# Stability of singularly perturbed hybrid systems with restricted systems
evolving on boundary layer manifolds

Suad Krilašević  Sergio Grammatico  Delft Center for Systems and Control, TU
Delft, The Netherlands

###### Abstract

We present a singular perturbation theory applicable to systems with hybrid
boundary layer systems and hybrid reduced systems with jumps from the boundary
layer manifold. First, we prove practical attractivity of an adequate
attractor set for small enough tuning parameters and sufficiently long time
between almost all jumps. Second, under mild conditions on the jump mapping,
we prove semi-global practical asymptotic stability of a restricted attractor
set. Finally, for certain classes of dynamics, we prove semi-global practical
asymptotic stability of the restricted attractor set for small enough tuning
parameters and sufficiently long period between almost all jumps of the slow
states only.

###### keywords:

Singular perturbations, boundary layer, multi-agent game

††thanks: This work was partially supported by the ERC under research project
COSMOS (802348). E-mail addresses: {s.krilasevic-1<EMAIL_ADDRESS>
and

##  1 Introduction

A realistic modeling of many control systems requires high-order nonlinear
differential equations that might be difficult to fully analyze. To alleviate
this problem, we often design control systems with various parameters that
with proper tuning can effectively reduce the order of the model and thus
simplify the stability analysis. The main theoretical framework for such
analysis is singular perturbation theory [8], [4]. The associated model
reduction is accomplished by splitting the states into fast and slow states;
for each constant value of the slow states, the fast states should converge to
an equilibrium point defined by the slow states, and the union of these
equilibrium points for all possible slow states defines the so-called boundary
layer manifold. Then, the reduced system contains just the slow states and
their dynamics assuming they are evolving along that manifold.

Singular perturbation theory has been successfully applied to equilibrium
seeking in optimization and game theory. One common method of applying zeroth-
order algorithms to dynamical systems with cost measurements as output is
through a time-time scale separation of the controller and the plant, as
demonstrated in [7] and [14]. Time-scale separation can be useful for
algorithms where consensus on specific states must be reached before
initiating the equilibrium seeking process [1], [9], [23], [18]. Furthermore,
in some works [14], [12], [11], via singular perturbation analysis the
(pseudo)gradient estimate is filtered before being incorporated into the
algorithm. Singular perturbation theory is also used to demonstrate algorithm
convergence in problems with slowly varying parameters [2].

Several extensions of singular perturbation theory are known for hybrid
systems. In [16], the authors examine a singularly perturbed system in which
the boundary layer system is continuous, and the reduced system is hybrid, and
both render the corresponding sets globally asymptotically stable. While the
work in [21] proposes averaging theory results, in can also be used to prove
stability in singularly perturbed systems. Similarly to [16], the authors
assume that the boundary layer system is continuous and that the averaged
system, which plays the role of the reduced system, is hybrid. In [22], the
same authors extend the results for the case when the boundary layer system
itself is hybrid. In the aforementioned works, the reduced system is derived
by assuming that the slow states “flow” along the boundary layer manifold,
while the slow states _do not_ jump from that manifold. Therefore, the reduced
system jumps cannot use the properties of the boundary layer manifold to
support stabilization; essentially only the continuous dynamics are used to
prove stability, “despite” the jumps.
In order for the discrete-time dynamics to support stabilization of singularly
perturbed systems, we can design the dynamics so that we jump when we are in
the proximity of the manifold. This scenario in principle is similar to that
in [20], [5] where the authors prove that there exists a sampling period such
that a discrete-time optimization-based controller (the reduced system) can
find a neighborhood of the optimum of a steady-state output map of a
continuous system with an input (boundary layer system). In [15], the authors
take a step further and design an event-triggered framework to accomplish the
same task by measuring the changes in the output and in turn to determine when
the system has approached the boundary layer manifold. Although these methods
better incorporate discrete-time reduced system dynamics, the boundary layer
system is still only continuous. In this paper, we instead deal with a hybrid
boundary layer system and thus extend the current state of the art.
_Contribution_ : In view of the above literature, our theoretical
contributions are summarized next:

  * •

We propose a singular perturbation theory for hybrid systems, where the
reduced system takes into account the jumps from the boundary layer manifold,
differently from [16], [21] where jumps are assumed not to interfere with
stability. Furthermore, we allow for the set of fast variables not to be
bounded a priori, thus enabling the use of reference trajectories and counter
variables in the boundary layer system.

  * •

We prove semi-global practical asymptotic stability of the restricted
attractor set, under certain mild assumptions on the jump mapping. This
attractor set includes only the steady-state values of the fast states that
correspond to the slow attractor states, rather than the complete range of
possible fast variables.

  * •

We show that, in a system resembling the one described in [22], where a
distinction is made between jumps in the slow and fast states, the
aforementioned results remain valid if there are sufficiently long intervals
between nearly all jumps in the _slow states_.

Our theory enables the analysis of multiple timescale control systems where
both the controller and the plant are hybrid. Furthermore, as the jumps occur
at the boundary layer, it would be also possible to incorporate state/output
feedback into the controller jump mappings.

_Notation_ : The set of real numbers and the set of nonnegative real numbers
are denoted by $\mathbb{R}$ and $\mathbb{R}_{+}$, respectively. Given a set
$\mathcal{Z}$, $\mathcal{Z}^{n}$ denotes the Cartesian product of $n$ sets
$\mathcal{Z}$. For vectors $x,y\in\mathbb{R}^{n}$ and
$\mathcal{A}\subset\mathbb{R}^{n}$, $\left\langle x\ \middle|\
y\right\rangle$, $\|x\|$ and $\|x\|_{\mathcal{A}}$ denote the Euclidean inner
product, norm, weighted norm and distance to set respectively. Given $N$
vectors $x_{1},\dots,x_{N}$, possibly of different dimensions,
$\operatorname{col}\left({x_{1},\dots
x_{N}}\right)\coloneqq\left[x_{1}^{\top},\dots,x_{N}^{\top}\right]^{\top}$.
Collective vectors are denoted in bold, i.e,
$\boldsymbol{x}\coloneqq\operatorname{col}\left({x_{1},\dots,x_{N}}\right)$ as
they collect vectors from multiple agents. We use
$\mathbb{S}^{1}:=\left\\{z\in\mathbb{R}^{2}:z_{1}^{2}+z_{2}^{2}=1\right\\}$ to
denote the unit circle in $\mathbb{R}^{2}$. $\operatorname{Id}$ is the
identity operator; $I_{n}$ is the identity matrix of dimension $n$ and
$\boldsymbol{0}_{n}$ is vector column of $n$ zeros; their index is omitted
where the dimensions can be deduced from context. The unit ball of appropriate
dimensions depending on context is denoted with $\mathbb{B}$. A continuous
function $\gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is of class
$\mathcal{K}$ if it is zero at zero and strictly increasing. A continuous
function $\alpha:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is of class
$\mathcal{L}$ if is non-increasing and converges to zero as its arguments
grows unbounded. A continuous function
$\beta:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is of
class $\mathcal{KL}$ if it is of class $\mathcal{K}$ in the first argument and
of class $\mathcal{L}$ in the second argument. UGAS refers to uniform global
asymptotic stability, as defined in [13, Def. 2.2, 2.3]. We define semi-global
practical asymptotic stability (SGPAS) similarly as in [19].

######  Definition 1 (SGPAS).

The set $\mathcal{A}$ is SGPAS as
$(\varepsilon_{1},\dots,\varepsilon_{k})\rightarrow 0$ for the parametrized
hybrid system $\mathcal{H}_{\varepsilon}$, if for each given
$\Delta>\delta>0$, there exists a parameter $\varepsilon_{1}^{*}$ such that
for each $\varepsilon_{1}\in\left(0,\varepsilon_{1}^{*}\right)$ there exists
$\varepsilon_{2}^{*}\left(\varepsilon_{1}\right)>0$ such that for each
$\varepsilon_{2}\in(0,\varepsilon_{2}^{*}\left(\varepsilon_{1}\right))$
$\ldots$ there exists $\varepsilon_{k}^{*}\left(\varepsilon_{k-1}\right)>0$
such that for each
$\varepsilon_{k}\in$$\left(0,\varepsilon_{k}^{*}\left(\varepsilon_{k-1}\right)\right)$
it holds:

  1. 1.

(Semi-global stability) for each $R\geq\delta$, there exists $r>0$, such that
$\left\|\phi(l,i)\right\|_{\mathcal{A}}\leq
r\implies\left\|\phi(t,j)\right\|_{\mathcal{A}}\leq R$ for $l+i\leq t+j$ and
each solution $\phi$.

  2. 2.

(Practical attractivity) for each $R,r$ that satisfy $\Delta\geq R\geq
r\geq\delta$, there exists a period $T(r,R)\geq 0$, such that
$\left\|\phi(l,i)\right\|_{\mathcal{A}}\leq
R\implies\left\|\phi(t,j)\right\|_{\mathcal{A}}\leq r$ for all $t+j\geq
T(r,R)+l+i$ and each solution $\phi$. ∎

##  2 Singular perturbation theory for hybrid systems

We consider two different system setups, with the first case featuring a
hybrid reduced system and a continuous boundary layer system. In the second,
both the reduced system and the boundary layer system are hybrid. Despite the
different scenarios, we require similar assumptions in all configurations.
Notably, we provide the most comprehensive coverage of the first case.

