsymbol{x}}\left(t\right)\right)={\mathrm{e}}^{-{\boldsymbol{A}}t}{\boldsymbol{b}}\left(t\right)$
implies that
${\mathrm{e}}^{-{\boldsymbol{A}}t}{\boldsymbol{x}}\left(t\right)-{\mathrm{e}}^{-{\boldsymbol{A}}t_{0}}{\boldsymbol{x}}_{0}=\int_{t_{0}}^{t}{\mathrm{e}}^{-{\boldsymbol{A}}\tau}{\boldsymbol{b}}\left(\tau\right)\mathrm{d}\tau$.
∎

Lemma 1 implies that for an unconstrained arc where $u\equiv
u_{0}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$,

${\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}\left(t-t_{0}\right)}{\boldsymbol{x}}\left(t_{0}\right)+u_{0}\left(\int_{t_{0}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}\left(t-\tau\right)}\mathrm{d}\tau\right){\boldsymbol{b}}.$
(16)

For a constrained arc,

${\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{\widehat{{\boldsymbol{A}}}\left(t-t_{0}\right)}{\boldsymbol{x}}\left(t_{0}\right).$
(17)

For a given system behavior, ${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$
has an explicit elementary expression since every scalar element of
${\boldsymbol{e}}^{\widehat{{\boldsymbol{A}}}t}$ is composed of exponentials
and polynomials w.r.t. $t$.

###  III-D Switching Law

Based on the analysis in Section III-C, if the optimal trajectory consists of
a finite number of arcs, then the optimal trajectory can be represented by a
sequence of system behaviors. The switching law is defined as follows:

######  Definition 2.

Assume that the optimal trajectory in problem (1a) consists of a finite number
of arcs represented as
${\mathcal{S}}_{1},{\mathcal{S}}_{2},\dots,{\mathcal{S}}_{N}$ successively.
The switching law of the optimal trajectory is defined as
${\mathcal{L}}={\mathcal{S}}_{1}{\mathcal{S}}_{2}\dots\mathcal{S}_{N}$.

######  Remark.

The switching law focuses on the arcs that compose the optimal trajectory,
while the motion time of each arc is not included in a switching law.
Furthermore, Definition 2 does not characterize the chattering phenomenon
since chattering induces an infinite number of arcs.

The connection between arcs is analyzed in Proposition 3. Denote the two
adjacent arcs as ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$, where
${\mathcal{S}}_{1}\not={\mathcal{S}}_{2}$. $\forall i=1,2$,
${\mathcal{S}}_{i}=\left(\widehat{{\boldsymbol{A}}}_{i},\widehat{{\boldsymbol{b}}}_{i},\widehat{{\boldsymbol{F}}}_{i},\widehat{{\boldsymbol{g}}}_{i},\mathcal{P}_{i}\right)$.
Among them, ${\mathcal{S}}_{1}={\mathcal{S}}_{2}$ is defined as
$\widehat{{\boldsymbol{A}}}_{1}=\widehat{{\boldsymbol{A}}}_{2}$,
$\widehat{{\boldsymbol{b}}}_{1}=\widehat{{\boldsymbol{b}}}_{2}$, and
$\mathcal{P}_{1}=\mathcal{P}_{2}$ since $\widehat{{\boldsymbol{F}}}_{i}$ and
$\widehat{{\boldsymbol{g}}}_{i}$ can be induced by $\mathcal{P}_{i}$. Before
introducing Proposition 3, a lemma is provided as preparation.

######  Lemma 2.

Consider the IVP
$\dot{\boldsymbol{x}}\left(t\right)={\boldsymbol{A}}{\boldsymbol{x}}\left(t\right)+{\boldsymbol{b}}$,
$t\in{\mathbb{R}}$, where
${\boldsymbol{x}}\left(t_{0}\right)={\boldsymbol{x}}_{0}\in{\mathbb{R}}^{n}$,
${\boldsymbol{A}}\in{\mathbb{R}}^{n\times n}$,
${\boldsymbol{b}}\in{\mathbb{R}}^{n}$. If
${\boldsymbol{c}}\in{\mathbb{R}}^{n}$, s.t. $\forall 1\leq r\leq n$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}\right)}{\mathrm{d}t^{r}}\left(t_{0}\right)=0$,
then
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)\equiv{\mathrm{const}}$
on ${\mathbb{R}}$.

###### Proof.

According to Lemma 1, $\forall t\in{\mathbb{R}}$,
${\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}\left(t-t_{0}\right)}{\boldsymbol{x}}_{0}+\left(\int_{t_{0}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}\left(t-\tau\right)}\left(\tau\right)\mathrm{d}\tau\right){\boldsymbol{b}}$.
Considering the Jordan normal form of ${\boldsymbol{A}}$, it has $\forall
t\in{\mathbb{R}}$,
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)=\sum_{i=1}^{l}p_{i}\left(t-t_{0}\right){\boldsymbol{e}}^{\lambda_{i}\left(t-t_{0}\right)}$,
where $\left\\{\lambda_{i}\right\\}_{i=1}^{l}$ are the eigenvalues of
${\boldsymbol{A}}$, and $p_{i}$ is a polynomial of degree not greater than
$n$. Evidently, ${\boldsymbol{c}}^{\top}{\boldsymbol{x}}$ is an analytic
function; hence, $\forall t\in{\mathbb{R}}$, it has

${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)=\sum_{r=0}^{\infty}\frac{\left(t-t_{0}\right)^{r}}{r!}\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}\right)\right|_{t_{0}}.$
(18)

The case where ${\boldsymbol{A}}={\boldsymbol{0}}$ is trivial. Assume that
${\boldsymbol{A}}\not={\boldsymbol{0}}$. Considering the minimal polynomial of
${\boldsymbol{A}}$, $\exists p^{*}$, s.t. $p^{*}$ is a polynomial of degree
less than $n$ and ${\boldsymbol{A}}^{n}=p^{*}\left({\boldsymbol{A}}\right)$.
Therefore, $\forall r\geq n$, $\exists p_{n}^{*}$ s.t. $p_{n}^{*}$ is a
polynomial of degree less than $n$ and
${\boldsymbol{A}}^{r}=p_{n}^{*}\left({\boldsymbol{A}}\right)$.

Note that $\forall r\in{\mathbb{N}}^{*}$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}\right)}{\mathrm{d}t^{r}}={\boldsymbol{c}}^{\top}{\boldsymbol{A}}^{r-1}\left({\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{b}}\right)$.
Since $\forall 1\leq r\leq n$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}\right)}{\mathrm{d}t^{r}}\left(t_{0}\right)=0$,
it holds that $\forall r\geq n+1$,

$\displaystyle\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}\right)}{\mathrm{d}t^{r}}\left(t_{0}\right)={\boldsymbol{c}}^{\top}{\boldsymbol{A}}^{r-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{0}\right)+{\boldsymbol{b}}\right)$
(19) $\displaystyle=$
$\displaystyle{\boldsymbol{c}}^{\top}p_{r-1}^{*}\left({\boldsymbol{A}}\right)\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{0}\right)+{\boldsymbol{b}}\right)=0.$

By (18), $\forall t\in{\mathbb{R}}$,
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)\equiv{\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t_{0}\right)$.
∎

######  Proposition 3.

Assume that $\forall i=1,2$, ${\mathcal{S}}_{i}$ occurs for
$t\in\left[t_{i-1},t_{i}\right]$ in the above notation. ${\boldsymbol{x}}$
enters ${\mathcal{S}}_{2}$ from ${\mathcal{S}}_{1}$ at $t_{1}$.

  1. 1.

If ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ are unconstrained arcs, then
${\boldsymbol{b}}^{\top}{\boldsymbol{\lambda}}$ crosses 0 at $t_{1}$.
$\widehat{{\boldsymbol{b}}}_{1}=u_{1}{\boldsymbol{b}}$ and
$\widehat{{\boldsymbol{b}}}_{2}=-u_{1}{\boldsymbol{b}}$, where
$u_{1}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$.

  2. 2.

If ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ are constrained arcs, then
$\mathcal{P}_{1}\cap\mathcal{P}_{2}=\varnothing$. Furthermore, $\forall
p\in\mathcal{P}_{1}$, $\exists r_{p}\leq\hat{r}_{p}\leq n$, s.t.

$\begin{dcases}{\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}_{2}^{\hat{r}_{p}}{\boldsymbol{x}}\left(t_{1}\right)<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,{\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}_{2}^{r}{\boldsymbol{x}}\left(t_{1}\right)=0.\end{dcases}$
(20)

$\forall p\in\mathcal{P}_{2}$, $\exists r_{p}\leq\hat{r}_{p}\leq n$, s.t.

$\begin{dcases}\left(-1\right)^{\hat{r}_{p}}{\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}_{1}^{\hat{r}_{p}}{\boldsymbol{x}}\left(t_{1}\right)<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,{\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}_{1}^{r}{\boldsymbol{x}}\left(t_{1}\right)=0.\end{dcases}$
(21)

Conversely, assume that
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{1}{\boldsymbol{x}}$ for
$\left(t_{0},t_{1}\right)$ and
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{2}{\boldsymbol{x}}$ for
$\left(t_{1},t_{2}\right)$. If ${\boldsymbol{x}}\left(t_{1}\right)$ satisfies
(20) for $p\in\mathcal{P}_{1}$, (21) for $p\in\mathcal{P}_{2}$, and (12) for
$p\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$, then $\exists\varepsilon>0$,
${\mathcal{S}}_{1}$ occurs for $\left(t_{1}-\varepsilon,t_{1}\right)$ and
${\mathcal{S}}_{2}$ occurs for $\left(t_{1},t_{1}+\varepsilon\right)$.

  3. 3.

If ${\mathcal{S}}_{1}$ is a constrained arc and ${\mathcal{S}}_{2}$ is an
unconstrained arc, then $\forall p,q\in\mathcal{P}_{1}$,
${\mathrm{sgn}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\right)={\mathrm{sgn}}\left({\boldsymbol{c}}_{q}^{\top}{\boldsymbol{A}}^{r_{q}-1}{\boldsymbol{b}}\right)$.
On ${\mathcal{S}}_{2}$, $\forall p\in\mathcal{P}_{1}$, it has

$u\equiv
u_{2}=-{\mathrm{sgn}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\right){u_{\mathrm{m}}}.$
(22)

Furthermore, $\forall p\in\mathcal{P}_{1}$, $\exists r_{p}\leq\hat{r}_{p}\leq
n$, s.t.

$\begin{dcases}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{\hat{r}_{p}-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{2}{\boldsymbol{b}}\right)<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{2}{\boldsymbol{b}}\right)=0.\end{dcases}$
(23)

Conversely, assume that
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{1}{\boldsymbol{x}}$ for
$\left(t_{0},t_{1}\right)$ and
$\dot{\boldsymbol{x}}={\boldsymbol{A}}{\boldsymbol{x}}+u_{2}{\boldsymbol{b}}$
for $\left(t_{1},t_{2}\right)$ where $u_{2}$ is given in (22). If
${\boldsymbol{x}}\left(t_{1}\right)$ satisfies (23) and (12) for
$p\in\mathcal{P}_{1}$, then $\exists\varepsilon>0$, ${\mathcal{S}}_{1}$ occurs
for $\left(t_{1}-\varepsilon,t_{1}\right)$ and ${\mathcal{S}}_{2}$ occurs for
$\left(t_{1},t_{1}+\varepsilon\right)$.

  4. 4.

If ${\mathcal{S}}_{1}$ is an unconstrained arc and ${\mathcal{S}}_{2}$ is a
constrained arc, then $\forall p,q\in\mathcal{P}_{2}$,
${\mathrm{sgn}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\right)={\mathrm{sgn}}\left({\boldsymbol{c}}_{q}^{\top}{\boldsymbol{A}}^{r_{q}-1}{\boldsymbol{b}}\right)$.
On ${\mathcal{S}}_{1}$, $\forall p\in\mathcal{P}_{2}$, it has

$u\equiv
u_{1}=\left(-1\right)^{r_{p}-1}{\mathrm{sgn}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\right){u_{\mathrm{m}}}.$
(24)

Furthermore, $\forall p\in\mathcal{P}_{2}$, $\exists r_{p}\leq\hat{r}_{p}\leq
n$, s.t.

$\begin{dcases}\left(-1\right)^{\hat{r}_{p}}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{\hat{r}_{p}-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{1}{\boldsymbol{b}}\right)<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{1}{\boldsymbol{b}}\right)=0.\end{dcases}$
(25)

Conversely, assume that
$\dot{\boldsymbol{x}}={\boldsymbol{A}}{\boldsymbol{x}}+u_{1}{\boldsymbol{b}}$
for $\left(t_{0},t_{1}\right)$ and
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{1}{\boldsymbol{x}}$ for
$\left(t_{1},t_{2}\right)$ where $u_{1}$ is given in (24). If
${\boldsymbol{x}}\left(t_{1}\right)$ satisfies (25) and (12) for
$p\in\mathcal{P}_{2}$, then $\exists\varepsilon>0$, ${\mathcal{S}}_{1}$ occurs
for $\left(t_{1}-\varepsilon,t_{1}\right)$ and ${\mathcal{S}}_{2}$ occurs for
$\left(t_{1},t_{1}+\varepsilon\right)$.

###### Proof.

Consider the connection between two unconstrained arcs. By Proposition 1,
assume that $\forall i=1,2$, $u\equiv u_{i}$ on ${\mathcal{S}}_{i}$, where
$u_{i}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$. Then,
$\widehat{{\boldsymbol{b}}}_{i}=u_{i}{\boldsymbol{b}}$. Since
$\widehat{{\boldsymbol{A}}}_{1}=\widehat{{\boldsymbol{A}}}_{2}={\boldsymbol{A}}$
and $\mathcal{P}_{1}=\mathcal{P}_{2}=\varnothing$, it holds that
$\widehat{{\boldsymbol{b}}}_{1}\not=\widehat{{\boldsymbol{b}}}_{2}$; hence,
$u_{1}=-u_{2}$. Therefore, Proposition 3-1 holds.

Consider the connection between two constrained arcs. Assume that
$\mathcal{P}_{1}\cap\mathcal{P}_{2}\not=\varnothing$. Arbitrarily take a
$p^{*}\in\mathcal{P}_{1}\cap\mathcal{P}_{2}$ into consideration. According to
Proposition 2, $\forall t\in\left[t_{0},t_{2}\right]$,
$\dot{\boldsymbol{x}}=\left({\boldsymbol{A}}-\frac{{\boldsymbol{b}}{\boldsymbol{c}}_{p^{*}}^{\top}{\boldsymbol{A}}^{r_{p^{*}}}}{{\boldsymbol{c}}_{p^{*}}^{\top}{\boldsymbol{A}}^{r_{p^{*}}-1}{\boldsymbol{b}}}\right){\boldsymbol{x}}$
holds. Evidently, $\forall p\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d_{p}\equiv 0$,
and $\forall 1\leq r<r_{p}$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r}{\boldsymbol{x}}\left(t_{1}\right)\equiv
0$. By Proposition 2, $\forall t\in\left[t_{0},t_{2}\right]$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$ holds. Therefore,
$\mathcal{P}_{1}=\mathcal{P}_{2}$, which contradicts
${\mathcal{S}}_{1}\not={\mathcal{S}}_{2}$. Hence,
$\mathcal{P}_{1}\cap\mathcal{P}_{2}=\varnothing$.

Furthermore, consider the monotonicity of
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+{\boldsymbol{d}}_{p}$ at $t_{1}$
to guarantee the feasibility of ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$.
$\exists\varepsilon>0$, s.t. $\forall p\in\mathcal{P}_{1}$,
$t\in\left(t_{1},t_{1}+\varepsilon\right)$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+{\boldsymbol{d}}_{p}<0$.
Note that on ${\mathcal{S}}_{2}$,
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{2}{\boldsymbol{x}}$ and
$\forall r\in{\mathbb{N}}^{*}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}={\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}_{2}^{r}{\boldsymbol{x}}\left(t_{1}\right)$.
Lemma 2 implies that $\exists 1\leq\hat{r}_{p}\leq n$, s.t.
$\left.\frac{\mathrm{d}^{\hat{r}_{p}}}{\mathrm{d}t^{\hat{r}_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}<0$
and $\forall 1\leq r<\hat{r}_{p}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0$.
Since $\forall 1\leq r<r_{p}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0$,
it has $\hat{r}_{p}\geq r_{p}$. In other words, (20) holds. For the same
reason, (21) holds.

Conversely, assume that
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{1}{\boldsymbol{x}}$ for
$\left(t_{0},t_{1}\right)$ and
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{2}{\boldsymbol{x}}$ for
$\left(t_{1},t_{2}\right)$. Assume that ${\boldsymbol{x}}\left(t_{1}\right)$
satisfies (20) for $p\in\mathcal{P}_{1}$, (21) for $p\in\mathcal{P}_{2}$, and
(12) for $p\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$. Then, $\forall
p\in\mathcal{P}_{1}$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d_{p}=0$ and

$\begin{dcases}\left.\frac{\mathrm{d}^{\hat{r}_{p}}}{\mathrm{d}t^{\hat{r}_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0.\end{dcases}$
(26)

Hence, $\exists\varepsilon>0$, $\forall p\in\mathcal{P}_{1}$,
$t\in\left(t_{1},t_{1}+\varepsilon\right)$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$. For the
same reason, $\exists\varepsilon^{\prime}\in\left(0,\varepsilon\right)$,
$\forall p\in\mathcal{P}_{2}$,
$t\in\left(t_{1}-\varepsilon^{\prime},t_{1}\right)$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$. Hence,
Proposition 3-2 holds.

Consider the connection between an unconstrained arc and a constrained arc.
Assume that ${\boldsymbol{x}}$ enters an unconstrained arc ${\mathcal{S}}_{2}$
from a constrained arc ${\mathcal{S}}_{1}$. By Proposition 2, $\forall
p\in\mathcal{P}_{1}$, $r\in{\mathbb{N}}^{*}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{-}}=0$;
hence,

$\begin{dcases}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d_{p}=0,\\\
\forall 1\leq
r<r_{p},\,{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r}{\boldsymbol{x}}\left(t_{1}\right)=0,\\\
{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u\left(t_{1}^{-}\right){\boldsymbol{b}}\right)=0.\end{dcases}$
(27)

Assume that $u\equiv u_{2}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$ on
${\mathcal{S}}_{2}$. Then, $\forall 1\leq r<r_{p}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r}{\boldsymbol{x}}\left(t_{1}\right)=0$
since ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r-1}{\boldsymbol{b}}=0$.
Since $\exists\varepsilon>0$, s.t. $\forall
t\in\left(t_{1},t_{1}+\varepsilon\right)$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$, it has
$\left.\frac{\mathrm{d}^{r_{p}}}{\mathrm{d}t^{r_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{2}{\boldsymbol{b}}\right)\leq
0$. Therefore,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\left(u_{2}-u\left(t_{1}^{-}\right)\right)\leq
0$. Since
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}\not=0$
and
$\left|u\left(t_{1}^{-}\right)\right|\leq\left|u_{2}\right|={u_{\mathrm{m}}}$,
(22) holds.

Furthermore, consider the monotonicity of
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+{\boldsymbol{d}}_{p}$ at $t_{1}$
to guarantee the feasibility of ${\mathcal{S}}_{2}$ where
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+{\boldsymbol{d}}_{p}<0$ for
$\left(t_{1},t_{1}+\varepsilon\right)$. Note that on ${\mathcal{S}}_{2}$,
$\dot{\boldsymbol{x}}={\boldsymbol{A}}{\boldsymbol{x}}+u_{2}{\boldsymbol{b}}$;
hence, $\forall r\in{\mathbb{N}}^{*}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r-1}\left({\boldsymbol{A}}{\boldsymbol{x}}\left(t_{1}\right)+u_{2}{\boldsymbol{b}}\right)$.
Since $\forall 1\leq r<r_{p}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0$,
Lemma 2 implies that $\exists r_{p}\leq\hat{r}_{p}\leq n$, s.t.
$\left.\frac{\mathrm{d}^{\hat{r}_{p}}}{\mathrm{d}t^{\hat{r}_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}<0$
and $\forall 1\leq r<\hat{r}_{p}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0$.

Conversely, assume that
$\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}_{1}{\boldsymbol{x}}$ for
$\left(t_{0},t_{1}\right)$ and
$\dot{\boldsymbol{x}}={\boldsymbol{A}}{\boldsymbol{x}}+u_{2}{\boldsymbol{b}}$
for $\left(t_{1},t_{2}\right)$. Assume that
${\boldsymbol{x}}\left(t_{1}\right)$ satisfies (23) and (12) for
$p\in\mathcal{P}_{1}$. Then,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d_{p}=0$ and

$\begin{dcases}\left.\frac{\mathrm{d}^{\hat{r}_{p}}}{\mathrm{d}t^{\hat{r}_{p}}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}<0,\\\
\forall 1\leq
r<\hat{r}_{p},\,\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\right)\right|_{t_{1}^{+}}=0.\end{dcases}$
(28)

Hence, $\exists\varepsilon>0$, $\forall p\in\mathcal{P}_{1}$,
$t\in\left(t_{1},t_{1}+\varepsilon\right)$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$. By
Proposition 2, $\forall p\in\mathcal{P}_{1}$, $t\in\left[t_{0},t_{1}\right]$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}\equiv 0$. So
Proposition 3-3 holds. Similarly, Proposition 3-4 holds. ∎

######  Remark.

Proposition 3 provides a necessary and sufficient condition for the
feasibility of adjacent arcs near the connection. Note that the equality
constraints at the connection time, i.e., (12), (21), (23), and (25), are not
independent since (12) implies the latter three equations for $1\leq r<r_{p}$.
When examining feasibility near the connection time, it is sufficient to
consider all inequality constraints and the maximal linearly independent set
of the above equality constraints. Furthermore, Proposition 3 temporarily does
not consider constraints on $u$ and ${\boldsymbol{x}}$, except for those in
$\mathcal{P}_{1}\cup\mathcal{P}_{2}$ which will be discussed in Section IV-A.

##  IV Keypoints for the Feasibility of the Optimal Trajectory

The switching law proposed in Section III can describe a given optimal
trajectory and the feasibility near connections of arcs, but it fails to fully
present information on the feasibility during an arc. To address this
limitation, this section investigates the feasibility of the optimal
trajectory on each arc, especially of the “keypoints” in an arc. Section IV-A
analyzes some constraints of the end of an arc besides Proposition 3. The
tangent marker is proposed in Section IV-B to describe the feasibility of the
optimal trajectory tangent to the constrained boundary. Finally, Section IV-C
proposes the augmented switching law and provides a sufficient and necessary
condition for the feasibility near the keypoints.

###  IV-A Additional Constraints at the End of An Arc

Proposition 3 fully discusses the feasibility of the connection of two
adjacent arcs w.r.t. constraints induced by $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$. However, the constraints on $u$ and ${\boldsymbol{x}}$,
except for those in $\mathcal{P}_{1}\cup\mathcal{P}_{2}$ still lacks
discussion. The following proposition provides a sufficient and necessary
condition for the feasibility at the end of an arc.

######  Proposition 4.

Assume that
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$
is a system behavior for $t\in\left[t_{0},t_{1}\right]$.

  1. 1.

$\forall p\not\in\mathcal{P}$, $\exists\varepsilon>0$, s.t.
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}\leq 0$ on
$\left[t_{0},t_{0}+\varepsilon\right]$ if and only if one of the following
conditions holds:

    1. (a)

${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{0}\right)+d_{p}<0$.

    2. (b)

${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{0}\right)+d_{p}=0$.
$\exists 1\leq\hat{r}_{p}\leq n$, s.t. $\forall 1\leq r<\hat{r}_{p}$,
${\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}^{r-1}\left(\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}\left(t_{0}\right)+\widehat{{\boldsymbol{b}}}\right)=0$,
and
${\boldsymbol{c}}_{p}^{\top}\widehat{{\boldsymbol{A}}}^{\hat{r}_{p}-1}\left(\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}\left(t_{0}\right)+\widehat{{\boldsymbol{b}}}\right)<0$.

Similar conclusions hold for
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{1}^{-}\right)+d_{p}\leq
0$.

  2. 2.

Assume that ${\mathcal{S}}$ is a constrained arc. Then,
$\exists\varepsilon>0$, s.t. $u\left(t\right)\leq{u_{\mathrm{m}}}$ on
$\left[t_{0},t_{0}+\varepsilon\right]$ if and only if one of the following
conditions holds, where $p\in\mathcal{P}$:

    1. (a)

$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)<{u_{\mathrm{m}}}$.

    2. (b)

$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)={u_{\mathrm{m}}}$.
$\exists 1\leq\hat{r}_{p}\leq n$, s.t. $\forall 1\leq r<\hat{r}_{p}$,
$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}\widehat{{\boldsymbol{A}}}^{r}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)=0$,
and
$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}\widehat{{\boldsymbol{A}}}^{\hat{r}_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)<0$.

    3. (c)

$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)={u_{\mathrm{m}}}$,
and $\forall 1\leq r\leq n$,
$-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}\widehat{{\boldsymbol{A}}}^{r}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}\left(t_{0}\right)=0$.

Similar conclusions hold for
$\left|u\left(t_{0}^{+}\right)\right|,\left|u\left(t_{1}^{-}\right)\right|\leq{u_{\mathrm{m}}}$.

###### Proof.

Proposition 4-1 holds for the same reason as that of Proposition 3.

Consider Proposition 4-2. Proposition 2 implies that
$u=-\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}{\boldsymbol{x}}$.
$\exists\varepsilon>0$, s.t. $u\left(t\right)\leq{u_{\mathrm{m}}}$ on
$\left[t_{0},t_{0}+\varepsilon\right]$ if and only if (1)
$\exists\varepsilon>0$, s.t. $u<{u_{\mathrm{m}}}$ for
$\left(t_{0},t_{0}+\varepsilon\right)$, or (2) $\exists\varepsilon>0$, s.t.
$u\left(t\right)\equiv{u_{\mathrm{m}}}$ on
$\left[t_{0},t_{0}+\varepsilon\right]$. By Proposition 2, (1) is equivalent to
$\exists 0\leq\hat{r}_{p}\leq n$, s.t. $\forall 0\leq r<\hat{r}_{p}$,
$\frac{\mathrm{d}^{r}\left(u-{u_{\mathrm{m}}}\right)}{\mathrm{d}t^{r}}\left(t_{0}\right)=0$,
and
$\frac{\mathrm{d}^{\hat{r}_{p}}\left(u-{u_{\mathrm{m}}}\right)}{\mathrm{d}t^{\hat{r}_{p}}}\left(t_{0}\right)<0$.
(2) is equivalent to $\forall 0\leq r\leq n$,
$\frac{\mathrm{d}^{r}\left(u-{u_{\mathrm{m}}}\right)}{\mathrm{d}t^{r}}\left(t_{0}\right)=0$.
Hence, Proposition 4-2 holds. ∎

######  Remark.

Proposition 4 encompasses discussions on the cases covered by Proposition 3,
while Proposition 3 provides more details for
$\mathcal{P}_{1}\cup\mathcal{P}_{2}$.

Based on Proposition 4, the additional end-constraint is defined as follows:

######  Definition 3.

Assume that ${\mathcal{S}}$ is a given arc for $t\in\left[t_{0},t_{1}\right]$.
At an end of ${\mathcal{S}}$, i.e., $t_{i}$, $i\in\left\\{0,1\right\\}$,
${\mathcal{S}}$ and its adjacent arc, if it exists, correspond to the active
constraints $\mathcal{P}$ and $\mathcal{P}^{\prime}$. $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ induce the equality constraints
${\boldsymbol{F}}{\boldsymbol{x}}\left(t_{i}\right)+{\boldsymbol{g}}={\boldsymbol{0}}$
where ${\boldsymbol{F}}$ has full row rank. The additional end-constraint of
${\boldsymbol{x}}\left(t_{i}\right)$ is defined as
${\mathcal{E}}=\left(\widehat{{\boldsymbol{C}}},\widehat{{\boldsymbol{d}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}}\right)$.
Among them,
$\widehat{{\boldsymbol{C}}}{\boldsymbol{x}}\left(t_{i}\right)+\widehat{{\boldsymbol{d}}}<{\boldsymbol{0}}$
and
$\widehat{{\boldsymbol{F}}}{\boldsymbol{x}}\left(t_{i}\right)+\widehat{{\boldsymbol{g}}}={\boldsymbol{0}}$
are induced by Proposition 4 for the active constraints of ${\mathcal{S}}$ and
its adjacent arc at $t_{i}$, except for those in
$\mathcal{P}\cup\mathcal{P}^{\prime}$.
$\left[\begin{array}[]{c}{\boldsymbol{F}}\\\
\widehat{{\boldsymbol{F}}}\end{array}\right]$ has full row rank.