###  2.1 Continuous boundary layer dynamics

We consider the following hybrid dynamical system, denoted by
$\mathcal{H}_{1}$:

$\displaystyle\dot{x}$ $\displaystyle\in\begin{bmatrix}I_{n_{1}}&0\\\
0&\tfrac{1}{\varepsilon}I_{n_{2}}\end{bmatrix}F(x),$ $\displaystyle\text{if
}x\in\mathcal{X}_{1}\times\mathcal{X}_{2},$ (1a) $\displaystyle x^{+}$
$\displaystyle\in G(x),$ $\displaystyle\text{if }x\in D_{1}\times D_{2},$ (1b)

where
$x\coloneqq\operatorname{col}\left({x_{1},x_{2}}\right)\in\mathcal{X}_{1}\times\mathcal{X}_{2}\subset\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$
are the system states, $\varepsilon>0$ is small parameter used to speed up the
$x_{2}$ dynamics, $\mathcal{X}_{1},D_{1}\subset\mathcal{X}_{1}$,
$\mathcal{X}_{2},D_{2}\subset\mathcal{X}_{2}$ are flow and jump sets for the
slow states $x_{1}$ and the fast states $x_{2}$, respectively. Other than
$\varepsilon$, the system is implicitly parametrized by parameters
$\beta,\gamma$ and $\tau$ i.e. $F=F_{\beta,\gamma,\tau}$ and
$G=G_{\beta,\gamma,\tau}$. As it is common for hybrid dynamical systems, we
postulate certain regularity assumptions that provide useful properties.

######  Assumption 1.

The hybrid dynamical system in (1) satisfies the basic regularity assumptions
for hybrid systems [3, Assum. 6.5] for all parameters
$\beta\in(0,\overline{\beta}],\gamma\in(0,\overline{\gamma}]$,
$\tau\in(0,\overline{\tau}]$. The mapping $G$ satisfies item [3, Assum. 6.5,
A3] also for $\beta=0,\gamma=0,\tau=0$. Furthermore, all of systems’s
solutions are complete. ∎

Furthermore, we define two auxiliary systems in view of that in (1), the
boundary layer system and the reduced system. The former,
$\mathcal{H}_{1}^{\rho}$, for any given constant $\rho>0$, is defined as

$\displaystyle\dot{x}$ $\displaystyle\in\begin{bmatrix}0&0\\\
0&I_{n_{2}}\end{bmatrix}F(x)\,$ $\displaystyle
x\in\left(\left(\mathcal{A}+\rho\mathbb{B}\right)\cap\mathcal{X}_{1}\right)\times\mathcal{X}_{2},$
(2)

where $\mathcal{A}\subset\mathbb{R}^{n}$ is the equilibrium set of a reduced
system, to be introduced later on. Furthermore, the system dynamics are
parametrized by a small parameter $\beta$ which is used for tuning the desired
convergence radius. In (2), the dynamics of $x_{1}$ are frozen, i.e.
$\dot{x}_{1}=0$, thus they approximate the behavior of those in (1) when
$\varepsilon>0$ is chosen very small. Since the first state is constant, it is
natural to assume that the equilibrium set, if it exists, contains all
possible $x_{1}$, i.e. the ones contained in the set
$(\mathcal{A}+\rho\mathbb{B})\cap\mathcal{X}_{1}$, and that for every $x_{1}$,
there exists a specific set of equilibrium points $x_{2}$. We characterize
this dependence with the “steady-state” mapping $H$, and assume that it
satisfies certain regularity properties [16, Assum. 2], [14, Assum. 2].

######  Assumption 2.

The set-valued mapping $H:\mathcal{X}_{1}\rightrightarrows\mathcal{X}_{2}$,

$\displaystyle H(\overline{x}_{1})\coloneqq\\{\overline{x}_{2}\mid
F(\overline{x}_{1},\overline{x}_{2})=0\\}$ (3)

is outer semicontinuous and locally bounded; for each
$\overline{x}_{1}\in\mathcal{X}_{1},H(\overline{x}_{1})$ is a non-empty subset
of $\mathcal{X}_{2}$. ∎

Now, we can define the complete equilibrium set of the system in (2), the
boundary layer manifold, as

$\displaystyle\mathcal{M}_{\rho}\coloneqq\\{(x_{1},x_{2})\mid
x_{1}\in\left(\mathcal{A}+\rho\mathbb{B}\right)\cap\mathcal{X}_{1},\,x_{2}\in
H(x_{1})\\}.$ (4)

It is possible that the set ${\mathcal{M}_{\rho}}$ contains some unbounded
states corresponding to the logic states or reference trajectories of the
boundary layer system. We denote the bounded states with
$x_{2}{{}^{\prime}}\in\mathcal{X}_{2}^{\prime}$ and the unbounded states with
$x_{2}^{\prime\prime}\in\mathcal{X}_{2}^{\prime\prime}$,
$\mathcal{X}_{2}^{\prime}\times\mathcal{X}_{2}^{\prime\prime}=\mathcal{X}_{2}$.
Furthermore, we assume that these unbounded states only affect each other
during jumps, and that the bounded states are a priori contained in a compact
set.

######  Assumption 3.

The jump mapping $G$ in (1b), and the steady-state mapping $H$ in 3 are
decomposed as follows:

$\displaystyle G(x)$
$\displaystyle=\begin{bmatrix}G_{1}(x_{1},x_{2}^{\prime})\\\
G_{2}^{\prime}(x)\end{bmatrix},$ (5) $\displaystyle H(x_{1})$
$\displaystyle=H_{1}(x_{1})\times\mathcal{X}_{2}^{\prime\prime},$ (6)

where
$G_{1}:\mathcal{X}_{1}\times\mathcal{X}_{2}^{\prime}\rightrightarrows\mathcal{X}_{1}\times\mathcal{X}_{2}^{\prime}$,
$G_{2}^{\prime}:\mathcal{X}\rightrightarrows\mathcal{X}_{2}^{\prime\prime}$,
and $H_{1}:\mathcal{X}_{1}\rightrightarrows\mathcal{X}_{2}^{\prime}$. ∎

######  Assumption 4.

The set $\mathcal{X}^{\prime}_{2}$ in Assumption 3 is compact. ∎

Furthermore, we assume that the set ${\mathcal{M}_{\rho}}$ is SGPAS for
boundary layer dynamics in Equation (2).

######  Assumption 5.

The set $\mathcal{M}_{\rho}$ in (4) is SGPAS as $\beta\rightarrow 0$ for the
dynamics in (2). Let $\Delta>\delta>0$ be given by the definition of SGPAS.
For every $\Delta>0$, the corresponding Lyapunov function is given by

$\displaystyle\underline{\alpha_{2,\rho}}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)\leq
V_{2,\rho}(x)\leq\overline{\alpha_{2,\rho}}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)$
(7a) $\displaystyle\sup_{\begin{array}[]{l}\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)\end{array}}\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}0\\\
f_{2}\end{smallmatrix}\right]\right\rangle\leq-\alpha_{2,\rho}(\left\|x\right\|_{\mathcal{M}_{\rho}})$
(7c) $\displaystyle\text{for all }x\text{ such that
}\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\alpha_{\beta}(\beta),$ (7d)
$\displaystyle\sup_{\begin{array}[]{l}\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)\end{array}}\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}0\\\
f_{2}\end{smallmatrix}\right]\right\rangle\leq\hat{\alpha}_{\beta}(\beta)$
(7f) $\displaystyle\text{for all }x\text{ such that
}\left\|x\right\|_{\mathcal{M}_{\rho}}\leq\alpha_{\beta}(\beta)$ (7g)
$\displaystyle\nabla V_{2,\rho}(x)=0\text{ for all
}{x}\in{\mathcal{M}_{\rho}},$ (7h)

where
$\underline{\alpha_{2,\rho}},\overline{\alpha_{2,\rho}},\alpha_{2,\rho},\alpha_{\beta},\hat{\alpha}_{\beta}$
are functions of class $\mathcal{K}$, where $\alpha_{2,\rho},\alpha_{\beta}$
are possibly parametrized by $\Delta$. Furthermore, for each compact set
$K\in\mathcal{X}_{1}$, there exists $M>0$, such that

$\displaystyle\sup_{x\in
K\times\mathcal{X}_{2}}\left\|V_{2,\rho}(x)\right\|+\left\|\nabla_{x_{1}}V_{2,\rho}(x)\right\|\leq
M.\qed$ (8)

######  Reamrk 1.

In Assumption 7, we allow the set $\mathcal{X}_{2}$ to be unbounded.
Nevertheless, the Lyapunov function is assumed to take bounded values, as in
(8). ∎

On the other hand, since the $x_{2}$ dynamics are much faster than those of
$x_{1}$ in (1), from the time scale of the latter, it seems that the $x_{2}$
dynamics are evolving on the manifold defined by the mapping $H$. To
characterize this behaviour, we can define the reduced system
$\mathcal{H}_{1}^{\text{r}}$ as:

$\displaystyle\dot{x}_{1}$ $\displaystyle\in F_{\text{r}}(x_{1})\,$
$\displaystyle\text{ if }x_{1}$ $\displaystyle\in\mathcal{X}_{1}$ (9a)
$\displaystyle x_{1}^{+}$ $\displaystyle\in G_{\text{r}}(x_{1})\,$
$\displaystyle\text{ if }x_{1}$ $\displaystyle\in D_{1},$ (9b)

where
$F_{\text{r}}(x_{1})\coloneqq\overline{\operatorname{co}}\\{v_{1}\mid(v_{1},v_{2})\in
F(x_{1},x_{2}),x_{2}\in H(x_{1})\\}$,
$G_{\text{r}}(x_{1})\coloneqq\\{v_{1}\mid(v_{1},v_{2})\in
G(x_{1},x_{2}),x_{2}\in H(x_{1})\\}$. Furthermore, the system dynamics are
parametrized by the parameter $\gamma$, which is used for the tuning of the
convergence radius to the attractor set, and the parameter $\tau$ adjusts the
minimum time interval between consecutive jumps, for _almost all_ jumps of the
systems in (1) (consequently also the reduced system in (9)), as formalized
next:

######  Definition 2 ($\tau$-regular jump).

A jump $j$ in a solution trajectory $\phi$ is a $\tau$-_regular jump_ if it
occurs after an interval of flowing greater or equal than $\tau$, i.e.
$\tau_{j}\coloneqq\sup\\{\left|t-t^{\prime}\right|:(t,j-1),\left(t^{\prime},j-1\right)\in\operatorname{dom}\phi\\}\geq\tau$.
Otherwise, the jump $j$ is called $\tau$-irregular. ∎

######  Assumption 6.

Let $\phi$ be any solution of the system in (1) with
$\left\|\phi(0,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\Delta$. Then,
there exists a finite number of jumps $N^{*}$ and finite time interval
$T^{*}$, such that $\phi$ has at most $N^{*}$
$\underline{\sigma}(\tau)$-irregular jumps, and they all occur before $t\leq
T^{*}$, where $\underline{\sigma}$ is a function of class $\mathcal{L}$, and
$\tau$ is the parameter of the system. ∎

Differently from [16], where the reduced mapping is defined as
$G_{\text{r}}(x_{1})\coloneqq\\{v_{1}\mid(v_{1},v_{2})\in
G(x_{1},x_{2}),x_{2}\in\mathcal{X}_{2}\\}$, the mapping in (9b) only includes
the jumps from the stead-state “pairs” $(x_{1},H(x_{1}))$ that belong to the
manifold. Thus, our next assumption is weaker than [16, Assum. 4], as it
requires that the jumps stabilize the set $\mathcal{A}$ via a much more
restricted set of dynamics. This is due to the fact that the reduced mapping
$G_{\text{r}}$ does not contain all possible jumps from the set $D_{1}$, but
only those from the boundary layer manifold ${\mathcal{M}_{\rho}}$.