######  Remark.

Given a feasible arc, the order $\hat{r}_{p}$ in Proposition 4 is directly
determined. In Definition 3, the additional end-constraint ${\mathcal{E}}$
does not include the equality constraints induced by
$\mathcal{P}\cup\mathcal{P}^{\prime}$. To avoid over-determination, the
equality constraints in ${\mathcal{E}}$ and
$\mathcal{P}\cup\mathcal{P}^{\prime}$ are made linearly independent through
elimination. The elimination process is guaranteed by the feasibility of the
given trajectory.

###  IV-B Tangent Markers in An Arc

Proposition 3 and Proposition 4 analyzed the feasibility of an arc near the
end points. An arc might be tangent to some constrained boundary, where light
perturbations may affect the feasibility of the arc. The following proposition
provides a sufficient and necessary condition for the feasibility of the
optimal trajectory tangent to the constrained boundary.

######  Proposition 5.

Assume that
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$
is an arc for $t\in\left[t_{0},t_{2}\right]$. ${\mathcal{S}}$ is tangent to
the boundary of the constraint ${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\leq
0$ at $t_{1}\in\left(t_{0},t_{2}\right)$ if and only if $\exists
1\leq\hat{r}\leq n$ is even, s.t.

$\begin{dcases}{\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d=0,\\\
\forall 1\leq
r<\hat{r},\,{\boldsymbol{c}}^{\top}\widehat{{\boldsymbol{A}}}^{r-1}\left(\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}\left(t_{1}\right)+\widehat{{\boldsymbol{b}}}\right)=0,\\\
{\boldsymbol{c}}^{\top}\widehat{{\boldsymbol{A}}}^{\hat{r}-1}\left(\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}\left(t_{1}\right)+\widehat{{\boldsymbol{b}}}\right)<0.\end{dcases}$
(29)

###### Proof.

${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t_{1}\right)+d=0$.
$\exists\varepsilon>0$, s.t.
$B_{\varepsilon}\left(t_{1}\right)\subset\left(t_{0},t_{2}\right)$ and
$\forall
t\in{\left.B_{\varepsilon}\left(t_{1}\right)\middle\backslash\left\\{t_{1}\right\\}\right.}$,
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)+d<0$. Note that
$\forall r\in{\mathbb{N}}^{*}$,
$\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)={\boldsymbol{c}}^{\top}\widehat{{\boldsymbol{A}}}^{r-1}\left(\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}+\widehat{{\boldsymbol{b}}}\right)$.

Assume that $\forall 1\leq r\leq n$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)\right|_{t_{1}}=0$.
By Lemma 2, $\forall t\in B_{\varepsilon}\left(t_{1}\right)$,
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}\left(t\right)+d\equiv 0$, which
contradicts the tangent condition. Hence, $\exists 1\leq\hat{r}\leq n$, s.t.
$\forall 1\leq r<\hat{r}$,
$\left.\frac{\mathrm{d}^{r}}{\mathrm{d}t^{r}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)\right|_{t_{1}}=0$,
and
$\left.\frac{\mathrm{d}^{\hat{r}}}{\mathrm{d}t^{\hat{r}}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)\right|_{t_{1}}\not=0$.

If $\hat{r}$ is odd, then ${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d$ strictly
crosses $0$ at $t_{1}$, which contradicts the tangent condition. Hence,
$\hat{r}$ is even. If
$\left.\frac{\mathrm{d}^{\hat{r}}}{\mathrm{d}t^{\hat{r}}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)\right|_{t_{1}}>0$,
then ${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d$ achieves a strict local
minimum at $t_{1}$, which contradicts the maximum condition. Hence,
$\left.\frac{\mathrm{d}^{\hat{r}}}{\mathrm{d}t^{\hat{r}}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)\right|_{t_{1}}<0$.
Therefore, (29) holds.

Conversely, assume that (29) holds. Considering the Taylor of
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d$ at $t_{1}$, it holds that
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d<0$ at a deleted neighborhood of
$t_{1}$. Hence, ${\mathcal{S}}$ is tangent to the boundary of the constraint
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d$ at $t_{1}$. ∎

No matter the arc
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$
is constrained or unconstrained, ${\mathcal{S}}$ can be tangent to the
boundary of the state constraints
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\leq 0$ where
$p\not\in\mathcal{P}$. It is noteworthy that ${\mathcal{S}}$ can also be
tangent to the boundary of the control constraints $u\leq{u_{\mathrm{m}}}$ and
$-u\leq{u_{\mathrm{m}}}$. For this case, ${\mathcal{S}}$ should be
constrained. Arbitrarily consider $p\in\mathcal{P}$. According to Proposition
2, $\pm u\leq{u_{\mathrm{m}}}$ is equivalent to
$\mp\frac{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}{\boldsymbol{x}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}\leq{u_{\mathrm{m}}}$.
In other words, the constraints on $u$ are equivalent to the constraints on
${\boldsymbol{x}}$. Hence, Proposition 5 can apply for both state constraints
and control constraints. For convenience of discussion, $\forall 1\leq p\leq
P$, the $p$-th constraint refers to
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\leq 0$. The
$\left(P+1\right)$-th constraint refers to $u\leq{u_{\mathrm{m}}}$, and the
$\left(P+2\right)$-th constraint refers to $-u\leq{u_{\mathrm{m}}}$.

Another noteworthy fact is that an arc ${\mathcal{S}}$ can be tangent to
multiple constraints $\forall p\in\mathcal{P}^{\prime}\subset\left[P+2\right]$
at a time point $t_{1}$. Then, the necessary and sufficient condition for
feasibility is $\forall p\in\mathcal{P}^{\prime}$, (29) holds. The tangent
marker is defined as follows:

######  Definition 4.

Assume that $\forall p\in\mathcal{P}^{\prime}\subset\left[P+2\right]$, an arc
${\mathcal{S}}$ is tangent to the $p$-th constraint at $t_{1}$. Denote the
tangent marker as
${\mathcal{T}}=\left(\widehat{{\boldsymbol{C}}},\widehat{{\boldsymbol{d}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}^{\prime}\right)$.
$\widehat{{\boldsymbol{C}}}{\boldsymbol{x}}\left(t_{1}\right)\widehat{{\boldsymbol{d}}}<{\boldsymbol{0}}$
and
$\widehat{{\boldsymbol{F}}}{\boldsymbol{x}}\left(t_{1}\right)+\widehat{{\boldsymbol{g}}}={\boldsymbol{0}}$
are induced by (29), where $\widehat{{\boldsymbol{C}}}$ has full row rank.

######  Remark.

In Definition 4, the equality constraints induced by (29) can be linearly
dependent. However, for a given feasible trajectory, the equalities should
have a feasible solution; hence, equality constraints with full row rank,
i.e.,
$\widehat{{\boldsymbol{F}}}{\boldsymbol{x}}\left(t_{1}\right)+\widehat{{\boldsymbol{g}}}={\boldsymbol{0}}$,
can be constructed through elimination. Therefore, the tangent marker is well-
defined.

###  IV-C Augmented Switching Law

Section III, Section IV-A, and Section IV-B analyze the feasibility of the
optimal trajectory at the end points of arcs and the tangent points to the
constrained boundaries. To characterize the optimal trajectory, this section
proposes the augmented switching law, which represents the input control and
the feasibility of the trajectory in a compact form.

######  Definition 5.

Assume that the optimal trajectory of problem (1a) satisfies Assumption 1;
hence, it contains a finite number of arcs. The augmented switching law is
defined as
${\mathcal{L}}={\mathcal{E}}_{0}\left({\mathcal{S}}_{1},\left\\{{\mathcal{T}}_{k}^{\left(1\right)}\right\\}_{k=1}^{M_{1}}\right){\mathcal{E}}_{1}\dots\mathcal{E}_{N-1}\left({\mathcal{S}}_{N},\left\\{{\mathcal{T}}_{k}^{\left(N\right)}\right\\}_{k=1}^{M_{N}}\right){\mathcal{E}}_{N}$.
$\forall i\in\left[N\right]$, ${\mathcal{S}}_{i}$ occurs on
$\left[t_{0}^{\left(i\right)},t_{0}^{\left(i+1\right)}\right]$. $\forall
k\in\left[M_{i}\right]$, the tangent marker
${\mathcal{T}}_{k}^{\left(i\right)}$ occurs at $t_{k}^{\left(i\right)}$ during
${\mathcal{S}}_{i}$, where
$t_{0}^{\left(i\right)}<t_{1}^{\left(i\right)}<t_{2}^{\left(i\right)}\dots<t_{M_{i}}^{\left(i\right)}<t_{0}^{\left(i+1\right)}$.
The additional end-constraint ${\mathcal{E}}_{i}$ is induced by Proposition 4
at $t_{0}^{\left(i\right)}$. Among them, ${\mathcal{E}}_{N}$ additionally
contains the constraints
${\boldsymbol{x}}\left(t_{0}^{\left(N+1\right)}\right)={{\boldsymbol{x}}_{\mathrm{f}}}$,
where $t_{0}^{\left(N+1\right)}={t_{\mathrm{f}}}$. $\forall
i\in\left[N+1\right]$, $k\in\left[M_{i}\right]\cup\left\\{0\right\\}$, the
points $t_{k}^{\left(i\right)}$ are called keypoints for the feasibility of
the trajectory.

######  Remark.

If a feasible trajectory with a finite number of arcs follows the Bang-
Singular-Bang control law (7), then Proposition 3, Proposition 4, and
Proposition 5 hold for the trajectory. For this case, the augmented switching
law can also represent the feasible trajectory.

######  Remark.

If a non-chattering Bang-Singular-Bang feasible trajectory is disturbed by a
sufficiently small perturbation, then the disturbed trajectory can only exceed
the constrained boundary at the keypoints since the distance between the
original trajectory and the constrained boundary is 0 at the keypoints, while
it is strictly positive at other points. Therefore, the augmented switching
law can not only characterize the input control but also describe the
feasibility of the trajectory.

######  Remark.

The switching law in Definition 2 corresponding to the augmented switching law
in Definition 5 is ${\mathcal{S}}_{1}{\mathcal{S}}_{2}\dots\mathcal{S}_{N}$.
It can be observed that the augmented switching law provides more detailed
information on the feasibility of the trajectory at keypoints than the
switching law. Notably, neither the switching law nor the augmented switching
law provides the motion time of each arc.

To simplify the notation, denote an arc without tangent markers
$\left({\mathcal{S}},\varnothing\right)$ as ${\mathcal{S}}$ in an augmented
switching law. If no additional end-constraint is induced at the end of an
arc, i.e., ${\mathcal{E}}=\left(\sim,\sim,\sim,\sim\right)$, then
${\mathcal{E}}$ is omitted.

For convenience of understanding, the chain-of-integrators system with full
box state constraints in Example 2 is considered as an example that was
discussed in our previous works [8, 18]. The system behaviors where
$u\equiv{u_{\mathrm{m}}}$ and $u\equiv-{u_{\mathrm{m}}}$, i.e., unconstrained
arcs, are denoted as $\overline{0}$ and $\underline{0}$, respectively. The
system behaviors where $x_{k}\equiv x_{{\mathrm{m}}k}$ and $x_{k}\equiv-
x_{{\mathrm{m}}k}$, i.e., constrained arcs, are denoted as $\overline{k}$ and
$\underline{k}$, respectively. Evidently, during $\overline{k}$ or
$\underline{k}$, $u\equiv 0$ and $\forall 1\leq j<k$, $x_{j}\equiv 0$. The
tangent marker where ${\boldsymbol{x}}$ is tangent to $x_{{\mathrm{m}}k}$ and
$-x_{{\mathrm{m}}k}$ of an even order $2l<k$ in Proposition 5 is denoted as
$\left(\overline{k},2l\right)$ and $\left(\underline{k},2l\right)$,
respectively. The additional end-constraint
${\mathcal{E}}=\left(\left(s_{k},r_{k}\right)\right)_{k=1}^{K}$ means that the
additional state constraints $s_{k}$ is active with order $r_{k}$ in
Proposition 4, where $s_{k}=\overline{j}$ and $s_{k}=\underline{j}$ refer to
the constraints $x_{j}\leq x_{{\mathrm{m}}j}$ and $-x_{j}\leq
x_{{\mathrm{m}}j}$, respectively. Note that the constraints on control are
inactive at the ends of constrained arcs since $u\equiv 0$ in a constrained
arc. Based on the above simplified notations, an example is provided to
illustrate the augmented switching law.

######  Example 3.

Consider a position-to-position problem for 4th order chain-of-integrators
systems with full box constraints shown in Fig. 1. A feasible suboptimal
trajectory with a Bang-Singular-Bang form, planned by the manifold-intercepted
method (MIM) [8], is shown in Fig. 1(a). It can be observed that the augmented
switching law is
${\mathcal{L}}=\overline{01}\underline{0}\overline{2}\underline{01}\overline{03}\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}$.

Figure 1: Two examples of augmented switching laws. For a 4th order chain-of-
integrators system with full box state constraints, let ${u_{\mathrm{m}}}=1$,
$x_{{\mathrm{m}}1}=1$, $x_{{\mathrm{m}}2}=1.5$, $x_{{\mathrm{m}}3}=4$,
$x_{{\mathrm{m}}4}=15$,
${\boldsymbol{x}}_{0}=-x_{{\mathrm{m}}4}{\boldsymbol{e}}_{4}$, and
${{\boldsymbol{x}}_{\mathrm{f}}}=x_{{\mathrm{m}}4}{\boldsymbol{e}}_{4}$. (a) A
non-chattering feasible trajectory planned by our previous work [8]. (b) A
chattering optimal trajectory planned by our previous work [18]. In (b2),
$\hat{x}_{3}\left(t\right)=\left(x_{{\mathrm{m}}3}-x_{3}\left(t\right)\right)\left(t_{\infty}-t\right)^{-3}$
where $t_{\infty}\approx 6.0732$ is a chattering limit time.