######  Assumption 7.

The set $\mathcal{A}$ is SGPAS as $\gamma\rightarrow 0$ for the reduced system
in (9). Let $\Delta>\delta>0$ be given by the definition of SGPAS. For every
$\Delta>0$, the corresponding Lyapunov function is given by

$\displaystyle\underline{\alpha_{1}}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)\leq
V_{1}(x_{1})\leq\overline{\alpha_{1}}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
(10a) $\displaystyle\sup_{f_{1r}\in F_{\text{r}}(x_{1})}\left\langle\nabla
V_{1}(x_{1})\ \middle|\
f_{1r}\right\rangle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
(10b) $\displaystyle\sup_{g_{1r}\in
G_{\text{r}}(x_{1})}V_{1}(g_{r1})-V_{1}(x_{1})\leq-\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
(10c) $\displaystyle\text{for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma),$ (10d) where
$\underline{\alpha_{1}},\overline{\alpha_{1}},\alpha_{1},\alpha_{\gamma},\hat{\alpha}_{\gamma}$
are functions of class $\mathcal{K}$, where $\alpha_{1},\alpha_{\gamma}$ are
possibly parametrized by $\Delta$, and $\hat{\sigma}_{\tau}$ is a function of
class $\mathcal{L}$.

∎

We claim that our original system in (1) renders the set
$\mathcal{A}\times\mathcal{X}_{2}$ practically attractive, if for almost all
intervals of flow we allow the state of the system to converge to the
neighborhood of the ${\mathcal{M}_{\rho}}$ manifold. The intuition is that in
the neighborhood of the manifold, “the jumps of the _reduced system_ ” also
contribute to the stabilization.

######  Theorem 1.

Let Assumptions 1—7 hold. Then the set $\mathcal{A}\times\mathcal{X}_{2}$ is
practically attractive as
$(\gamma,\tfrac{1}{\tau},\varepsilon,\beta)\rightarrow 0$ for the hybrid
system in (1). ∎

See Appendix A. $\blacksquare$

######  Example 1.

Consider the hybrid dynamical system

$\displaystyle\left\\{\begin{array}[]{rl}\dot{u}&=\gamma\max\\{0,1-\tfrac{\left\|u\right\|}{R}\\}\\\
\dot{v}&=\tfrac{1}{\tau}\\\
\dot{x}&=-\tfrac{1}{\varepsilon}(x-u)\end{array}\right.$ (11d)
$\displaystyle\text{if }(u,v,x)\in{[0,R]\times[0,1]\times[0,R]};$ (11e)
$\displaystyle\left\\{\begin{array}[]{rl}{u}^{+}&=\tfrac{x}{2}\\\
{v}^{+}&=0\\\ {x}^{+}&=R\end{array}\right.$ (11i) $\displaystyle\text{if
}(u,v,x)\in{[0,R]\times\\{1\\}\times[0,R]},$ (11j)

where $\gamma,\tau,\varepsilon$ are tuning parameters, and $R>0$ is the
maximal trajectory radius. We show that the set
$\\{0\\}\times[0,1]\times[0,R]$ is practically attractive. First, we see that
the boundary layer system reads as

$\displaystyle\left\\{\begin{array}[]{rl}\dot{u}&=0\\\ \dot{v}&=0\\\
\dot{x}&=-(x-u)\end{array}\right.$ (15) $\displaystyle\text{if
}(u,v,x)\in{[0,R]\times[0,1]\times[0,R]},$ (16)

while the reduced system is given by

$\displaystyle\left\\{\begin{array}[]{rl}\dot{u}&=\gamma\max\\{0,1-\tfrac{\left\|u\right\|}{R}\\}\\\
\dot{v}&=\tfrac{1}{\tau}\\\ \end{array}\right.$ (17c) $\displaystyle\text{if
}(u,v)\in{[0,R]\times[0,1]};$ (17d)
$\displaystyle\left\\{\begin{array}[]{rl}{u}^{+}&=\tfrac{u}{2}\\\
{v}^{+}&=0\\\ \end{array}\right.$ (17g) $\displaystyle\text{if
}(u,v)\in{[0,R]\times\\{1\\}}.$ (17h)

Assumptions 1—6 are satisfied. Regarding Assumption 7, let the Lyapunov
function of the reduced system be $V_{1}(u,v)=(2-v)u^{2}$. It follows that

$\displaystyle\dot{V}_{1}(u,v)\leq-\tfrac{1}{\tau}u^{2}+4\gamma aR,$
$\displaystyle V_{1}(u^{+},v^{+})-V_{1}(u,v)\leq-\tfrac{1}{2}u^{2}.$ (18)

Since the reduced system satisfies Assumption 7, in view of Theorem 1,
practical attractivity is ensured. Unlike previous works [22], [16], and [21],
our reduced jump mapping includes jumps only from the boundary layer, which
allows us to establish stability results using jumps. In the aforementioned
works, the reduced system jump mapping includes all possible jumps [16, Equ.
13], [21, Equ. 17], [22, Equ. 13], and and for our example, it is given by
$u^{+}\in[-\tfrac{R}{2},\tfrac{R}{2}]$. Thus, the assumption on the stability
for reduced system dynamics [16, Assum. 4], [21, Thm. 2], [22, Thm. 2] does
not hold. ∎

We note that Theorem 1 gives us no guarantee on the stability of the state
$x_{1}$, due to the fact that jumps can move the state arbitrarily far away
from any set in $\mathcal{X}_{1}$ (also seen in Example 1 for
$u(0,0)=0,v(0,0)=0.99,x(0,0)=R$). Under an additional assumption, it is
possible to bound both the states $x_{1}$ and $x_{2}$ to a neighborhood of the
set ${\mathcal{M}_{\mathcal{A}}}\coloneqq\\{(x_{1},x_{2})\mid
x_{1}\in\mathcal{A},x_{2}\in H(x_{1})\\}$.

######  Assumption 8.

The jump mapping $G$ in (1b) is such that
$G({\mathcal{M}_{\mathcal{A}}})\subset{\mathcal{M}_{\mathcal{A}}}$. ∎

Assumption 8 is sufficient to guarantee that for any neighborhood of the
equilibrium set ${\mathcal{M}_{\mathcal{A}}}+\overline{r}\mathbb{B}$, there
exists a neighborhood ${\mathcal{M}_{\mathcal{A}}}+\underline{r}\mathbb{B}$,
such that jumps from the latter do not exit the former, i.e.
$G({\mathcal{M}_{\mathcal{A}}}+\underline{r}\mathbb{B})\subset{\mathcal{M}_{\mathcal{A}}}+\overline{r}\mathbb{B}$.
Lastly, we do not need to assume the compactness of the set
$\mathcal{X}_{2}^{\prime}$, as the distance from the set
${\mathcal{M}_{\mathcal{A}}}$ also bounds the values of the $x_{2}^{\prime}$
state.

######  Theorem 2.

Let Assumptions 1—3, 5—8 hold. Then the set ${\mathcal{M}_{\mathcal{A}}}$ is
SGPAS as $(\gamma,\tfrac{1}{\tau},\varepsilon,\beta)\rightarrow 0$ for the
hybrid system in (1). ∎

See Appendix B. $\blacksquare$

######  Example 2.

We consider a hybrid dynamical system similar to one in (11):

$\displaystyle\left\\{\begin{array}[]{rl}\dot{u}&=\gamma\\\
\dot{v}&=\tfrac{1}{\tau}\\\
\dot{x}&=-\tfrac{1}{\varepsilon}(x-u)\end{array}\right.$ (19d)
$\displaystyle\text{if }(u,v,x)\in{\mathbb{R}\times[0,1]\times\mathbb{R}};$
(19e) $\displaystyle\left\\{\begin{array}[]{rl}{u}^{+}&=\tfrac{x}{2}\\\
{v}^{+}&=0\\\ {x}^{+}&=2x\end{array}\right.$ (19i) $\displaystyle\text{if
}(u,v,x)\in{\mathbb{R}\times\\{1\\}\times\mathbb{R}},$ (19j)

where $\gamma,\tau,\varepsilon$ are tuning parameters. Differently from (11),
the jump mapping is such that Assumption 8 is satisfied. Furthermore, as
Theorem 2 does not require compactness of the set $\mathcal{X}_{2}^{\prime}$
in Assumption 4, the flow and jump sets are both unbounded. The boundary layer
system has the same dynamics as the system in (16), apart for the flow set
which now reads as $\mathbb{R}\times[0,1]\times\mathbb{R}$. The reduced system
is given by

$\displaystyle\begin{array}[]{l}\left\\{\begin{array}[]{rl}\dot{u}&=\gamma\\\
\dot{v}&=\tfrac{1}{\tau}\\\ \end{array}\right.\\\ \text{if
}(u,v)\in{\mathbb{R}\times[0,1]};\end{array}$ (20e)
$\displaystyle\begin{array}[]{l}\left\\{\begin{array}[]{rl}{u}^{+}&=\tfrac{u}{2}\\\
{v}^{+}&=0\\\ \end{array}\right.\\\ \text{if
}(u,v)\in{\mathbb{R}\times\\{1\\}}.\end{array}$ (20j)

Similarly to the previous example, all the Assumptions hold, thus due to
Theorem 2, the set $\\{0\\}\times[0,1]\times\\{0\\}$ is SGPAS as
$(\gamma,\tfrac{1}{\tau},\varepsilon,\beta)\rightarrow 0$ for the dynamics in
(20e). Differently from [22, Thm. 2], [16, Thm. 1], and [21, Thm. 2, Cor. 2 ]
where the _fast states_ are only a priori bounded to a compact set, here we
can prove their convergence to the equilibrium set. ∎

###  2.2 Hybrid boundary layer dynamics

Theorems 1 and 2 assume a lower limit on the time between _all consecutive
jumps_ that occur in the system in (1). However, under certain conditions, it
is possible to make a distinction between consecutive jumps of $x_{1}$, and
the consecutive jumps of $x_{2}$. This is useful when the convergence of the
boundary layer system is in fact driven by jumps in $x_{2}$, and imposing a
high lower limit on the period between consecutive jumps slows down
convergence. Consider the following hybrid dynamical system, denoted with
$\mathcal{H}_{2}$:

$\displaystyle\dot{x}$ $\displaystyle\in\begin{bmatrix}I_{n_{1}}&0\\\
0&\tfrac{1}{\varepsilon}I_{n_{2}}\end{bmatrix}F(x),\text{ if
}x\in\mathcal{X}_{1}\times\mathcal{X}_{2}$ (21a) $\displaystyle x^{+}$
$\displaystyle\in\left\\{\begin{array}[]{l}{\left[\begin{array}[]{l}x_{1}\\\
G_{2}(x)\end{array}\right],\text{ if }x\in\mathcal{X}_{1}\times D_{2}}\\\
{\left[\begin{array}[]{c}G_{1}\left(x\right)\\\ x_{2}\end{array}\right],\text{
if }x\in D_{1}\times\mathcal{X}_{2}}\\\ {\left[\begin{array}[]{l}x_{1}\\\
G_{2}(x)\end{array}\right]\cup\left[\begin{array}[]{l}G_{1}\left(x\right)\\\
x_{2}\end{array}\right],\text{ if }x\in D_{1}\times D_{2}}.\end{array}\right.$
(21m)

In this formulation, the distinction between the jumps of states $x_{1}$ and
$x_{2}$ are highlighted, because during the jumps of $x_{1}$, $x_{2}$ stays
constant, and vice versa. Furthermore, we define the boundary layer system,
$\mathcal{H}_{2}^{\rho}$, as

$\displaystyle\dot{x}$ $\displaystyle\in\begin{bmatrix}0&0\\\
0&I_{n_{2}}\end{bmatrix}F(x)\,$ $\displaystyle\text{if
}x\in\mathcal{X}_{1}\times\mathcal{X}_{2},$ (22a) $\displaystyle x^{+}$
$\displaystyle\in\begin{bmatrix}x_{1}\\\ G_{2}(x)\end{bmatrix}\,$
$\displaystyle\text{if }x\in\mathcal{X}_{1}\times D_{2},$ (22b)

and the reduced system, $\mathcal{H}_{2}^{r}$, as

$\displaystyle\dot{x}_{1}$ $\displaystyle\in F_{\text{r}}(x_{1})\,$
$\displaystyle\text{ if }x_{1}$ $\displaystyle\in\mathcal{X}_{1},$ (23a)
$\displaystyle x_{1}^{+}$ $\displaystyle\in G_{\text{r}}(x_{1})\,$
$\displaystyle\text{ if }x_{1}$ $\displaystyle\in D_{1},$ (23b)

where
$F_{\text{r}}(x_{1})\coloneqq\overline{\operatorname{co}}\\{v_{1}\mid(v_{1},v_{2})\in
F(x_{1},x_{2}),x_{2}\in H(x_{1})\\}$,
$G_{\text{r}}(x_{1})\coloneqq\\{v_{1}\mid v_{1}\in
G_{1}(x_{1},x_{2}),\,x_{2}\in H(x_{1})\\}$. Differently from the boundary
layer system in (2), jumps are also included in this formulation, while the
formulation of the reduced system is the same. Next, we pose analogous
technical assumptions as for the system in (1) and in turn provide results
analogous to Theorem 2.

######  Assumption 9.

The hybrid dynamical system in (21) satisfies the same conditions as in
Assumption 1. ∎

######  Assumption 10.

The jump mapping $G$ in (21m), and the steady-state mapping $H$ in 3 are
decomposed as in Equations (5) and (6). ∎

######  Assumption 11.

The set $\mathcal{M}_{\rho}$ is SGPAS as $\beta\rightarrow 0$ for the dynamics
in (22). Let $\Delta>\delta>0$ be given by the definition of SGPAS. The
corresponding Lyapunov function is given by (7), with the additional equation

$\displaystyle\sup_{g_{1}=x_{1},g_{2}\in
G_{2}(x)}V_{2,\rho}(g)-V_{2,\rho}(x)\leq 0,$

and for each compact set $K\in\mathcal{X}_{1}\times\mathcal{X}_{2}^{\prime}$,
there exists $M>0$, such that

$\displaystyle\sup_{x\in
K\times\mathcal{X}_{2}^{\prime\prime}}\left\|V_{2,\rho}(x)\right\|+\left\|\nabla_{x_{1}}V_{2,\rho}(x)\right\|\leq
M.\qed$ (24)

######  Assumption 12.

The set $\mathcal{A}$ is SGPAS as $\gamma\rightarrow 0$ for the reduced system
in (23). Let $\Delta>\delta>0$ be given by the definition of SGPAS. For every
$\Delta>0$, the corresponding Lyapunov function is given by Equation (10) with
the redefined mappings in (23). ∎

######  Definition 3 ($\tau$-regular jump in $x_{1}$).

A jump $j$ in a solution trajectory $\phi$ of the system in (21) is a
$\tau$-_regular in $x_{1}$ jump_ if it occurs after an interval of flowing in
the $x_{1}$ state greater or equal than $\tau$, i.e.
$\tau_{j}^{1}\coloneqq\min\\{\left|t-t^{\prime}\right|:\phi_{1}(t,j+1)\in
G_{1}(\phi(t,j));\phi_{2}(t,j+1)=\phi_{2}(t,j);(t,j),(t,j+1)\in\operatorname{dom}\phi;\phi_{1}(t^{\prime},j^{\prime}+1)\in
G_{1}(\phi(t^{\prime},j^{\prime}));\phi_{2}(t^{\prime},j^{\prime}+1)=\phi_{2}(t^{\prime},j^{\prime});(t^{\prime},j^{\prime}),(t^{\prime},j^{\prime}+1)\in\operatorname{dom}\phi;j^{\prime}>j\\}\geq\tau$.
Otherwise, if $\tau_{j}^{1}$ exists and $\tau_{j}^{1}<\tau$, jump $j$ is
called $\tau$-irregular in $x_{1}$. ∎

######  Assumption 13.

Let $\phi$ be any solution of the system in (21) with
$\left\|\phi(0,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\Delta$. Then,
there exists a finite number of jumps $N^{*}$ and finite time interval
$T^{*}$, such that $\phi$ has at most $N^{*}$
$\underline{\sigma}(\tau)$-irregular jumps in $x_{1}$, and they all occur
before $t\leq T^{*}$, where $\underline{\sigma}$ is a function of class
$\mathcal{L}$. ∎

######  Corollary 1.

Let Assumptions 2, 8—13 hold. Then the set ${\mathcal{M}_{\mathcal{A}}}$ is
SGPAS as $(\gamma,\tfrac{1}{\tau},\varepsilon,\beta)\rightarrow 0$ for the
hybrid system in (21). ∎

The proof is analogous to the proofs of Theorems 1 and 2. An equivalent for
Lemma 2 can be constructed with jumps of the $x_{2}$ state. The rest of the
proof is essentially the same. $\blacksquare$

##  3 Illustrative example

In [17], the issue of connectivity control was approached as a Nash
equilibrium problem. In numerous practical situations, multi-agent systems are
constructed with the goal of maintaining specific connectivity as a secondary
objective in addition to their primary objective. In the subsequent
discussion, we consider a comparable problem in which each agent is
responsible for detecting an unknown signal source while also preserving a
certain level of connectivity. Unlike [17], both the robots and the
controllers have hybrid dynamics in our example.

Consider a multi-agent system consisting of unicycle vehicles, indexed by
$i\in\mathcal{I}\coloneqq\\{1,\dots N\\}$. Each agent is tasked with locating
a source of a unique unknown signal. The strength of all signals abides by the
inverse-square law, i.e. proportional to $1/r^{2}$. Therefore, the inverse of
the signal strength can be used as a cost function. Additionally, the agents
must not drift apart from each other too much, as they should provide quick
assistance to each other in case of critical failure. This is enforced by
incorporating the signal strength of the fellows agents in the cost functions.
Thus, we design the cost functions as follows:

$\displaystyle\forall
i\in\mathcal{I}:h_{i}(u)=\|u_{i}-u_{i}^{\textup{s}}\|^{2}+c\sum_{j\in\mathcal{I}_{-i}}\|u_{i}-u_{j}\|^{2}.$
(25)

where $\mathcal{I}_{-i}\coloneqq\mathcal{I}\setminus\\{i\\}$, $c,b>0$ and
$u_{i}^{\textup{s}}$ represents the position of the source assigned to agent
$i$. Goal of each agent is to minimize their cost function, and the solution
to this problem is a Nash equilibrium.