######  Example 4.

Our previous work [18] pointed out that the chattering phenomenon occurs in
the optimal trajectory for the problem in Example 3. The chattering optimal
trajectory is shown in Fig. 1(b1). Fig. 1(b2) plots the trajectory of $x_{3}$
during the chattering period with amplitude compensation applied. It can be
observed that the trajectory during the left chattering period consists of a
cycle of
$\left(\overline{0},\left\\{\left(\overline{3},2\right)\right\\}\right)\underline{0}$.
According to Bellman’s principle of optimality [20], the trajectory in
$\left[0,t_{\infty}-\varepsilon\right]$ is also time-optimal, where
$\varepsilon>0$ is small enough. The augmented switching of the above sub-arc
begins with
$\overline{01}\underline{0}\overline{2}\underline{01}\left(\overline{0},\left\\{\left(\overline{3},2\right)\right\\}\right)\underline{0}\left(\overline{0},\left\\{\left(\overline{3},2\right)\right\\}\right)\underline{0}\dots$

##  V A State-Centric Necessary Condition for Optimal Control of Problem (1a)

The theoretical framework for the augmented switching law of time-optimal
control for linear systems is established in Section IV. Based on the
established framework, this section provides a state-centric necessary
condition for the optimal control of problem (1a). Section V-A discusses the
uniqueness of the time-optimal control. Then, given a feasible Bang-Singular-
Bang trajectory, Section V-B analyzes the feasibility of the disturbed Bang-
Singular-Bang trajectory. Based on the conclusions in Section V-A and Section
V-B, Section V-C provides a necessary condition for the optimal trajectory
represented by the augmented switching law, which constrains the difference
between the switching times and the sum of orders of keypoints in a way.

###  V-A Uniqueness of Time-Optimal Control

The uniqueness of the time-optimal control for problem (1a) is proved in
Theorem 1 as preliminaries.

######  Theorem 1.

Under Assumption 1, the time-optimal control of problem (1a) is unique in an
almost everywhere sense. In other words, if $u=u_{1}^{*}\left(t\right)$ and
$u=u_{2}^{*}\left(t\right)$, $t\in\left[0,t_{\mathrm{f}}^{*}\right]$, are both
optimal controls of problem (1a), then
$u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)$ almost everywhere; hence,
$\forall t\in\left[0,t_{\mathrm{f}}^{*}\right]$,
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$.

###### Proof.

If $\forall t\in\left[0,t_{\mathrm{f}}^{*}\right]$,
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$,
then evidently it has $u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)$ almost
everywhere since
$\dot{\boldsymbol{x}}={\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{b}}u$ and
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{0}$.

Assume that $\exists t^{\prime}\in\left(0,t_{\mathrm{f}}^{*}\right)$, s.t.
${\boldsymbol{x}}_{1}^{*}\left(t^{\prime}\right)\not={\boldsymbol{x}}_{2}^{*}\left(t^{\prime}\right)$.
Let
$t_{0}=\arg\min\left\\{t\in\left(0,t_{\mathrm{f}}^{*}\right):{\boldsymbol{x}}_{1}^{*}\left(t\right)\not={\boldsymbol{x}}_{2}^{*}\left(t\right)\right\\}$.
Evidently, $0\leq t_{0}<t_{\mathrm{f}}^{*}$. Assume that
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)\not={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)$.
Then, $t_{0}>0$. Due to the continuity of ${\boldsymbol{x}}$, $\forall 0\leq
t<t_{0}$,
${\boldsymbol{x}}_{1}^{*}\left(t\right)\equiv{\boldsymbol{x}}_{2}^{*}\left(t\right)$
implies that
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)$,
which contradicts the assumption. Therefore,
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)$
holds. Furthermore, $\forall t\in\left[0,t_{0}\right]$,
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$
implies that $u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)$ for
$t\in\left[0,t_{0}\right]$ almost everywhere.

According to Bellman’s principle of optimality [20],
${\boldsymbol{x}}={\boldsymbol{x}}_{1}^{*}\left(t\right)$ and
${\boldsymbol{x}}={\boldsymbol{x}}_{2}^{*}\left(t\right)$,
$t\in\left[t_{0},t_{\mathrm{f}}^{*}\right]$ are still the time-optimal
trajectories between
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)$
and
${\boldsymbol{x}}_{1}^{*}\left(t_{\mathrm{f}}^{*}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{\mathrm{f}}^{*}\right)={{\boldsymbol{x}}_{\mathrm{f}}}$.
Let
$u_{3}^{*}\left(t\right)=\frac{1}{2}\left(u_{1}^{*}\left(t\right)+u_{2}^{*}\left(t\right)\right)$
and
${\boldsymbol{x}}_{3}^{*}\left(t\right)=\frac{1}{2}\left({\boldsymbol{x}}_{1}^{*}\left(t\right)+{\boldsymbol{x}}_{2}^{*}\left(t\right)\right)$.
Evidently, $u_{3}^{*}\left(t\right)$ is feasible for problem (1a) since the
feasible set of $\left(u,{\boldsymbol{x}}\right)$ is convex; hence,
$u_{3}^{*}\left(t\right)$ is optimal.

Denote that $\forall k=1,2$, the switching law of
${\boldsymbol{x}}={\boldsymbol{x}}_{i}^{*}\left(t\right)$,
$t\in\left[t_{0},t_{\mathrm{f}}^{*}\right]$ is
${\mathcal{L}}_{i}={\mathcal{S}}_{1}^{\left(i\right)}{\mathcal{S}}_{2}^{\left(i\right)}\dots\mathcal{S}_{N_{i}}^{\left(i\right)}$,
where $\forall j\in\left[N_{i}\right]$,
${\mathcal{S}}_{j}^{\left(i\right)}=\left(\widehat{{\boldsymbol{A}}}_{j}^{\left(i\right)},\widehat{{\boldsymbol{b}}}_{j}^{\left(i\right)},\widehat{{\boldsymbol{F}}}_{j}^{\left(i\right)},\widehat{{\boldsymbol{g}}}_{j}^{\left(i\right)},\mathcal{P}_{j}^{\left(i\right)}\right)$
lasts for $t\in\left(t_{j-1}^{\left(i\right)},t_{j}^{\left(i\right)}\right)$.
Among them, $t_{0}^{\left(i\right)}=t_{0}$ and
$t_{N_{i}}^{\left(i\right)}=t_{\mathrm{f}}^{*}$. Evidently, if
${\mathcal{S}}_{1}^{\left(1\right)}={\mathcal{S}}_{1}^{\left(2\right)}$, then
$\forall
t\in\left(t_{0},\min\left\\{t_{1}^{\left(1\right)},t_{1}^{\left(2\right)}\right\\}\right)$,
$u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)$ and
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$,
which contradicts the definition of $t_{0}$. Hence,
${\mathcal{S}}_{1}^{\left(1\right)}\not={\mathcal{S}}_{1}^{\left(2\right)}$.

Assume that both ${\mathcal{S}}_{1}^{\left(1\right)}$ and
${\mathcal{S}}_{1}^{\left(2\right)}$ are unconstrained arcs. Without loss of
generality, assume that $u_{1}^{*}\left(t\right)\equiv{u_{\mathrm{m}}}$ in
${\mathcal{S}}_{1}^{\left(1\right)}$; hence,
$u_{2}^{*}\left(t\right)\equiv-{u_{\mathrm{m}}}$ in
${\mathcal{S}}_{1}^{\left(2\right)}$. Then, $\exists\varepsilon>0$, s.t.
$\forall t\in\left(t_{0},t_{0}+\varepsilon\right)$,
${\boldsymbol{C}}{\boldsymbol{x}}_{1}^{*}\left(t\right)+{\boldsymbol{d}}<{\boldsymbol{0}}$
and
${\boldsymbol{C}}{\boldsymbol{x}}_{2}^{*}\left(t\right)+{\boldsymbol{d}}<{\boldsymbol{0}}$.
Therefore,
${\boldsymbol{C}}{\boldsymbol{x}}_{3}^{*}\left(t\right)+{\boldsymbol{d}}={\boldsymbol{C}}\frac{{\boldsymbol{x}}_{1}^{*}\left(t\right)+{\boldsymbol{x}}_{2}^{*}\left(t\right)}{2}+{\boldsymbol{d}}<{\boldsymbol{0}}$.
In other words, ${\boldsymbol{x}}_{3}^{*}$ is strictly unconstrained in
$\in\left(t_{0},t_{0}+\varepsilon\right)$. According to Proposition 2,
$\forall t\in\left(t_{0},t_{0}+\varepsilon\right)$,
$u_{3}^{*}\left(t\right)\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$, which
contradicts the fact that
$u_{3}^{*}\left(t\right)=\frac{u_{1}^{*}\left(t\right)+u_{2}^{*}\left(t\right)}{2}\equiv
0$. Therefore, one of ${\mathcal{S}}_{1}^{\left(1\right)}$ and
${\mathcal{S}}_{1}^{\left(2\right)}$ is constrained. Without loss of
generality, assume that ${\mathcal{S}}_{1}^{\left(1\right)}$ is constrained.

Assume that ${\mathcal{S}}_{1}^{\left(2\right)}$ is an unconstrained arc.
Then,
$\exists\varepsilon\in\left(0,\min\left\\{t_{1}^{\left(1\right)},t_{1}^{\left(2\right)}\right\\}-t_{0}\right)$,
s.t. $\forall t\in\left(t_{0},t_{0}+\varepsilon\right)$,
${\boldsymbol{C}}{\boldsymbol{x}}_{2}^{*}\left(t\right)+{\boldsymbol{d}}<{\boldsymbol{0}}$.
Since
${\boldsymbol{C}}{\boldsymbol{x}}_{1}^{*}\left(t\right)+{\boldsymbol{d}}\leq{\boldsymbol{0}}$,
it has
${\boldsymbol{C}}{\boldsymbol{x}}_{3}^{*}\left(t\right)+{\boldsymbol{d}}={\boldsymbol{C}}\frac{{\boldsymbol{x}}_{1}^{*}\left(t\right)+{\boldsymbol{x}}_{2}^{*}\left(t\right)}{2}+{\boldsymbol{d}}<{\boldsymbol{0}}$.
In other words, ${\boldsymbol{x}}_{3}^{*}$ is strictly unconstrained in
$\in\left(t_{0},t_{0}+\varepsilon\right)$. According to Proposition 2,
$\forall t\in\left(t_{0},t_{0}+\varepsilon\right)$,
$u_{3}^{*}\left(t\right)=\frac{u_{1}^{*}\left(t\right)+u_{2}^{*}\left(t\right)}{2}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$.
Note that
$u_{1}^{*}\left(t\right)\in\left[-{u_{\mathrm{m}}},{u_{\mathrm{m}}}\right]$
and $u_{2}^{*}\left(t\right)\equiv
u_{0}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$; hence, $\forall
t\in\left(t_{0},t_{0}+\varepsilon\right)$, $u_{1}^{*}\left(t\right)\equiv
u_{0}$. Therefore,
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$
for $\left[t_{0},t_{0}+\varepsilon\right]$, which contradicts the definition
of $t_{0}$.

Therefore, both ${\mathcal{S}}_{1}^{\left(1\right)}$ and
${\mathcal{S}}_{1}^{\left(2\right)}$ are constrained arcs. Assume that
$\mathcal{P}_{1}^{\left(1\right)}\cap\mathcal{P}_{1}^{\left(2\right)}\not=\varnothing$.
Arbitrarily consider
$p\in\mathcal{P}_{1}^{\left(1\right)}\cap\mathcal{P}_{1}^{\left(2\right)}$.
$\forall
t\in\left(t_{0},\min\left\\{t_{1}^{\left(1\right)},t_{1}^{\left(2\right)}\right\\}\right)$,
$i=1,2$,
$\dot{\boldsymbol{x}}_{i}^{*}\left(t\right)=\left({\boldsymbol{A}}-\frac{{\boldsymbol{b}}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}}}{{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}^{r_{p}-1}{\boldsymbol{b}}}\right){\boldsymbol{x}}_{i}^{*}\left(t\right)$.
${\boldsymbol{x}}_{1}^{*}\left(t_{0}\right)={\boldsymbol{x}}_{2}^{*}\left(t_{0}\right)$
implies that
${\boldsymbol{x}}_{1}^{*}\left(t\right)\equiv{\boldsymbol{x}}_{2}^{*}\left(t\right)$,
which contradicts the definition of $t_{0}$. Therefore,
$\mathcal{P}_{1}^{\left(1\right)}\cap\mathcal{P}_{1}^{\left(2\right)}=\varnothing$.
Then,
$\exists\varepsilon\in\left(0,\min\left\\{t_{1}^{\left(1\right)},t_{1}^{\left(2\right)}\right\\}-t_{0}\right)$,
s.t. $\forall t\in\left(t_{0},t_{0}+\varepsilon\right)$, $i=1,2$,
$p\not\in\mathcal{P}_{1}^{\left(i\right)}$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}_{i}^{*}\left(t\right)+d_{p}<0$.
Hence, $\forall p\in\left[P\right]$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}_{3}^{*}\left(t\right)+d_{p}={\boldsymbol{c}}_{p}^{\top}\frac{{\boldsymbol{x}}_{1}^{*}\left(t\right)+u{\boldsymbol{x}}_{2}^{*}\left(t\right)}{2}+d_{p}<0$.
According to Proposition 2,
$u_{3}^{*}\left(t\right)\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$; hence,
$u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$,
which contradicts the definition of $t_{0}$. Therefore, $\forall
t\in\left[t_{0},t_{\mathrm{f}}^{*}\right]$,
${\boldsymbol{x}}_{1}^{*}\left(t\right)={\boldsymbol{x}}_{2}^{*}\left(t\right)$;
hence, $u_{1}^{*}\left(t\right)=u_{2}^{*}\left(t\right)$ almost everywhere. ∎

######  Remark.

Based on Theorem 1, it can be proved that the optimal control of problem (1a)
is unique in an almost everywhere sense when there exists at most one
chattering limit point. The proof is omitted since it does not contribute to
the proposed state-centric necessary condition in Section V-C.

According to Theorem 1, if the time-optimal trajectory is disturbed by a
sufficiently small perturbation, then the resulting disturbed trajectory
achieves a strictly shorter terminal time than the optimal terminal time. The
above observation will be applied in Section V-B.