###  3.1 Unicycle dynamics

As the unicycles are dynamical systems, a reference tracking controller is
necessary in order to move them to the desired positions. In our example, let
each agent implement a hybrid feedback controller similar to one in [10] for
trajectory tracking:

$\displaystyle{\chi}_{i}^{\textup{u}}=\operatorname{col}\left({{x}_{i},{y}_{i},\theta_{i}^{e},{\tau}_{i},\theta_{i},\hat{v}_{i},\hat{\omega}_{i}}\right),$
$\displaystyle\dot{\chi}_{i}^{\textup{u}}=F_{i}^{\textup{u}}(\chi_{i}^{\textup{u}})\coloneqq$
$\displaystyle\operatorname{col}\left({\hat{v}_{i}\cos\left(\theta_{i}\right),\hat{v}_{i}\sin\left(\theta_{i}\right),\omega_{\text{r}}-\hat{\omega}_{i},\tfrac{1}{\sigma_{i}},\hat{\omega}_{i},0,0}\right)$
$\displaystyle\text{ if }\chi_{i}^{\textup{u}}\in
C_{i}^{\textup{u}}\coloneqq\mathbb{R}^{3}\times[0,1]\times\mathbb{R}^{3},$
(26a)
$\displaystyle{\chi_{i}^{\textup{u}}}^{+}=G_{i}^{\textup{u}}(\chi_{i}^{\textup{u}})\coloneqq\operatorname{col}\left({x_{i},y_{i},\theta_{i}^{e},0,\theta_{i},v_{i},\omega_{i}}\right)$
$\displaystyle\text{ if }\chi_{i}^{\textup{u}}\in
D_{i}^{\textup{u}}\coloneqq\mathbb{R}^{3}\times\\{1\\}\times\mathbb{R}^{3},$
(26b)

where
$v_{i}=c_{1}(x_{i}^{e}-c_{3}\omega_{i}y_{i}^{e})-c_{3}c_{2,i}(\omega_{\textup{r}}-\omega_{i})y_{i}^{e}+c_{3}\omega_{i}^{2}x_{i}^{e}$,
$x_{i}^{e}\coloneqq\cos(\theta_{i})(u_{i}^{1}-x_{i})+\sin(\theta_{i})(u_{i}^{2}-y_{i})$,
$y_{i}^{e}\coloneqq-\sin(\theta_{i})(u_{i}^{1}-x_{i})+\cos(\theta_{i})(u_{i}^{2}-y_{i})$,
$\theta_{i}^{e}=\theta_{\textup{r}}-\theta_{i}$,
$\omega_{i}\coloneqq\omega_{\textup{r}}+c_{2,i}\theta_{i}^{e}$,
$\dot{\theta}_{\textup{r}}=\omega_{\textup{r}}=const.$,
$c_{1},c_{2,i},c_{3}>0$ are tuning parameters, $\sigma_{i}$ is the sampling
period parameter, $u_{i}^{1}$ and $u_{i}^{2}$ are the reference positions.
Differently from [10], the jumps are triggered by a timer, and the reference
trajectory is that of a unicycle with a fixed position $(u_{i}^{1},u_{i}^{2})$
and constant rotational velocity $\omega_{\textup{r}}$. Similarly to [10,
Lemma 4., Thm. 5], it is possible to prove that the dynamics in (26) render
the set
$\\{\operatorname{col}\left({u_{i}^{1},u_{i}^{2},0}\right)\\}\times\tilde{\mathcal{T}}_{i}\times\mathbb{R}^{3}$
SGPAS as $\sigma_{i}\rightarrow 0$.

######  Theorem 3.

For $c_{2,i}=\sigma_{i}$, $c_{3}=\tfrac{1}{3{{\omega}}_{\textup{r}}}$,
$c_{1}=\tfrac{1}{2c_{3}}$, the dynamics in (26) render the set
$\\{\operatorname{col}\left({u_{i}^{1},u_{i}^{2},0}\right)\\}\times\tilde{\mathcal{T}}_{i}\times\mathbb{R}^{3}$
SGPAS as $\sigma_{i}\rightarrow 0$. ∎

See Appendix C. $\blacksquare$From the proof of Theorem 3, it follows that
system in (26), for all $i\in\mathcal{I}$, satisfies Assumptions 11.

###  3.2 Nash equilibrium seeking reference controller

To steer the reference positions towards the Nash equilibrium, we implement
the following asynchronous zeroth-order controller:

$\displaystyle\chi^{\textup{c}}=\operatorname{col}\left({\boldsymbol{u},\boldsymbol{\xi},\boldsymbol{\mu},\boldsymbol{t}}\right),$
$\displaystyle\dot{\chi}^{\textup{c}}=F^{\textup{c}}(\chi^{\textup{c}})\coloneqq\operatorname{col}\left({\boldsymbol{0},\boldsymbol{0},\boldsymbol{0},\boldsymbol{\tau}^{-1}}\right)$
$\displaystyle\text{ if }\chi^{\textup{c}}\in
C^{\textup{c}}\coloneqq\mathbb{R}^{m}\times\mathcal{N}\times\mathbb{S}^{m}\times[0,1]^{N},$
(27a)
$\displaystyle{\chi^{\textup{c}}}^{+}=G^{\textup{c}}(\chi^{\textup{c}}),\text{i.e}.$
$\displaystyle\left\\{\begin{array}[]{ll}\boldsymbol{u}^{+}=\boldsymbol{u}-\alpha\beta
S_{x}(\boldsymbol{\tau})\boldsymbol{\xi}\\\
\boldsymbol{\xi}^{+}=\boldsymbol{\xi}+\alpha
S_{x}(\boldsymbol{t})\left(2A^{-1}J(\boldsymbol{x}+A\mathbb{D}\boldsymbol{\mu})\mathbb{D}\boldsymbol{\mu}-\boldsymbol{\xi}\right)\\\
\boldsymbol{\mu}^{+}=(I-S_{\mu}(\boldsymbol{t}))\boldsymbol{\mu}+S_{\mu}(\boldsymbol{t}))\mathcal{R}\boldsymbol{\mu}\\\
\boldsymbol{t}^{+}=(I-S_{\tau}(\boldsymbol{t}))\boldsymbol{t}\end{array}\right.$
(27f) $\displaystyle\text{ if }\chi^{\textup{c}}\in
D^{\textup{c}}\coloneqq\mathbb{R}^{m}\times\mathcal{N}\times\mathbb{S}^{m}\times\mathcal{T}_{\textup{R}},$
(27g)

where
$\boldsymbol{u}=\operatorname{col}\left({\left(u_{i}^{1},u_{i}^{2}\right)_{i\in\mathcal{I}}}\right)$
is used as the reference position for the systems in (26)$,\boldsymbol{\xi}$
is the collective filter state bound in a compact set
$\mathcal{N}\subset\mathbb{R}^{N}$ chosen large enough to encompass all
possible values of the state for all practical applications,
$\boldsymbol{\mu}\in\mathbb{S}^{2N}$ are oscillator states, $\boldsymbol{t}$
are the timer states that control the sampling of each individual robot,
$\boldsymbol{\tau}^{-1}=\tau_{0}\operatorname{col}\left({\left(\tau_{i}^{-1}\right)_{i\in\mathcal{I}}}\right)$
are the sampling periods that satisfy [6, Assum. 9], $\boldsymbol{x}$ are the
positions of the unicycles, $\alpha,\beta>0$ are small time-scale separation
parameters,
$\mathcal{R}\coloneqq\operatorname{Diag}\left({(\mathcal{R}_{i})_{i\in\mathcal{I}}}\right)$,
$\mathcal{R}_{i}\coloneqq\operatorname{Diag}\left({\left[\begin{smallmatrix}\cos(\omega_{i}^{j})&-\sin(\omega_{i}^{j})\\\
\sin(\omega_{i}^{j})&\cos(\omega_{i}^{j})\end{smallmatrix}\right]_{j\leq
m_{i}}}\right)$, $\omega_{i}^{j}>0$ for all $i$ and $j$ are rotational
frequencies and they satisfy [6, Assum. 8], $\mathbb{D}\in\mathbb{R}^{2N\times
4N}$ is a matrix that selects every odd row from the vector of size $2N$,
$a_{i}>0$ are small perturbation amplitude parameters,
$A\coloneqq\operatorname{diag}\left({(a_{i})_{i\leq m}}\right)$,
$J(\boldsymbol{x})=\operatorname{Diag}\left({(J_{i}(x_{i},\boldsymbol{x}_{-i})I_{m_{i}})_{i\in\mathcal{I}}}\right)$,
$\mathcal{T}\subset[0,1]^{N}$ is a closed invariant set in which all of the
timers evolve and it excludes the initial conditions and their neighborhood
for which we have concurrent sampling,
$\mathcal{T}_{\textup{R}}\coloneqq\left(\cup_{i\in\mathcal{I}}[0,1]^{i-1}\times\\{1\\}\times[0,1]^{N-i}\right)\cap\mathcal{T}$
is the set of timer intervals where one agent has triggered its sampling,
$S_{x}:\mathcal{T}\rightarrow\mathbb{R}^{m\times m}$ and
$S_{\tau}:\mathcal{T}\rightarrow\mathbb{R}^{N\times N}$ are continuous
functions that output diagonal matrices with ones on the positions that
correspond to states and timers of agents with $t_{i}=1$, respectively, while
other elements are equal to zero, when evaluating at
$\boldsymbol{t}\in\mathcal{T}_{\textup{R}}$.

###  3.3 The full system

We define the collective state
$\chi\coloneqq\operatorname{col}\left({\chi^{\textup{c}},(\chi^{\textup{u}}_{i})_{i\in\mathcal{I}}}\right)$,
collective flow map
$F(\chi)\coloneqq\operatorname{col}\left({F^{\textup{c}}(\chi^{\textup{c}}),\tfrac{1}{\varepsilon}(F^{\textup{u}}_{i}(\chi^{\textup{u}}_{i}))_{i\in\mathcal{I}}}\right)$,
collective flow set
$C\coloneqq{C^{\textup{c}}\times(C^{\textup{u}}_{i})_{i\in\mathcal{I}}}$,
collective jump map
$G(\chi)\coloneqq\operatorname{col}\left({G^{\textup{c}}(\chi^{\textup{c}}),(G^{\textup{u}}_{i}(\chi^{\textup{u}}_{i}))_{i\in\mathcal{I}}}\right)$,
collective flow set
$D\coloneqq(D^{\textup{c}}\times(C^{\textup{u}}_{i})_{i\in\mathcal{I}})\cup(C^{\textup{c}}\times(D^{\textup{u}}_{i})_{i\in\mathcal{I}})$,
and the equilibrium set
$\mathcal{A}_{\chi}\coloneqq\\{\boldsymbol{u}^{*}\\}\times\mathcal{N}\times\mathbb{S}^{N}\times\mathcal{T}\times\\{\operatorname{col}\left({\boldsymbol{u}^{*},\boldsymbol{0}}\right)\\}\times[0,1]^{N}\times\mathbb{R}^{3N}$.
We see that the steady state mapping is given by
$H(\chi^{\textup{c}})=\operatorname{col}\left({\boldsymbol{u},\boldsymbol{0}}\right)\times[0,1]^{N}\times\mathbb{R}^{3N}$.
Hence, the restricted system is equivalent to the one in [6, Equ. 22]. To show
that Assumption 12 is satisfied, we note that [6, Thm. 1] and [6, Equ. E.10]
assure that the fully discrete-time zeroth-order variant of the algorithm in
[6, Equ. 22], has a Lyapunov function of the form

$\displaystyle\underline{\alpha}_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)\leq
V_{\textup{a}}(\boldsymbol{z})\leq\overline{\alpha}_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)$
$\displaystyle
V_{\textup{a}}(\boldsymbol{z}^{+})-V_{\textup{a}}(\boldsymbol{z})\leq-\hat{\alpha}_{\alpha}\left({\alpha}\right)\alpha_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)$
$\displaystyle\text{ for
}\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\geq\max\\{\alpha_{\beta}(\beta),{\alpha}_{\alpha}(\alpha)\\},$

where $\boldsymbol{z}\coloneqq\boldsymbol{u}+\boldsymbol{\eta}$, and
$\boldsymbol{\eta}$ is a state of a bounded discrete system [6, Equ. 7]. For
the sampled variant we have as our restricted system, we propose the following
Lyapunov function

$\displaystyle V_{1}(\boldsymbol{z})\coloneqq$
$\displaystyle\tfrac{1}{2}\left\langle\boldsymbol{1}-\boldsymbol{t}\ \middle|\
\boldsymbol{1}\right\rangle\,\hat{\alpha}_{\alpha}\left({\alpha}\right)\alpha_{\textup{a}}\left(\overline{\alpha}_{\textup{a}}^{-1}\left(\underline{\alpha}_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)\right)\right)$
$\displaystyle+V_{\textup{a}}(\boldsymbol{z}).$ (28)

Hence, it holds

$\displaystyle\underline{\alpha}_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)\leq
V_{1}(\boldsymbol{z})\leq(\overline{\alpha}_{\textup{a}}+\alpha_{\textup{a}}\circ\overline{\alpha}_{\textup{a}}^{-1}\circ\underline{\alpha}_{\textup{a}})\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)$
$\displaystyle\dot{V}_{1}(z)\leq-\tfrac{1}{2\tau_{0}}\sum_{i\in\mathcal{I}}\tau_{i}^{-1}\hat{\alpha}_{\alpha}\left({\alpha}\right)\alpha_{\textup{a}}\left(\overline{\alpha}_{\textup{a}}^{-1}\left(\underline{\alpha}_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)\right)\right)$
$\displaystyle
V_{1}(\boldsymbol{z}^{+})-V_{1}(\boldsymbol{z})\leq-\tfrac{1}{2}\hat{\alpha}_{\alpha}\left({\alpha}\right)\alpha_{\textup{a}}\left(\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\right)$
$\displaystyle\text{ for
}\left\|\boldsymbol{z}-\boldsymbol{u}^{*}\right\|\geq\max\\{\alpha_{\beta}(\beta),{\alpha}_{\alpha}(\alpha)\\},$

which satisfies Assumption 12. Furthermore, it is easy to show that
Assumptions 2, 8, 9, 10 hold as well. Since $\tau_{0}$ can be considered a
tuning parameter for jump periods in the timers states $\boldsymbol{t}$ in
(27), we can guarantee satisfaction of Assumption 13. Hence, we satisfy all
the Assumptions of the Corollary 1, and for small enough parameters, the
combined dynamics render the set $\mathcal{A}_{\chi}$ SGPAS as
$(\alpha,\beta,\max{\tau_{i}^{-1}},\varepsilon,\max{\sigma_{i}})\rightarrow
0$.
For our numerical simulations, we choose the parameters: $u^{s}_{1}=(-4,-8)$,
$u^{s}_{2}=(-12,-3)$, $u^{s}_{3}=(1,7)$, $u^{s}_{4}=(16,8)$,
$(\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4})=\operatorname{col}\left({2,3,4,2}\right)\times
10^{-3}$, $c_{1}=\tfrac{1}{3}$, $c_{3}=1.5$, $\alpha=0.05$, $\beta=0.003$,
$c_{2,i}=\sigma_{i}$, $a_{i}=0.1$ for all $i$,
$\boldsymbol{t}(0,0)=(0,0.002,0.004,0.006)$, the perturbation frequencies
$\omega_{i}^{j}$ were chosen as different natural numbers with added random
numbers of maximal amplitude of 0.5, and the sampling of the Nash equilibrium
seeking controller in (27) is five time slower than the sampling of the
unicycle controller in (26a), i.e.
$\boldsymbol{\tau}=\operatorname{col}\left({1,1.5,2,1}\right)\times 10^{-2}$.
The numerical results are illustrated on Figures 1 and 2. We note that the
trajectories converge to the neighborhood of the Nash equilibrium.

Figure 1: State trajectories in the $x-y$ plane. Circle symbols represent
locations of the sources, while the $\times$ symbols represent locations of
the NE. Figure 2: Time response of the unicycle position coordinates. The
dashed lines correspond to the corresponding state of the Nash equilibrium.

##  4 Conclusion

The application of singular perturbation theory can be extended to systems
where the restricted system evolves on the boundary layer manifold through
both flows and jumps. Moreover, by introducing some mild tehnical assumptions,
one can show convergence of the fast state components towards a restricted
attractor set that does not encompass the complete space of fast variables.
With this theoretical extension, we can examine control systems that employ
hybrid plants, along with controllers that are “jump-driven” such as sampled
controllers.

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##  Appendix A Proof of Theorem 1

Let $\Delta>\delta>0$ be given. We denote with $\rho$ the maximum distance to
the equilibrium set $\mathcal{A}\times\mathcal{X}_{2}$ for trajectories
starting in $(\mathcal{A}+\Delta\mathbb{B})\times\mathcal{X}_{2}$, which we
characterize later on. Next, due to the fact that both
$\mathcal{A}\times\mathcal{X}_{2}$ and ${\mathcal{M}_{\rho}}$ are unbounded in
the dimensions corresponding to the same states, it follows that for any
$\rho>0$, there exists a $P>0$ such that
$\left\|x\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\rho$ implies that
$\left\|x\right\|_{{\mathcal{M}_{\rho}}}\leq P$. We consider the system in (1)
with restricted flow and jump sets:

$\displaystyle C$
$\displaystyle\coloneqq\left(\left(\mathcal{A}+\rho\mathbb{B}\right)\cap\mathcal{X}_{1}\right)\times\mathcal{X}_{2}$
(29a) $\displaystyle D$
$\displaystyle\coloneqq\left(\left(\mathcal{A}+\rho\mathbb{B}\right)\cap
D_{1}\right)\times D_{2}.$ (29b)

By plugging in $\Delta=P$ in Assumption 5, and $\Delta=\rho$ in Assumptions
10, we construct the following Lyapunov function candidate:

$\displaystyle V(x)=V_{1}(x_{1})+\sqrt{\varepsilon}V_{2,\rho}(x).$ (30)

###  A.1 Analysis of the jumps

The Lyapunov function after jumps equals to

$\displaystyle V(g)=V_{1}(g_{1})+\sqrt{\varepsilon}V_{2,\rho}(g),$ (31)

where $\left[\begin{smallmatrix}g_{1}\\\ g_{2}\end{smallmatrix}\right]=g\in
G(x)$. We prove the following Lemma:

######  Lemma 1.

Consider the hybrid system in (1) with restricted flow and jump sets in (29),
and let Assumptions 1—7 hold. For every $e>0$ and $\Delta>0$, there exists
$v>0$, such that $\left\|x\right\|_{\mathcal{M}_{\rho}}\leq v$ implies that

$\displaystyle\sup_{\left[\begin{smallmatrix}g_{1}\\\
g_{2}\end{smallmatrix}\right]\in G(x)}V_{1}(g_{1})-V_{1}(x_{1})\leq$
$\displaystyle\sup_{g_{1}^{\textup{r}}\in
G_{\textup{r}}(x_{1})}V_{1}(g_{1}^{\textup{r}})-V_{1}(x_{1})+\tfrac{e}{2}.\qed$