###  V-B Feasibility of the Disturbed Bang-Singular-Bang Trajectory

Consider a feasible Bang-Singular-Bang trajectory
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ of problem (1a) without
chattering. The augmented switching law is denoted as ${\mathcal{L}}$. For
convenience of discussion, the keypoints of ${\mathcal{L}}$ occur at $t_{i}$,
where $0=t_{0}<t_{1}<\dots<t_{M}=t_{\mathrm{f}}$. In other words,
$\left\\{t_{i}\right\\}_{i=0}^{M}=\bigcup_{i=1}^{N+1}\left\\{t_{k}^{\left(i\right)}\right\\}_{k=0}^{M_{i}}$.
Denote ${\boldsymbol{t}}=\left(t_{i}\right)_{i=1}^{M}$, and
$\dot{\boldsymbol{x}}={\boldsymbol{A}}_{i}{\boldsymbol{x}}+{\boldsymbol{b}}$
in $\left(t_{i-1},t_{i}\right)$. This section discusses the feasibility of a
disturbed trajectory
${\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$ represented by the
augmented switching law ${\mathcal{L}}^{\prime}$, where the time vector
${\boldsymbol{t}}^{\prime}=\left(t_{i}^{\prime}\right)_{i=1}^{M}$ is close to
${\boldsymbol{t}}$, i.e.,
$\left\|{\boldsymbol{t}}^{\prime}-{\boldsymbol{t}}\right\|$ is small enough.
For convenience, the notation $\left\|\bullet\right\|$ refers to
$\left\|\bullet\right\|_{\infty}$ in this section if not specified.

The feasibility of the disturbed trajectory is proved by two steps. Firstly,
the consistent error between the disturbed trajectory and the original
trajectory is bounded by
$C\left\|{\boldsymbol{t}}-{\boldsymbol{t}}^{\prime}\right\|$ where $C$ is a
constant dependent on the original trajectory. In this way, the disturbed
trajectory can be proved feasible at the points far from keypoints. Secondly,
the feasibility of the disturbed trajectory near the keypoints is proved based
on Proposition 3, Proposition 4, and Proposition 5. As a result, the
feasibility of the disturbed trajectory is proved. The two steps correspond to
Proposition 6 and Theorem 2, respectively.

######  Proposition 6.

${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ is driven by $\forall
i\in\left[M\right]$, $t\in\left(t_{i-1},t_{i}\right)$,
$\dot{\boldsymbol{x}}={\boldsymbol{A}}_{i}{\boldsymbol{x}}+{\boldsymbol{b}}_{i}$.
$\exists C>0$, s.t.
$\forall{\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$,

$\displaystyle\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}\triangleq\sup_{t\in\left[t_{0},\min\left\\{t_{M},t_{M}^{\prime}\right\\}\right]}\left\|{\boldsymbol{x}}\left(t\right)-{\boldsymbol{x}}^{\prime}\left(t\right)\right\|\leq
C\varepsilon,$ (30a)
$\displaystyle\max_{i\in\left[M\right]}\left\|{\boldsymbol{x}}\left(t_{i}\right)-{\boldsymbol{x}}^{\prime}\left(t_{i}^{\prime}\right)\right\|\leq
C\varepsilon.$ (30b)

Among them, $\forall i\in\left[M\right]$,
$t\in\left(t_{i-1}^{\prime},t_{i}^{\prime}\right)$,
$\dot{\boldsymbol{x}}^{\prime}={\boldsymbol{A}}_{i}{\boldsymbol{x}}^{\prime}+{\boldsymbol{b}}_{i}$.
${\boldsymbol{x}}^{\prime}\left(t_{0}\right)={\boldsymbol{x}}\left(t_{0}\right)$
and $t_{0}^{\prime}=t_{0}$. ${\boldsymbol{t}}=\left(t_{i}\right)_{i=1}^{M}$
and ${\boldsymbol{t}}^{\prime}=\left(t_{i}^{\prime}\right)_{i=1}^{M}$.
$0<\left\|{\boldsymbol{t}}-{\boldsymbol{t}}^{\prime}\right\|<\varepsilon<\frac{1}{2}\min_{i\in\left[M-1\right]}\left\\{\left|t_{i+1}-t_{i}\right|\right\\}$.

###### Proof.

Since
$\varepsilon<\frac{1}{2}\min_{i\in\left[M-1\right]}\left\\{\left|t_{i+1}-t_{i}\right|\right\\}$
and
$\left|t_{i}-t_{i}^{\prime}\right|\leq\left\|{\boldsymbol{t}}-{\boldsymbol{t}}^{\prime}\right\|<\varepsilon$,
it has $\forall i\in\left[M-1\right]$,
$\max\left\\{t_{i},t_{i}^{\prime}\right\\}<\min\left\\{t_{i+1},t_{i+1}^{\prime}\right\\}$.
$\forall i\in\left[M\right]$, let
$\hat{t}_{2i-1}=\min\left\\{t_{i},t_{i}^{\prime}\right\\}$,
$\hat{t}_{2i}=\max\left\\{t_{i},t_{i}^{\prime}\right\\}$, and $\hat{t}_{0}=0$.
Then $\left\\{\hat{t}_{i}\right\\}_{i=0}^{2M}$ increases monotonically.

Note that $\forall{\boldsymbol{A}}\in{\mathbb{R}}^{n\times n}$,
${\mathrm{e}}^{{\boldsymbol{A}}t}$ is locally Lipschitz continuous w.r.t. $t$.
Therefore, $\exists C_{1}>0$, s.t. $\forall t\in\left[0,t_{M}-t_{0}\right]$,
$\max_{i\in\left[M\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{i}t}-{\boldsymbol{I}}\right\|\leq
C_{1}t$. Let
$C_{2}=\max_{i\in\left[M\right]}\left\|{\boldsymbol{b}}_{i}\right\|$ and
$C_{3}=\max_{i\in\left[M\right]}\sup_{t\in\left[0,t_{M}-t_{0}\right]}\left\|{\boldsymbol{e}}^{{\boldsymbol{A}}_{i}t}\right\|\geq
1$.

By Lemma 1, $\forall i\in\left[M-1\right]\cup\left\\{0\right\\}$,
$t\in\left[\hat{t}_{2i},\hat{t}_{2i+1}\right]$,

$\begin{dcases}{\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\hat{t}_{2i}\right)}{\boldsymbol{x}}\left(\hat{t}_{2i}\right)+\int_{\hat{t}_{2i}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\tau\right)}{\boldsymbol{b}}_{i}\mathrm{d}\tau,\\\
{\boldsymbol{x}}^{\prime}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\hat{t}_{2i}\right)}{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i}\right)+\int_{\hat{t}_{2i}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\tau\right)}{\boldsymbol{b}}_{i}\mathrm{d}\tau.\end{dcases}$
(31)

Therefore,

$\displaystyle\sup_{t\in\left[\hat{t}_{2i},\hat{t}_{2i+1}\right]}\left\|{\boldsymbol{x}}\left(t\right)-{\boldsymbol{x}}^{\prime}\left(t\right)\right\|$
(32) $\displaystyle=$
$\displaystyle\sup_{t\in\left[\hat{t}_{2i},\hat{t}_{2i+1}\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\hat{t}_{2i}\right)}\left({\boldsymbol{x}}\left(\hat{t}_{2i}\right)-{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i}\right)\right)\right\|$
$\displaystyle\leq$
$\displaystyle\sup_{t\in\left[\hat{t}_{2i},\hat{t}_{2i+1}\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t-\hat{t}_{2i}\right)}\right\|\left\|{\boldsymbol{x}}\left(\hat{t}_{2i}\right)-{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i}\right)\right\|$
$\displaystyle\leq$ $\displaystyle
C_{3}\left\|{\boldsymbol{x}}\left(\hat{t}_{2i}\right)-{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i}\right)\right\|.$

$\forall i\in\left[M\right]$, let

$\begin{dcases}j=i+1,\,j^{\prime}=i,&\text{if }t_{i}\leq t_{i}^{\prime},\\\
j=i,\,j^{\prime}=i+1,&\text{if }t_{i}>t_{i}^{\prime}.\end{dcases}$ (33)

Then, $\forall t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]$,
$\dot{\boldsymbol{x}}={\boldsymbol{A}}_{j}{\boldsymbol{x}}+{\boldsymbol{b}}_{j}$
and
$\dot{\boldsymbol{x}}^{\prime}={\boldsymbol{A}}_{j^{\prime}}{\boldsymbol{x}}+{\boldsymbol{b}}_{j^{\prime}}$.
Lemma 1 implies that $\forall t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]$,

$\begin{dcases}{\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}_{j}\left(t-\hat{t}_{2i-1}\right)}{\boldsymbol{x}}\left(\hat{t}_{2i-1}\right)+\int_{\hat{t}_{2i-1}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}_{j}\left(t-\tau\right)}{\boldsymbol{b}}_{j}\mathrm{d}\tau,\\\
{\boldsymbol{x}}^{\prime}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}_{j^{\prime}}\left(t-\hat{t}_{2i-1}\right)}{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i-1}\right)+\int_{\hat{t}_{2i-1}}^{t}{\mathrm{e}}^{{\boldsymbol{A}}_{j^{\prime}}\left(t-\tau\right)}{\boldsymbol{b}}_{j^{\prime}}\mathrm{d}\tau.\end{dcases}$
(34)

Therefore,

$\displaystyle\sup_{t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]}\left\|{\boldsymbol{x}}\left(t\right)-{\boldsymbol{x}}^{\prime}\left(t\right)\right\|$
(35) $\displaystyle\leq$
$\displaystyle\sup_{t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{j}\left(t-\hat{t}_{2i-1}\right)}-{\mathrm{e}}^{{\boldsymbol{A}}_{j^{\prime}}\left(t-\hat{t}_{2i-1}\right)}\right\|\left\|{\boldsymbol{x}}\left(\hat{t}_{2i-1}\right)\right\|$
$\displaystyle+$
$\displaystyle\sup_{t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{j^{\prime}}\left(t-\hat{t}_{2i-1}\right)}\right\|\left\|{\boldsymbol{x}}\left(\hat{t}_{2i-1}\right)-{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i-1}\right)\right\|$
$\displaystyle+$
$\displaystyle\varepsilon\sup_{t\in\left[\hat{t}_{2i-1},\hat{t}_{2i}\right]}\left\|{\mathrm{e}}^{{\boldsymbol{A}}_{j}\left(t-\hat{t}_{2i-1}\right)}{\boldsymbol{b}}_{j}-{\mathrm{e}}^{{\boldsymbol{A}}_{j^{\prime}}\left(t-\hat{t}_{2i-1}\right)}{\boldsymbol{b}}_{j^{\prime}}\right\|$
$\displaystyle\leq$ $\displaystyle
2\left(C_{1}\left\|{\boldsymbol{x}}\right\|+C_{2}C_{3}\right)\varepsilon+C_{3}\left\|{\boldsymbol{x}}\left(\hat{t}_{2i-1}\right)-{\boldsymbol{x}}^{\prime}\left(\hat{t}_{2i-1}\right)\right\|.$

Since
${\boldsymbol{x}}\left(\hat{t}_{0}\right)={\boldsymbol{x}}^{\prime}\left(\hat{t}_{0}\right)$,
(32) and (35) implies that

$\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}\leq
2\left(C_{3}^{2M-2}-1\right)\left(C_{1}\left\|{\boldsymbol{x}}\right\|+C_{2}C_{3}\right)\varepsilon.$
(36)

Therefore, (30a) holds. Similarly, $\forall i\in\left[M\right]$,

$\displaystyle\left\|{\boldsymbol{x}}\left(t_{i}\right)-{\boldsymbol{x}}^{\prime}\left(t_{i}^{\prime}\right)\right\|$
(37) $\displaystyle\leq$
$\displaystyle\left\|{\boldsymbol{x}}\left(t_{i}^{\prime}\right)-{\boldsymbol{x}}^{\prime}\left(t_{i}^{\prime}\right)\right\|+\left\|{\boldsymbol{x}}\left(t_{i}\right)-{\boldsymbol{x}}\left(t_{i}^{\prime}\right)\right\|$
$\displaystyle\leq$
$\displaystyle\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}+C_{1}\left\|{\boldsymbol{x}}\right\|\varepsilon+C_{2}C_{3}\varepsilon$
$\displaystyle\leq$
$\displaystyle\left(2C_{3}^{2M-2}-1\right)\left(C_{1}\left\|{\boldsymbol{x}}\right\|+C_{2}C_{3}\right)\varepsilon.$

Therefore, (30b) holds. ∎

######  Theorem 2.

${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ and
${\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$ are Bang-Singular-
Bang trajectories of problem (1a) without chattering. Assume that
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ and
${\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$ are represented by
the same augmented switching law ${\mathcal{L}}={\mathcal{L}}^{\prime}$.
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ is feasible for problem
(1a). $\exists\varepsilon>0$ only dependent on
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$, s.t. if
$\left\|{\boldsymbol{t}}-{\boldsymbol{t}}^{\prime}\right\|<\varepsilon$, then
${\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$ is feasible for
problem (1a).

###### Proof.

Fix ${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$. Firstly, let

$\varepsilon<\frac{1}{2}\min_{i\in\left[M-1\right]}\left\\{\left|t_{i+1}-t_{i}\right|\right\\}.$
(38)

According to Proposition 6, $\exists C>0$ only dependent on
${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$, s.t. if
$\left\|{\boldsymbol{t}}-{\boldsymbol{t}}^{\prime}\right\|<\varepsilon$, then
(30a) holds.

Secondly, consider the inequality constraints
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}+d_{p}\leq 0$. Denote
that
$\left\\{i^{\left(p\right)}_{j}\right\\}_{j=0}^{N_{p}}\subset\left[M\right]\cup\left\\{0\right\\}$
increasing strictly monotonically, s.t. $i^{\left(p\right)}_{0}=0$,
$i^{\left(p\right)}_{N_{p}}=M$, and $\forall j\in\left[N_{p}\right]$,
$t\in\left(t_{i^{\left(p\right)}_{j-1}},t_{i^{\left(p\right)}_{j}}\right)$,
either ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}\equiv
0$ or ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$
holds. If ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$ in
$\left(t_{i^{\left(p\right)}_{j-1}},t_{i^{\left(p\right)}_{j}}\right)$, then
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}\equiv 0$ in
$\left(t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{i^{\left(p\right)}_{j}}^{\prime}\right)$
since ${\mathcal{L}}={\mathcal{L}}^{\prime}$.

Consider the case where ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}<0$
in $\left(t_{i^{\left(p\right)}_{j-1}},t_{i^{\left(p\right)}_{j}}\right)$.
Denote
$f_{p}\left(t\right)={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}$,
and then $\forall r\in{\mathbb{N}}^{*}$,
$\frac{\mathrm{d}^{r}f_{p}}{\mathrm{d}t^{r}}={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{A}}_{i_{j}^{\left(p\right)}}^{r-1}\left({\boldsymbol{A}}_{i_{j}^{\left(p\right)}}{\boldsymbol{x}}+{\boldsymbol{b}}_{i_{j}^{\left(p\right)}}\right)$
is continuous in
$\left(t_{i^{\left(p\right)}_{j-1}},t_{i^{\left(p\right)}_{j}}\right)$. If
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t_{i^{\left(p\right)}_{j-1}}\right)+d_{p}<0$,
then let $r_{j}^{\left(p\right)}=0$; otherwise, according to Proposition 3,
Proposition 4, and Proposition 5, $\exists 1\leq r_{j}^{\left(p\right)}\leq
n$, s.t.
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)<0$
and $\forall 1\leq r<r_{j}^{\left(p\right)}$,
$\frac{\mathrm{d}^{r}f_{p}}{\mathrm{d}t^{r}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)=0$.
Let $t_{1,j}^{\left(p\right)}>t_{i^{\left(p\right)}_{j-1}}$ is small enough,
s.t.
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t\right)<\frac{1}{2}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)<0$
and
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t\right)\leq\left|\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\right|+1$
for $t\in\left(t_{i^{\left(p\right)}_{j-1}},t_{1,j}^{\left(p\right)}\right)$.
$t_{2,j}^{\left(p\right)}<t_{i^{\left(p\right)}_{j}}$ is constructed
similarly. Then, $f_{p}\left(t\right)$ decreases strictly monotonically in
$\left(t_{i^{\left(p\right)}_{j-1}},t_{1,j}^{\left(p\right)}\right)$ and
increases strictly monotonically in
$\left(t_{2,j}^{\left(p\right)},t_{i^{\left(p\right)}_{j}}\right)$. In both
case,
$\sup_{t\in\left[t_{1,j}^{\left(p\right)},t_{2,j}^{\left(p\right)}\right]}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}<0$
due to continuity. Let

$\varepsilon<-\frac{\sup_{t\in\left[t_{1,j}^{\left(p\right)},t_{2,j}^{\left(p\right)}\right]}{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}}{2C\left\|{\boldsymbol{c}}_{p}\right\|}.$
(39)

Then, $\forall
t\in\left[t_{1,j}^{\left(p\right)},t_{2,j}^{\left(p\right)}\right]$,

${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)+d_{p}\leq{\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}\left(t\right)+d_{p}+\left\|{\boldsymbol{c}}_{p}\right\|\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}<0.$
(40)

In other words,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)$ is
feasible in $\left[t_{1,j}^{\left(p\right)},t_{2,j}^{\left(p\right)}\right]$.

The feasibility of ${\boldsymbol{x}}^{\prime}$ in
$\left(t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{1,j}^{\left(p\right)}\right)\cup\left(t_{2,j}^{\left(p\right)},t_{i^{\left(p\right)}_{j}}^{\prime}\right)$
needs further discussion. Let

$\begin{dcases}\varepsilon<t_{1,j}^{\left(p\right)}-t_{i^{\left(p\right)}_{j-1}},\\\
\varepsilon<-\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\left(4C\left\|{\boldsymbol{c}}_{p}\right\|\left(\left\|{\boldsymbol{A}}_{i_{j}^{\left(p\right)}}\right\|^{r_{j}^{\left(p\right)}}+1\right)\right)^{-1},\\\
\varepsilon<\left(C\left\|{\boldsymbol{c}}_{p}\right\|\left(\left\|{\boldsymbol{A}}_{i_{j}^{\left(p\right)}}\right\|^{r_{j}^{\left(p\right)}+1}+1\right)\right)^{-1},\\\
\varepsilon<-\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\left(4\left|\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\right|+8\right)^{-1}.\end{dcases}$
(41)

Denote
$f_{p}^{\prime}\left(t\right)={\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)+d_{p}$.
Then, $\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}},t_{1,j}^{\left(p\right)}\right]$,

$\displaystyle\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t\right)\leq\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t\right)+\left\|{\boldsymbol{c}}_{p}\right\|\left\|{\boldsymbol{A}}_{i_{j}^{\left(p\right)}}\right\|^{r_{j}^{\left(p\right)}}\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}$
(42) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)-\frac{1}{4}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)<0.$

$\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}}-\varepsilon,t_{i^{\left(p\right)}_{j-1}}\right]$,
it has

$\displaystyle\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t\right)$
(43) $\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t\right)+\left\|{\boldsymbol{c}}_{p}\right\|\left\|{\boldsymbol{A}}_{i_{j}^{\left(p\right)}}\right\|^{r_{j}^{\left(p\right)+1}}\left\|{\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\left|\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\right|+2.$

Therefore, if $t_{i^{\left(p\right)}_{j-1}}^{\prime}\geq
t_{i^{\left(p\right)}_{j-1}}$, then $\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{1,j}^{\left(p\right)}\right]$,
it has
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t\right)<0$.
If $t_{i^{\left(p\right)}_{j-1}}^{\prime}<t_{i^{\left(p\right)}_{j-1}}$, then
(30b) implies that
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\leq\frac{3}{4}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)<0$.
$\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{i^{\left(p\right)}_{j-1}}\right]$,

$\displaystyle\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t\right)$
(44) $\displaystyle\leq$
$\displaystyle\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)+\left(\left|\frac{\mathrm{d}^{r_{j}^{\left(p\right)}+1}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}+1}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)\right|+2\right)\varepsilon$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)<0.$

Therefore, $\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{1,j}^{\left(p\right)}\right]$,
it has
$\frac{\mathrm{d}^{r_{j}^{\left(p\right)}}f_{p}^{\prime}}{\mathrm{d}t^{r_{j}^{\left(p\right)}}}\left(t\right)<0$.

Since ${\mathcal{L}}={\mathcal{L}}^{\prime}$, $\forall 1\leq
r<r_{j}^{\left(p\right)}$,
$\frac{\mathrm{d}^{r}f_{p}^{\prime}}{\mathrm{d}t^{r}}\left(t_{i^{\left(p\right)}_{j-1}}^{+}\right)=0$;
hence, $f_{p}^{\prime}$ decreases strictly monotonically in
$\left(t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{1,j}^{\left(p\right)}\right)$.
Therefore, $\forall
t\in\left[t_{i^{\left(p\right)}_{j-1}}^{\prime},t_{1,j}^{\left(p\right)}\right]$,
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)+d_{p}<0$.
Similarly, let $\varepsilon$ is small enough, and it has
${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)+d_{p}<0$
in
$\left[t_{2,j}^{\left(p\right)},t_{i^{\left(p\right)}_{j}}^{\prime}\right]$.

In summary, when $\varepsilon$ is small enough, $\forall p\in\left[P\right]$,
$t\in\left[t_{0}^{\prime},t_{M}^{\prime}\right]$, it has
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}^{\prime}\left(t\right)+d_{p}\leq 0$
holds.

Thirdly, the inequality $\left|u^{\prime}\right|\leq{u_{\mathrm{m}}}$ can be
proved by applying a similar analysis when $\varepsilon$ is small enough. ∎

Theorem 2 points out that if the disturbed trajectory is represented by the
same augmented switching law to the original trajectory and the disturbance is
light enough, then the disturbed trajectory is feasible. In other words,
Theorem 2 provides a novel variational approach under fixed augmented
switching laws with a feasibility guarantee near the constrained boundary.

###  V-C Necessary Condition for the Optimal Trajectory

Based on Theorem 2, this section provides an approach to constructing a
feasible disturbed trajectory with the same augmented switching law. According
to Theorem 1, this section shows that the existence of such a disturbed
trajectory implies that the original trajectory is not optimal, resulting in a
state-centric necessary for non-chattering optimal control of problem (1a).

The Jacobian matrix of equality constraints induced by the augmented switching
law can be calculated by Proposition 7.

######  Proposition 7.

${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$ is driven by $\forall
i\in\left[M\right]$, $t\in\left(t_{i-1},t_{i}\right)$,
$\dot{\boldsymbol{x}}={\boldsymbol{A}}_{i}{\boldsymbol{x}}+{\boldsymbol{b}}_{i}$,
and ${\boldsymbol{x}}\left(t_{0}\right)={\boldsymbol{x}}_{0}$ is fixed.
Consider the mapping
${\boldsymbol{x}}_{i}:{\boldsymbol{t}}=\left(t_{i}\right)_{i=1}^{M}\mapsto{\boldsymbol{x}}\left(t_{i}\right)$.
Then, $\forall i,j\in\left[M\right]$,

$\frac{\partial{\boldsymbol{x}}_{i}}{t_{j}}=\begin{dcases}\prod_{k=i}^{j+1}{\mathrm{e}}^{{\boldsymbol{A}}_{k}\left(t_{k}-t_{k-1}\right)}\left(\left({\boldsymbol{A}}_{j}-{\boldsymbol{A}}_{j+1}\right){\boldsymbol{x}}_{j}+{\boldsymbol{b}}_{j}-{\boldsymbol{b}}_{j+1}\right),\\\
\qquad j<i,\\\
{\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{b}}_{i},\,j=i,\\\
0,\,j>i,\end{dcases}$ (45)

where
$\prod_{k=i}^{j+1}{\mathrm{e}}^{{\boldsymbol{A}}_{k}\left(t_{k}-t_{k-1}\right)}$
refers to

${\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t_{i}-t_{i-1}\right)}{\mathrm{e}}^{{\boldsymbol{A}}_{i-1}\left(t_{i-1}-t_{i-2}\right)}\dots\mathrm{e}^{{\boldsymbol{A}}_{j+1}\left(t_{j+1}-t_{j}\right)}.$
(46)

###### Proof.

Evidently, $\forall j>i$, ${\boldsymbol{x}}_{i}$ is independent with $t_{j}$;
hence, $\frac{\partial{\boldsymbol{x}}_{i}}{t_{j}}={\boldsymbol{0}}$.
According to Lemma 1,

${\boldsymbol{x}}_{i}={\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t_{i}-t_{i-1}\right)}{\boldsymbol{x}}_{i-1}+\int_{t_{i-1}}^{t_{i}}{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t_{i}-\tau\right)}{\boldsymbol{b}}_{i}\mathrm{d}\tau.$
(47)

Therefore, $\forall i\in\left[M\right]$,

$\frac{\partial{\boldsymbol{x}}_{i}}{\partial
t_{i}}={\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{b}}_{i}.$ (48)

Applying (48), $\forall i>1$, it has

$\displaystyle\frac{\partial{\boldsymbol{x}}_{i}}{\partial
t_{i-1}}={\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t_{i}-t_{i-1}\right)}\left(\frac{\partial{\boldsymbol{x}}_{i-1}}{\partial
t_{i-1}}-{\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i-1}-{\boldsymbol{b}}_{i}\right)$
(49) $\displaystyle=$
$\displaystyle{\mathrm{e}}^{{\boldsymbol{A}}_{i}\left(t_{i}-t_{i-1}\right)}\left(\left({\boldsymbol{A}}_{i-1}-{\boldsymbol{A}}_{i}\right){\boldsymbol{x}}_{i-1}+{\boldsymbol{b}}_{i-1}-{\boldsymbol{b}}_{i}\right).$

$\forall i>1$, $j<i-1$, it has

$\displaystyle\frac{\partial{\boldsymbol{x}}_{i}}{\partial
t_{j}}=\prod_{k=i}^{j+2}{\mathrm{e}}^{{\boldsymbol{A}}_{k}\left(t_{k}-t_{k-1}\right)}\frac{\partial{\boldsymbol{x}}_{j+1}}{\partial
t_{j}}$ (50) $\displaystyle=$
$\displaystyle\prod_{k=i}^{j+1}{\mathrm{e}}^{{\boldsymbol{A}}_{k}\left(t_{k}-t_{k-1}\right)}\left(\left({\boldsymbol{A}}_{j}-{\boldsymbol{A}}_{j+1}\right){\boldsymbol{x}}_{j}+{\boldsymbol{b}}_{j}-{\boldsymbol{b}}_{j+1}\right).$

Therefore, (45) holds. ∎

Utilizing the notations in Section V-B, the augmented switching law provides a
system of equalities on every keypoints. Denote that $\forall
i\in\left[M\right]$,
${\boldsymbol{F}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{g}}_{i}={\boldsymbol{0}}$
and
${\boldsymbol{C}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{d}}_{i}<{\boldsymbol{0}}$,
where ${\boldsymbol{x}}_{i}={\boldsymbol{x}}\left(t_{i}\right)$ and
${\boldsymbol{F}}_{i}$ has full row rank. For a constrained arc with active
constraints $\mathcal{P}$, the equality constraints induced by $\mathcal{P}$
are added in only one end instead of both ends of the arcs since one end
satisfies $\mathcal{P}$ implies that the whole arc satisfies $\mathcal{P}$.
Note that the additional end-constraints at both ends are eliminated by
equality constraints induced by $\mathcal{P}$. Similar processes are applied
if $u\equiv{u_{\mathrm{m}}}$ or $u\equiv-{u_{\mathrm{m}}}$ during a
constrained arc.
${\boldsymbol{F}}_{i}{\boldsymbol{x}}_{i}\left({\boldsymbol{t}}\right)+{\boldsymbol{g}}_{i}$
for $i\in\left[M\right]$ induce a function
${\boldsymbol{H}}\left({\boldsymbol{t}}\right)$. Then, Theorem 3 provides a
necessary condition for the optimal trajectory.

Before providing Theorem 3, the implicit function theorem [29] should be
introduced.