For the sake of contradiction, we assume that there exists $e>0$ such it holds

$\displaystyle\sup_{\left[\begin{smallmatrix}g_{1}\\\
g_{2}\end{smallmatrix}\right]\in G(x)}V_{1}(g_{1})-V_{1}(x_{1})>$
$\displaystyle\sup_{g_{1}^{\textup{r}}\in
G_{\textup{r}}(x_{1})}V_{1}(g_{1}^{\textup{r}})-V_{1}(x_{1})+\tfrac{e}{2},$
$\displaystyle\forall
x\in\left(\left(\mathcal{A}+\rho\mathbb{B}\right)\cap\mathcal{X}_{1}\right)\times\mathcal{X}_{2}.$
(32)

We define a sequence
$\left(x^{i}\right)_{i\in\mathbb{N}}\in\mathcal{X}_{1}\times\mathcal{X}_{2}$
such that $\left\|x^{i}\right\|_{\mathcal{M}_{\rho}}\leq\tfrac{1}{i}$ and that
it satisfies the inequality in Equation (32). Let
$x^{\prime}\coloneqq\operatorname{col}\left({x_{1},x_{2}^{\prime}}\right)$,
and ${\mathcal{M}_{\rho}}^{\prime}$ be a projection of ${\mathcal{M}_{\rho}}$
onto the subspace of bounded states,
$\mathcal{X}_{1}\times\mathcal{X}_{2}^{\prime}$. It holds that
$\left\|x^{i}\right\|_{\mathcal{M}_{\rho}}\leq\tfrac{1}{i}$ implies
$\left\|{x^{\prime}}^{i}\right\|_{{\mathcal{M}_{\rho}}{{}^{\prime}}}\leq\tfrac{1}{i}$.
Furthermore, it follows that the sequence ${x^{\prime}}^{i}$ is bounded. Due
to Assumption 1, we conclude that the sequence
$\left({g^{\prime}}^{i}\right)_{i\in\mathbb{N}}$, where ${g^{\prime}}^{i}\in
G_{1}({x^{\prime}}^{i})$, is also bounded. Thus, due to the Weierstrass
theorem, there exists a convergent subsequence that converges to the point
$({x^{\prime}}^{*},{g^{\prime}}^{*})$, where
${x^{\prime}}^{*}\in{\mathcal{M}_{\rho}}^{\prime}$. Next, due to the outer
semi-continuity of the mappings $G$ and $H$, it holds that
${x_{2}^{\prime}}^{*}\in H_{1}(x_{1}^{*})$, ${g^{\prime}}^{*}\in
G_{1}({x^{\prime}}^{*})$ and $g_{1}^{*}\in G_{\text{r}}(x_{1}^{*})$.
Therefore, it follows that

$\displaystyle\sup_{\left[\begin{smallmatrix}g_{1}^{*}\\\
{g_{2}^{\prime}}^{*}\end{smallmatrix}\right]\in
G_{1}(x_{1}^{*},{x_{2}^{\prime}}^{*})}$ $\displaystyle
V_{1}(g_{1}^{*})-V_{1}(x_{1}^{*})>$ $\displaystyle\sup_{g_{1}^{\textup{r}}\in
G_{\textup{r}}(x_{1}^{*})}$ $\displaystyle
V_{1}(g_{1}^{\textup{r}})-V_{1}(x_{1}^{*})+\tfrac{e}{2}\implies
0>\tfrac{e}{2}$

which leads us to a contradiction and in turn proves the Lemma.
$\blacksquare$If $\varepsilon$ is chosen such that
$\sqrt{\varepsilon}\leq\tfrac{e}{2\overline{V}}$, where
$\overline{V}\coloneqq\sup_{x\in\mathcal{X}_{1}\cap(\mathcal{A}+\rho\mathbb{B})\times\mathcal{X}_{2},g\in
G(x)}\left\|V_{2,\rho}(g)\right\|$, then due to Lemma 1, it holds that for any
$e>0$ and $\Delta>0$, there exist $v>0$ and $\varepsilon^{*}$ such that for
every $\varepsilon\in(0,\varepsilon^{*})$, inequality
$\left\|x\right\|_{\mathcal{M}_{\rho}}\leq v$ implies that

$\displaystyle\sup_{g\in
G(x)}V(g)-V(x)\leq-\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}(\left\|x_{1}\right\|_{\mathcal{A}})+e$
$\displaystyle\text{for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma)$ (33)

The previous condition is always satisfied during jumps if
$\left\|x\right\|_{\mathcal{M}_{\rho}}\leq v$ holds true before jumps. Thus,
we have the following result:

######  Lemma 2.

Consider the hybrid system in (1) with restricted flow and jump sets in (29),
and let Assumptions 1, 2, 5 hold. Then, for every $v>0$, and $\rho>0$ there
exists $\tau^{*}>0$, $\varepsilon^{*}$ and $\beta^{*}$, such that for every
$\varepsilon\in(0,\varepsilon^{*})$, $\beta\in(0,\beta^{*})$, it holds that if
trajectory $x$ satisfies
$\left\|x(t,j)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\rho$ for all
$t\in\mathrm{dom}(x(\cdot,j))$, then
$\left\|x(t,j)\right\|_{\mathcal{M}_{\rho}}\leq v$ for all
$t\in\mathrm{dom}(x(\cdot,j))$ such that $t\geq\tau^{*}$. ∎

From Assumption 5, the derivative of the Lyapunov function candidate, for
$\Delta=P$, reads as

$\displaystyle\left\langle\nabla V_{2,\rho}(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\tfrac{1}{\varepsilon}\alpha_{2,\rho}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}f_{1}\\\
0\end{smallmatrix}\right]\right\rangle$ $\displaystyle\text{for
}\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\alpha_{\beta}(\beta).$

We define the constant

$\displaystyle\mu\coloneqq\sup_{x\in\mathcal{X}_{1}\cap\left(\mathcal{A}+\rho\mathbb{B}\right)\times\mathcal{X}_{2}}\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}f_{1}\\\
0\end{smallmatrix}\right]\right\rangle.$

Then, the Lyapunov derivative is given by

$\displaystyle\left\langle\nabla V_{2,\rho}(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\tfrac{1}{\varepsilon}\alpha_{2,\rho}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)+\mu$
$\displaystyle\text{for
}\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\alpha_{\beta}(\beta).$

Let
$\beta^{*}=\alpha_{\beta}^{-1}\left(\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(v\right)\right)\right)$,
$m=\tfrac{1}{2}\alpha_{2,\rho}\left(\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(v\right)\right)\right)$
and $\varepsilon^{*}=\tfrac{m}{\mu}$. It follows that for any time interval
$(t,t+\tau)$ where only flowing occurred, for any
$\varepsilon\in(0,\varepsilon^{*})$, $\beta\in(0,\beta^{*})$ it holds

$\displaystyle\dot{V}_{2,\rho}^{P}(x(t,j))\leq-\mu\text{, for
$\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(v\right)\right)$}$
$\displaystyle\int_{t}^{t+\tau}dV_{2,\rho}(x(t,j))\leq-\mu\int_{t}^{t+\tau}dt$
$\displaystyle V_{2,\rho}(x(t+\tau,j))\leq V_{2,\rho}(x(t,j))-\mu\tau$
$\displaystyle\text{ for
$\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(v\right)\right)$}.$
(34)

As we assume $\left\|x(t,j)\right\|_{\mathcal{M}_{\rho}}\leq P$, from the
bounds of the Lyapunov function in 5, we have

$\displaystyle\left\|x(t+\delta t,j)\right\|_{\mathcal{M}_{\rho}}$
$\displaystyle\leq\underline{\alpha_{2,\rho}}^{-1}\left(\overline{\alpha_{2,\rho}}\left(\left\|x(t,j)\right\|\right)-\mu\delta
t\right)$
$\displaystyle\leq\underline{\alpha_{2,\rho}}^{-1}\left(\overline{\alpha_{2,\rho}}\left(P\right)-\mu\delta
t\right)\leq v.$ (35)

From the last inequality, it follows that
$\tau\geq\tau^{*}\coloneqq\underline{\sigma}^{-1}\left(\tfrac{\overline{\alpha_{2,\rho}}\left(P\right)-\underline{\alpha_{2,\rho}}\left(v\right)}{\mu}\right)$,
which proves the Lemma. $\blacksquare$It follows from (33) and Lemmas 1 and 2
that for any $e>0$, $\Delta>0$, there exist parameters $\varepsilon^{*}_{1}$,
$\beta^{*}_{1}$, and $\tau^{*}$ such that for any
$\varepsilon\in(0,\varepsilon^{*}_{1})$, $\beta\in(0,\beta^{*}_{1})$, if the
time between consecutive jumps is larger than $\tau^{*}$, it holds that

$\displaystyle\sup_{g\in
G(x)}V(g)-V(x)\leq-\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)+e$
(36) $\displaystyle\text{for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma).$

###  A.2 Analysis of the flows

The Lyapunov derivative is given by

$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left\langle\nabla V(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle=$
$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left(\left\langle\nabla V_{1}(x)\
\middle|\ f_{1}\right\rangle+\sqrt{\varepsilon}\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\right)$
$\displaystyle\leq\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left(\left\langle\nabla V_{1}(x)\
\middle|\ f_{1}\right\rangle+\left|\left\langle\nabla V_{2,\rho}(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
0\end{smallmatrix}\right]\right\rangle\right|\right.$
$\displaystyle\quad\left.+\tfrac{1}{\sqrt{\varepsilon}}\left\langle\nabla
V_{2,\rho}(x)\ \middle|\ \left[\begin{smallmatrix}0\\\
f_{2}\end{smallmatrix}\right]\right\rangle\right)$
$\displaystyle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)+\mu(x)$
$\displaystyle\quad+\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in
F(x)}\tfrac{1}{\sqrt{\varepsilon}}\left\langle\nabla V_{2,\rho}(x)\ \middle|\
\left[\begin{smallmatrix}0\\\ f_{2}\end{smallmatrix}\right]\right\rangle,$
$\displaystyle\text{ for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma),\left\|x\right\|_{\mathcal{M}_{\rho}}\geq\alpha_{\beta}(\beta),$
(37)

where $\mu(x)=-\sup_{f_{1}^{\textup{r}}\in
F_{\textup{r}}(x_{1})}\left(-\left\langle\nabla V_{1}(x)\ \middle|\
f_{1}^{\textup{r}}\right\rangle\right)$
$+\sup_{\left[\begin{smallmatrix}f_{1}\\\ f_{2}\end{smallmatrix}\right]\in
F(x),}\left(\left\langle\nabla V_{1}(x)\ \middle|\
f_{1}\right\rangle+\left|\left\langle\nabla V_{2,\rho}(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
0\end{smallmatrix}\right]\right\rangle\right|\right)$.
Let $v>\alpha_{\beta}(\beta)$ be chosen arbitrarily and let

$\displaystyle\varepsilon^{*}=\tfrac{\underline{\alpha_{2\rho}}^{2}\left(v\right)}{\sup_{x\in\mathcal{X}_{1}\cap\left(\mathcal{A}+\rho\mathbb{B}\right)\times\mathcal{X}_{2}}\left\|\mu(x)\right\|^{2}}.$
(38)

Then it holds that

$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left\langle\nabla V(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right),$
$\displaystyle\text{ for }\left\|x\right\|_{\mathcal{M}_{\rho}}\geq
v,\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma),$ (39)
$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left\langle\nabla V(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
$\displaystyle+\mu(x)+\tfrac{1}{\sqrt{\varepsilon}}\hat{\alpha}_{\beta}(\beta),\text{
for
}\left\|x\right\|_{\mathcal{M}_{\rho}}<v,\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma),$
(40)

which is combined into

$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left\langle\nabla V(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
$\displaystyle+\sup_{\left\|x\right\|_{\mathcal{M}_{\rho}}\leq
v}\mu(x)+\tfrac{1}{\sqrt{\varepsilon}}\hat{\alpha}_{\beta}(\beta),\text{ for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma).$ (41)

The next Lemma shows that the positive terms in the Lyapunov derivative, with
the proper choice of tuning parameters $\varepsilon$ and $\beta$, can be made
arbitrarily small.

######  Lemma 3.

Consider the hybrid system in (1) with restricted flow and jump sets in (29),
and let Assumptions 1—7 hold. For every $e>0$, $\Delta>0$, there exists
$\varepsilon^{*}>0$, $\beta^{*}(\varepsilon)>0$, such that for any
$\varepsilon\in(0,\varepsilon^{*})$, $\beta\in(0,\beta^{*}(\varepsilon))$, it
holds that $\sup_{\left\|x\right\|_{\mathcal{M}_{\rho}}\leq
v}\mu(x)+\tfrac{1}{\sqrt{\varepsilon}}\hat{\alpha}_{\beta}(\beta)\leq e$. ∎

We consider the following inequalities:

$\displaystyle\sup_{\left\|x\right\|_{\mathcal{M}_{\rho}}\leq
v}\mu(x)\leq\tfrac{e}{2}$ (42)
$\displaystyle\tfrac{1}{\sqrt{\varepsilon}}\hat{\alpha}_{\beta}(\beta)\leq\tfrac{e}{2}.$
(43)

If they hold, so does the inequality in the Lemma. The proof that inequality
in (42) can be made arbitrarily small by choice of small $v$, and hence
smaller $\varepsilon^{*}$ due to (38), is analogous to the proof of Lemma 1,
and thus it is omitted. Then, to satisfy inequality (43), it is sufficient to
have
$\beta^{*}\coloneqq\min\left(\hat{\alpha}_{\beta}^{-1}\left(e\sqrt{\varepsilon}\right),\alpha_{\beta}^{-1}(v)\right)$
and $\beta\in(0,\beta^{*})$. $\blacksquare$From Equations (41) and Lemma 3, it
follows that for any $e>0$, $\Delta>0$, there exists $\varepsilon^{*}_{2}$,
$\beta^{*}_{2}(\varepsilon_{2})$ such that for any
$\varepsilon\in(0,\varepsilon^{*}_{2})$ and
$\beta_{2}\in(0,\beta^{*}_{2}(\varepsilon_{2}))$, we have

$\displaystyle\sup_{\left[\begin{smallmatrix}f_{1}\\\
f_{2}\end{smallmatrix}\right]\in F(x)}\left\langle\nabla V(x)\ \middle|\
\left[\begin{smallmatrix}f_{1}\\\
\tfrac{1}{\varepsilon}f_{2}\end{smallmatrix}\right]\right\rangle\leq-\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)$
$\displaystyle+e,\text{ for
}\left\|x_{1}\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma).$ (44)

###  A.3 Complete Lyapunov analysis

We denote by $\phi(t,j)$ a solution of the system that contains only
$\underline{\sigma}(\tau)$ regular jumps. Let $\gamma$ be chosen so that
$\gamma\in(0,\gamma^{*})$, where
$\gamma^{*}\coloneqq\min(\alpha_{\gamma}^{-1}(\overline{\alpha_{1}}^{-1}(\tfrac{1}{2}\underline{\alpha_{1}}\left(\delta\right))),\overline{\gamma})$.
Next, $\eta$ is defined as
$\eta\coloneqq\tfrac{1}{2}\min\\{\underline{\alpha_{1}}\left(\delta\right),\alpha_{1}(\alpha_{\gamma}(\gamma^{*})),2\\}$.
Via Equation (33) and Lemmas 1 and 2, for
$e=\hat{\alpha}_{\gamma}(\gamma)\eta$, we have $\tau^{*}$,
$\varepsilon_{1}^{*}$, $\beta_{1}^{*}$. Next, we choose
$\tau\in(0,\min(\tau^{*},\overline{\tau}))$. From Equation (41) and Lemma 3
for $e=\hat{\sigma}(\tau)\hat{\alpha}_{\gamma}(\gamma)\eta$, we have
$\varepsilon_{2}^{*}$, $\beta_{2}^{*}(\varepsilon)$. Finally, let

$\displaystyle\varepsilon^{*}_{3}=\tfrac{\eta^{2}}{{\sup_{x\in\mathcal{X}_{1}\cap(\mathcal{A}+\rho\mathbb{B})\times\mathcal{X}_{2}}}\overline{\alpha_{2,\rho}}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)^{2}}.$
(45)

We define
$\varepsilon^{*}\coloneqq\min\\{\varepsilon^{*}_{1},\varepsilon^{*}_{2}\,\varepsilon^{*}_{3},\overline{\varepsilon}\\}$,
$\beta^{*}(\varepsilon)\coloneqq\min\\{\beta^{*}_{1},\beta^{*}_{2}(\varepsilon),$
$\overline{\beta}\\}$, and set the parameters as follows:
$\varepsilon\in(0,\varepsilon^{*})$, $\beta\in(0,\beta^{*}(\varepsilon))$.
From Equations (36) and (44), it follows that

$\displaystyle
V(\phi(t,j))+\sum_{i=0}^{j}\int_{t_{i}}^{t_{i+1}}\hat{\sigma}_{\tau}(\tau)\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|\phi_{1}(s,i)\right\|_{\mathcal{A}}\right)ds+$
$\displaystyle\sum_{i=1}^{j}\hat{\alpha}_{\gamma}(\gamma)\alpha_{1}\left(\left\|\phi_{1}\left(t_{i},i-1\right)\right\|_{\mathcal{A}}\right)\leq
V(\phi(0,0))$ $\displaystyle V(\phi(t,j))-V(\phi(0,0))\leq$
$\displaystyle-(\hat{\sigma}_{\tau}(\tau)t+j)\hat{\alpha}_{\gamma}(\gamma)\left(\alpha_{1}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)-\eta\right),$
$\displaystyle\text{ for
}\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma).$ (46)

As $\eta\leq\tfrac{1}{2}\alpha_{1}(\alpha_{\gamma}(\gamma))$ and
$\alpha_{\gamma}(\gamma^{*})<\alpha_{\gamma}(\gamma^{*})$, we rewrite the last
inequality as

$\displaystyle V(\phi(t,j))\leq
V(\phi(0,0))-\tfrac{1}{2}(\hat{\sigma}_{\tau}(\tau)t+j)\alpha_{1}\left(\alpha_{\gamma}\left(\gamma^{*}\right)\right),$
$\displaystyle\text{ for
}\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma^{*}),$
(47)

We can guarantee the decrease of the Lyapunov function up to the smallest
Lyapunov level set that contains the set
$(\mathcal{A}+\alpha_{\gamma}(\gamma^{*}))\mathbb{B})\times\mathcal{X}_{2}$.
Via equation 47, we move onto proving semi-global boudness and practical
attractivity of the equilibrium set.

#### Semi-global boundness

By definition, $\varepsilon\leq\varepsilon_{3}^{*}$. Thus, the upper and lower
bound of the Lyapunov function candidate are given by

$\displaystyle\underline{\alpha_{1}}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)\leq
V(x)$
$\displaystyle\leq\overline{\alpha_{1}}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)+\sqrt{\varepsilon}\overline{\alpha_{2,\rho}}\left(\left\|x\right\|_{\mathcal{M}_{\rho}}\right)$
$\displaystyle\leqq\overline{\alpha_{1}}\left(\left\|x_{1}\right\|_{\mathcal{A}}\right)+\eta$
(48)

From (47) and (48), for $\underline{\sigma}(\tau)$-reggular trajectories
$\phi$, it holds that

$\displaystyle\underline{\alpha_{1}}\left(\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\right)\leq\overline{\alpha_{1}}\left(\left\|\phi_{1}(0,0)\right\|_{\mathcal{A}}\right)+\eta,$
$\displaystyle\text{ for
}\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma^{*}).$
(49)

The maximal distance to the equilibrium of a
$\underline{\sigma}(\tau)$-regular trajectory $\phi(t,j)