######  Lemma 3 (The Implicit Function Theorem [29]).

Assume $S\subset{\mathbb{R}}^{p+q}$ is open and non-empty.
${\boldsymbol{F}}:S\to{\mathbb{R}}^{q}$ is $C^{1}$ continuous.
$\left({\boldsymbol{\xi}}_{0},{\boldsymbol{\eta}}_{0}\right)\in S$, satisfying
${\boldsymbol{F}}\left({\boldsymbol{\xi}}_{0},{\boldsymbol{\eta}}_{0}\right)={\boldsymbol{0}}$
and
$\det\frac{\partial{\boldsymbol{F}}}{\partial{\boldsymbol{\eta}}}\left({\boldsymbol{\xi}}_{0},{\boldsymbol{\eta}}_{0}\right)\not=0$.
Then, $\exists\delta_{1},\delta_{2}>0$, s.t. $\forall{\boldsymbol{\xi}}\in
B_{\delta_{1}}\left({\boldsymbol{\xi}}_{0}\right)$,
$\exists!{\boldsymbol{\eta}}\in
B_{\delta_{2}}\left({\boldsymbol{\eta}}_{0}\right)$ satisfying
${\boldsymbol{F}}\left({\boldsymbol{\xi}},{\boldsymbol{\eta}}\right)={\boldsymbol{0}}$.
The above relation induces a mapping
${\boldsymbol{f}}:B_{\delta_{1}}\left({\boldsymbol{\xi}}_{0}\right)\to
B_{\delta_{2}}\left({\boldsymbol{\eta}}_{0}\right)$. Then, ${\boldsymbol{f}}$
is $C^{1}$ continuous, where
$\frac{\mathrm{d}{\boldsymbol{f}}}{\mathrm{d}{\boldsymbol{\xi}}}\left({\boldsymbol{\xi}}\right)=-\frac{\partial{\boldsymbol{F}}}{\partial{\boldsymbol{\eta}}}\left({\boldsymbol{\xi}},{\boldsymbol{f}}\left({\boldsymbol{\xi}}\right)\right)^{-1}\frac{\partial{\boldsymbol{F}}}{\partial{\boldsymbol{\xi}}}\left({\boldsymbol{\xi}},{\boldsymbol{f}}\left({\boldsymbol{\xi}}\right)\right)$.

######  Theorem 3 (State-Centric Necessary Condition).

Assume that the optimal trajectory
${\boldsymbol{x}}={\boldsymbol{x}}^{*}\left(t\right)$ of problem (1a)
satisfies Assumption 1. Denote the equality constraints induced by the
augmented switching law as
${\boldsymbol{H}}\left({\boldsymbol{t}}^{*}\right)={\boldsymbol{0}}$, where
${\boldsymbol{t}}^{*}=\left(t_{i}^{*}\right)_{i=1}^{M}$ is the arriving time
of each keypoints and $t_{0}=0$. Then,
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}_{1:\left(M-1\right)}}\left({\boldsymbol{t}}^{*}\right)$
does not have full row rank.

###### Proof.

Assume that ${\boldsymbol{H}}:{\mathbb{R}}^{M}\to{\mathbb{R}}^{M^{\prime}}$.
$M^{\prime}$ is the number of linearly independent equality constraints on
keypoints, and $M$ is the number of keypoints. Assume that
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}_{1:\left(M-1\right)}}\left({\boldsymbol{t}}^{*}\right)$
has full row rank, i.e.,

${\mathrm{rank}}\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}_{1:\left(M-1\right)}}\left({\boldsymbol{t}}^{*}\right)=M^{\prime}\leq
M-1.$ (51)

Assume that the $i$-th row of
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}_{1:\left(M-1\right)}}\left({\boldsymbol{t}}^{*}\right)$
is linearly independent where $i\in\mathcal{I}\subset\left[M-1\right]$ and
$\\#\mathcal{I}=M^{\prime}$. Let $\varepsilon>0$ be small enough to satisfy
the condition of Theorem 2 and Lemma 3. The implicit function
${\boldsymbol{f}}:\left(t_{i}\right)_{i\in{\left.\left[M\right]\middle\backslash\mathcal{I}\right.}}\mapsto\left(t_{i}\right)_{i\in\mathcal{I}}$
is induced by
${\boldsymbol{H}}\left({\boldsymbol{t}}\right)={\boldsymbol{0}}$. Based on
${\boldsymbol{f}}$, $\exists{\boldsymbol{t}}^{\prime}$ satisfies
${\boldsymbol{H}}\left({\boldsymbol{t}}^{\prime}\right)={\boldsymbol{0}}$,
$t_{M}^{\prime}<t_{M}^{*}$, and
$\left\|{\boldsymbol{t}}^{\prime}-{\boldsymbol{t}}^{*}\right\|<\varepsilon$.
According to Theorem 2, the disturbed trajectory
${\boldsymbol{x}}={\boldsymbol{x}}^{\prime}\left(t\right)$ is feasible since
${\boldsymbol{H}}\left({\boldsymbol{t}}^{\prime}\right)={\boldsymbol{0}}$
implies that ${\boldsymbol{x}}^{\prime}$ and ${\boldsymbol{x}}^{*}$ have the
same augmented switching law. However, ${\boldsymbol{x}}^{\prime}$ achieves a
shorter terminal time $t_{M}^{\prime}<t_{M}^{*}$, which contradicts the
optimality of ${\boldsymbol{x}}^{*}$. ∎

######  Remark.

Theorem 3 provides a first order necessary condition for the optimal
trajectory of problem (1a). In fact, if
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}_{1:\left(M-1\right)}}\left({\boldsymbol{t}}^{*}\right)$
does not have full row rank but
$\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}}\left({\boldsymbol{t}}^{*}\right)$
has full row rank with $M^{\prime}<M$, then the solution
${\boldsymbol{t}}^{*}$ can be seen as a stationary point under the constraint
of $\mathcal{L}$. For this case, some high order necessary conditions can be
derived. Consider the implicit function
${\boldsymbol{f}}:\left(t_{i}\right)_{i\in{\left.\left[M\right]\middle\backslash\mathcal{I}\right.}}\mapsto\left(t_{i}\right)_{i\in\mathcal{I}}$
induced by ${\boldsymbol{H}}\left({\boldsymbol{t}}\right)={\boldsymbol{0}}$
where $M\in\mathcal{I}$. Then, the function
$\left(t_{i}\right)_{i\in{\left.\left[M\right]\middle\backslash\mathcal{I}\right.}}\mapsto
t_{M}$ should achieve a strictly local minimum at
$\left(t_{i}^{*}\right)_{i\in{\left.\left[M\right]\middle\backslash\mathcal{I}\right.}}$
according to Theorem 1. More high order necessary conditions can be derived
for this case.

######  Remark.

In fact, if a given non-chattering Bang-Singular-Bang trajectory does not
satisfy the necessary condition in Theorem 3, then one can perform numerical
local optimization of trajectories based on Theorem 3. For example, let
$t_{M}$ be the optimization objective and consider the equality and inequality
constraints of the augmented switching law as the constraints of the
optimization problem. In this way, it is hopeful to obtain a locally optimal
solution or a stationary point of problem (1a) through iteratively solving an
$M$-dimensional nonlinear problem. Note that when applying shooting methods
[30], problem (1a) is highly nonlinear and non-convex, and the long motion
time may result in unacceptable computation cost. In contrast, the local
optimization method on ${\boldsymbol{t}}$ based on Theorem 1a will be more
efficient due to the significantly lower dimensions and the suboptimal
feasible initial value. The above method will be developed in future works.

Theorem 3 provides a necessary condition for the optimal control of problem
(1a) under Assumption 1. Different from the existing costate-based necessary
conditions [14], Theorem 3 requires no information on the costates, which
significantly simplifies the computation of the necessary conditions. Compared
to existing state-centric necessary conditions [22], Theorem 3 is effective in
theoretical reasoning and has a lower computational complexity in numerical
computation.

Some further discussions on the Jacobian matrix are necessary. Consider a
keypoint $t_{i}$ with a constraint
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\leq{\boldsymbol{0}}$ active at
$t_{i}$. Assume that the constraint is of $\hat{r}$-th order w.r.t the arc
$\left(t_{i-1},t_{i}\right)$ where
$\dot{\boldsymbol{x}}={\boldsymbol{A}}_{i}{\boldsymbol{x}}+{\boldsymbol{b}}_{i}$.
In other words, if $1\leq\hat{r}<n$, then
$\left(-1\right)^{\hat{r}}\frac{\mathrm{d}^{\hat{r}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{\hat{r}}}\left(t_{i}^{-}\right)>0$
and $\forall 1\leq r<\hat{r}$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r}}\left(t_{i}^{-}\right)=0$.
If $\hat{r}=n$, then $\forall r\in{\mathbb{N}}^{*}$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r}}\left(t_{i}^{-}\right)=0$.
Among them. $\forall r\in{\mathbb{N}}^{*}$,
$\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r}}\left(t_{i}^{-}\right)={\boldsymbol{c}}^{\top}{\boldsymbol{A}}_{i}^{r-1}\left({\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{b}}_{i}\right)$.
According to Proposition 7, $\forall r\in{\mathbb{N}}$,

$\displaystyle\frac{\partial}{\partial
t_{i}}\left.\frac{\mathrm{d}^{r}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r}}\right|_{t_{i}^{-}}=\frac{\partial}{\partial
t_{i}}\left({\boldsymbol{c}}^{\top}{\boldsymbol{A}}_{i}^{r-1}\left({\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{b}}_{i}\right)\right)$
(52) $\displaystyle=$
$\displaystyle\left({\boldsymbol{c}}^{\top}{\boldsymbol{A}}_{i}^{r}\left({\boldsymbol{A}}_{i}{\boldsymbol{x}}_{i}+{\boldsymbol{b}}_{i}\right)\right)=\left.\frac{\mathrm{d}^{r+1}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r+1}}\right|_{t_{i}^{-}}.$

In other words, the Jacobian matrix of
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\leq{\boldsymbol{0}}$ w.r.t. $t_{i}$
is
$\left.\frac{\mathrm{d}^{r+1}\left({\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\right)}{\mathrm{d}t^{r+1}}\right|_{t_{i}^{-}}{\boldsymbol{e}}_{\hat{r}}$.
Specifically, if $\hat{r}=n$, then the Jacobian matrix of
${\boldsymbol{c}}^{\top}{\boldsymbol{x}}+d\leq{\boldsymbol{0}}$ w.r.t. $t_{i}$
is ${\boldsymbol{0}}$.

##  VI Applications in Chain-of-Integrators Systems and Simulation Experiments

This section provides some applications of the state-centric necessary
condition, i.e., Theorem 3, for the optimal control of problem (1a). For
convenience, the chain-of-integrators system with box constraints is
considered in this section, where the notations have been introduced in
Section IV-C.

Time-optimal control for chain-of-integrators systems with box constraints is
an open and challenging problem in the optimal control domain, yet to be
resolved. The problem can be summarized as follows:

$\displaystyle\min\quad$ $\displaystyle
J=\int_{0}^{{t_{\mathrm{f}}}}\mathrm{d}t,$ (53a)
$\displaystyle{\mathrm{s.t.}}\quad$
$\displaystyle\dot{x}_{k}\left(t\right)=x_{k-1}\left(t\right),\,\forall
1<k\leq n,\,t\in\left[0,t_{\mathrm{f}}\right],$ (53b)
$\displaystyle\dot{x}_{1}\left(t\right)=u\left(t\right),\,\forall
t\in\left[0,t_{\mathrm{f}}\right],$ (53c)
$\displaystyle{\boldsymbol{x}}\left(0\right)={\boldsymbol{x}}_{0},\,{\boldsymbol{x}}\left({t_{\mathrm{f}}}\right)={\boldsymbol{x}}_{\mathrm{f}},$
(53d) $\displaystyle\left|x_{k}\left(t\right)\right|\leq
x_{{\mathrm{m}}k},\,\forall 1\leq k\leq
n,\,t\in\left[0,t_{\mathrm{f}}\right],$ (53e)
$\displaystyle\left|u\left(t\right)\right|\leq{u_{\mathrm{m}}},\,\forall
t\in\left[0,t_{\mathrm{f}}\right].$ (53f)

The Jacobian matrix, except the last column induced by the augmented switching
law, denoted as ${\boldsymbol{J}}$, can be calculated as follows:

######  Corollary 1.

Assume that $\forall i\in\left[N\right]$, $u\equiv
u_{i}\in\left\\{\pm{u_{\mathrm{m}}},0\right\\}$ in
$\left(t_{i-1},t_{i}\right)$, and denote
${\boldsymbol{x}}_{i}={\boldsymbol{x}}\left(t_{i}\right)$. Then,

$\frac{\partial{\boldsymbol{x}}_{i}}{\partial t_{j}}=\begin{dcases}-\Delta
u_{j+1}{\boldsymbol{\phi}}\left(t_{i}-t_{j}\right),&j<i\\\
\left[\begin{array}[]{c}u_{i}\\\
{\boldsymbol{x}}_{i,1:\left(n-1\right)}\end{array}\right],&j=i\\\
{\boldsymbol{0}},&j>i.\end{dcases}$ (54)

Among them, $\Delta u_{i}=u_{i}-u_{i-1}$ and $u_{0}=0$.
${\boldsymbol{\phi}}\left(t\right)\triangleq\left(\phi_{k}\left(t\right)\right)_{k=1}^{n}$
where $\phi_{k}\left(t\right)=\frac{t^{k-1}}{\left(k-1\right)!}$.

###### Proof.

Proposition 7 directly implies Corollary 1. ∎

This section provides some applications of Theorem 3. Section VI-A proves a
trivial conclusion from a state-centric perspective. Section VI-B and Section
VI-C prove two corollaries that are challenging to prove by traditional
costate-based necessary conditions.

###  VI-A The Case where $\forall k\in\left[n\right]$,
$x_{{\mathrm{m}}k}=\infty$

The case where $\forall k\in\left[n\right]$, $x_{{\mathrm{m}}k}=\infty$ has a
well-known conclusion that the optimal control switches no more than
$\left(n-1\right)$ times [19]. This section proves the above conclusion based
on the proposed Theorem 3.

######  Corollary 2.

In problem (53a), assume that $\forall k\in\left[n\right]$,
$x_{{\mathrm{m}}k}=\infty$. Then, the optimal control switches no more than
$\left(n-1\right)$ times.

###### Proof.

Chattering does not occur since no state inequality constraints exist. Assume
that the optimal control switches $\left(N-1\right)$ times. In other words,
$\forall i\in\left[N\right]$, $t\in\left(t_{i-1},t_{i}\right)$,
$u\left(t\right)\equiv u_{i}\in\left\\{\pm{u_{\mathrm{m}}}\right\\}$.
$u_{i}\not=u_{i-1}$, implies that $\Delta u_{i}\not=0$. Then,

${\boldsymbol{J}}=\left[-\Delta
u_{i+1}{\boldsymbol{\phi}}\left(t_{N}-t_{i}\right)\right]_{i=1}^{N-1}\in{\mathbb{R}}^{n\times\left(N-1\right)}.$
(55)

The Vandermonde form of ${\boldsymbol{J}}$ implies that
${\mathrm{rank}}{\boldsymbol{J}}=\min\left\\{n,N-1\right\\}$. Theorem 3
implies that ${\mathrm{rank}}{\boldsymbol{J}}<n$; hence, $N-1<n$. ∎

Corollary 2 implies that the switching law is in the form of
$\overline{0}\underline{0}\overline{0}\underline{0}\dots$ or
$\underline{0}\overline{0}\underline{0}\overline{0}\dots$ with no more than
$n$ arcs. The case in Section VI-A is trivial since no state constraints exist
and the costate vector follows a fixed equation that $\forall
k\in\left[n-1\right]$, $\dot{\lambda}_{i}=-\lambda_{i+1}$ and
$\lambda_{n}\equiv{\mathrm{const}}$.

However, when the state constraints are induced, the behavior of the optimal
control can be complex. On the one hand, the multiplier ${\boldsymbol{\eta}}$
in (4) can be non-zero during a constrained arc. On the other hand, the
junction of ${\boldsymbol{\lambda}}$ can occur when a state constraint
switches between active and inactive [7]. For this case, the costate analysis
can be complex, while the proposed state-centric necessary condition provides
a simple and effective approach for analyzing the optimal control.

###  VI-B The Case where Only System Behaviors Occur

The case where only system behaviors occur, i.e., the optimal trajectory
consists of a finite number of unconstrained arcs and constrained arcs that
are not tangent to the constrained boundary, is widely applied in existing
works on trajectory planning [31, 32, 8].

Denote the augmented switching law of the trajectory as
${\mathcal{L}}={\mathcal{S}}_{1}{\mathcal{S}}_{2}\dots\mathcal{S}_{N}{\mathcal{E}}_{N}$,
where ${\mathcal{E}}_{N}$ is induced by
${\boldsymbol{x}}\left(t_{N}\right)={{\boldsymbol{x}}_{\mathrm{f}}}$. $\forall
k\in{\mathbb{N}}$, denote
$\left|\overline{k}\right|=\left|\underline{k}\right|=k$,
${\mathrm{sgn}}\left(\overline{k}\right)=+1$, and
${\mathrm{sgn}}\left(\underline{k}\right)=-1$. The non-existence of tangent
markers and additional end-constraints in ${\mathcal{L}}$ means that $\forall
i\in\left[N\right]$, the arc ${\mathcal{S}}_{i}$ achieves strictly
feasibility. An arc
${\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},\widehat{{\boldsymbol{F}}},\widehat{{\boldsymbol{g}}},\mathcal{P}\right)$
in $\left[t_{0},t_{1}\right]$ is strictly feasible if $\forall
p\not\in\mathcal{P}$, ${\boldsymbol{c}}_{p}^{\top}{\boldsymbol{x}}+d_{p}<0$
holds in $\left[t_{0},t_{1}\right]$.

######  Corollary 3.

Assume that the augmented switching law consists of system behaviors without
tangent markers and additional end-constraints except at ${t_{\mathrm{f}}}$.
Assume that:

  1. 1.

$\forall 1\leq j<i$, if
$\left|{\mathcal{S}}_{j}\right|\geq\left|{\mathcal{S}}_{i}\right|$, then
$\sum_{k=j+1}^{i}\left|{\mathcal{S}}_{k}\right|<i-j$.

  2. 2.

$\forall i<j\leq N$, if
$\left|{\mathcal{S}}_{j}\right|\geq\left|{\mathcal{S}}_{i}\right|$, then
$\sum_{k=i}^{j-1}\left|{\mathcal{S}}_{k}\right|<j-i$.

  3. 3.

$\forall i\in\left[N\right]$, if $\left|{\mathcal{S}}_{i}\right|>0$ and
$\forall i<j\leq N$,
$\left|{\mathcal{S}}_{j}\right|<\left|{\mathcal{S}}_{i}\right|$, then
$\sum_{k=i}^{N}\left|{\mathcal{S}}_{k}\right|\leq N-i$.

Then,

$N-\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|\leq n.$ (56)

###### Proof.

For a constrained arc ${\mathcal{S}}_{i}$, add the equality constraint at
$t_{i}$, i.e.,
${\boldsymbol{x}}_{i,1:\left|{\mathcal{S}}_{i}\right|}={\mathrm{sgn}}\left({\mathcal{S}}_{i}\right)x_{{\mathrm{m}}\left|{\mathcal{S}}_{i}\right|}{\boldsymbol{e}}_{\left|{\mathcal{S}}_{i}\right|}$.
Note that if ${\mathcal{S}}_{i}$ is constrained, i.e.,
$\left|{\mathcal{S}}_{i}\right|\not=0$, then $u\equiv 0$ in
${\mathcal{S}}_{i}$. Therefore, the constraints induced by ${\mathcal{L}}$ are

$\begin{dcases}{\boldsymbol{x}}_{i,1:\left|{\mathcal{S}}_{i}\right|}={\mathrm{sgn}}\left({\mathcal{S}}_{i}\right)x_{{\mathrm{m}}\left|{\mathcal{S}}_{i}\right|}{\boldsymbol{e}}_{\left|{\mathcal{S}}_{i}\right|},\,\forall
i\in\left[N\right],\,\left|{\mathcal{S}}_{i}\right|\not=0,\\\
{\boldsymbol{x}}_{N}={{\boldsymbol{x}}_{\mathrm{f}}}.\end{dcases}$ (57)

By Condition 3, $\left|{\mathcal{S}}_{N}\right|=0$. Denote
$\left|{\mathcal{S}}_{N+1}\right|=n$, and
$\mathcal{I}=\left\\{i\in\left[N+1\right]:\left|{\mathcal{S}}_{i}\right|\not=0\right\\}$.
According to Corollary 1, ${\boldsymbol{J}}$ contains the following rows:

${\boldsymbol{J}}=\left(-\Delta
u_{j+1}{\boldsymbol{\phi}}_{1:\left|{\mathcal{S}}_{i}\right|}\left(t_{i}-t_{j}\right)\delta_{j<i}\right)_{i\in\mathcal{I},\,j\in\left[N-1\right]},$
(58)

where $\delta_{A}$ is the indicator function of condition $A$, i.e.,

$\delta_{A}=\begin{dcases}1,&\text{condition }A\text{ holds},\\\
0,&\text{condition }A\text{ does not hold}.\end{dcases}$ (59)

Let ${\boldsymbol{A}}\leftrightarrow{\boldsymbol{B}}$ mean that
${\boldsymbol{A}}$ has full row rank if and only if ${\boldsymbol{B}}$ has
full row rank. Since $\forall j\in\left[N-1\right]$, $\Delta u_{j+1}\not=0$,
it has

${\boldsymbol{J}}\leftrightarrow\left({\boldsymbol{\phi}}_{1:\left|{\mathcal{S}}_{i}\right|}\left(t_{i}-t_{j}\right)\delta_{j<i}\right)_{i\in\mathcal{I},\,j\in\left[N-1\right]}.$
(60)

By iteratively taking differences between adjacent rows of (60), dividing by
the time difference, and eliminating, it can be proved that ${\boldsymbol{J}}$
has full rank based on Conditions 1, 2, and 3. The proof is similar to our
previous work [8] and is omitted due to space limitation. According to Theorem
3, the number of rows should be greater than the number of columns, i.e.,

$\sum_{i\in\mathcal{I}}\left|{\mathcal{S}}_{i}\right|=\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|+n>N-1.$
(61)

Therefore, (56) holds. ∎

######  Remark.

Further research suggests that Conditions 1, 2, and 3 in Corollary 3 might
hold almost everywhere. In other words,
$\left({\boldsymbol{x}}_{0},{\boldsymbol{x}}_{\mathrm{f}},{\boldsymbol{x}}_{\mathrm{m}},{u_{\mathrm{m}}}\right)$
whose resulting Bang-Singular-Bang trajectory does not satisfy Conditions 1,
2, and 3 forms a zero-measure set.

######  Remark.

In our previous work [8], a similar conclusion pointed out that when fixing
${{\boldsymbol{x}}_{\mathrm{f}}}$, then the set of
$\left({\boldsymbol{x}}_{0},{\boldsymbol{t}}\right)$ that satisfies the
augmented switching law
${\mathcal{L}}={\mathcal{S}}_{1}{\mathcal{S}}_{2}\dots\mathcal{S}_{N}{\mathcal{E}}_{N}$
locally forms a submanifold of $r$-dimensions. Among them,
$r=N-\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|$. However, the local
optimality is not discussed in [8]. Corollary 3 proves that the augmented
switching law of dimensions $r>n$ fails to achieve optimality.

${\boldsymbol{J}}=\left[\begin{array}[]{ccccccccc}-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}&-{u_{\mathrm{m}}}&0&0&0&\dots&0&0\\\
-2{u_{\mathrm{m}}}\left(t_{3}-t_{1}\right)&2{u_{\mathrm{m}}}\left(t_{3}-t_{2}\right)&0&0&0&0&\dots&0&0\\\
-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}&0&-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}&-{u_{\mathrm{m}}}&\dots&0&0\\\
-2{u_{\mathrm{m}}}\left(t_{6}-t_{1}\right)&2{u_{\mathrm{m}}}\left(t_{6}-t_{2}\right)&0&-2{u_{\mathrm{m}}}\left(t_{6}-t_{4}\right)&2{u_{\mathrm{m}}}\left(t_{6}-t_{5}\right)&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}&0&-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}&0&\dots&-2{u_{\mathrm{m}}}&2{u_{\mathrm{m}}}\\\
-2{u_{\mathrm{m}}}\left(t_{3N}-t_{1}\right)&2{u_{\mathrm{m}}}\left(t_{3N}-t_{2}\right)&0&-2{u_{\mathrm{m}}}\left(t_{3N}-t_{4}\right)&2{u_{\mathrm{m}}}\left(t_{3N}-t_{5}\right)&0&\dots&-2{u_{\mathrm{m}}}\left(t_{3N}-t_{3N-2}\right)&2{u_{\mathrm{m}}}\left(t_{3N}-t_{3N-1}\right)\\\
-2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{1}\right)&2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{2}\right)&{\boldsymbol{0}}&-2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{4}\right)&2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{5}\right)&{\boldsymbol{0}}&\dots&-2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{3N-2}\right)&2{u_{\mathrm{m}}}{\boldsymbol{\phi}}_{3:n}\left(t_{3N}-t_{3N-1}\right)\\\
\end{array}\right]$

(62)

$\left[\begin{array}[]{cc}\left(\delta_{j=i}-\delta_{j=i-1}\right)_{i\in\left[N\right],j\in\left[N\right]}&\\\
&{\boldsymbol{I}}_{n-1}\end{array}\right]\left[\begin{array}[]{cccc}{\boldsymbol{I}}_{N-1}&&&\\\
&0&1&\\\ &1&0&\\\
&&&{\boldsymbol{I}}_{n-2}\end{array}\right]\left[\begin{array}[]{cc}{\boldsymbol{I}}_{N+1}&\\\
&\left(\phi_{i+2-j}\left(t_{\infty}-t_{3N}\right)\right)_{i\in\left[n-2\right],j\in\left[n\right]}\end{array}\right]$
(63)

It is challenging to reason Corollary 3 by existing costate-based necessary
conditions [14] due to the complex behavior of the costates. In contrast,
existing costate-based necessary conditions can try to reason the costate for
a given trajectory case-by-case.

For example, in a 4th order problem,
$\mathcal{L}_{1}=\underline{01}\overline{01}\underline{0}\overline{2}\underline{01}\overline{01}\underline{0}$
can be feasible, but it cannot be optimal since
$N-\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|=5>4$.
$\mathcal{L}_{2}=\overline{01}\underline{0}\overline{2}\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}$
can serve as a candidate for the optimal control since
$N-\sum_{i=1}^{N}\left|{\mathcal{S}}_{i}\right|=4$. Of course, both
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ cannot be optimal in a 3rd order
problem. Though Corollary 3 is not a sufficient condition for optimal control,
it provides a simple and effective approach to analyzing the optimal control
of problem (53a).

###  VI-C The Chattering Phenomenon induced by $\left|x_{2}\right|\leq
x_{{\mathrm{m}}2}$

Our previous work [18] points out that if the chattering phenomenon occurs in
problem (53a), then there exists one and only one inequality state constraint
that switches between active and inactive during the chattering phenomenon. An
infinite number of unconstrained arcs are joined by the constrained boundary,
i.e., the unconstrained arcs are tangent to the boundary for infinite times.
In contrast, constrained arcs do not occur. This section discusses the
chattering phenomenon induced by $x_{2}\leq x_{{\mathrm{m}}2}$ based on the
proposed state-centric necessary condition. [18] provides the following
conclusion.

######  Lemma 4.

[18] Assume that the chattering phenomenon is induced by $x_{2}\leq
x_{{\mathrm{m}}2}$ in problem (1a) in a left neighborhood of $t_{\infty}$,
where $t_{\infty}$ is the chattering limit point. Then,
$\exists\left\\{t_{3k}\right\\}_{k=0}^{\infty}\subset\left[t_{0},t_{\infty}\right]$
increasing strictly monotonically and converging to $t_{\infty}$, s.t. $x_{2}$
is tangent to $x_{{\mathrm{m}}2}$ at $t_{3k}$. $\forall k\in{\mathbb{N}}^{*}$,
$u$ switches at most 2 times during $\left(t_{3k-3},t_{3k}\right)$.

Theorem 3 and Lemma 4 imply the following corollary.

######  Corollary 4.

Assume that the chattering phenomenon is induced by $x_{2}\leq
x_{{\mathrm{m}}2}$ in problem (1a) in a left neighborhood of $t_{\infty}$. Let
$\tau_{k}=t_{\infty}-t_{3k}$, $\forall k\in{\mathbb{N}}$. Denote
$f_{m}\left(a,b,c\right)$ as

$\left(b+3a\right)^{m}-3\left(3b+a\right)^{m}+3\left(c+3b\right)^{m}-\left(3c+b\right)^{m}.$
(64)

Then, $\forall N\in{\mathbb{N}}^{*}$, ${\boldsymbol{J}}^{\prime}$ does not
have full row rank, where

${\boldsymbol{J}}^{\prime}=\left(f_{i+1}\left(\tau_{j-1},\tau_{j},\tau_{j+1}\right)\right)_{i\in\left[n-2\right],j\in\left[N\right]}.$
(65)

Furthermore, $\forall k\geq n-2$, $\tau_{k}$ can be calculated by the
following recursive equation: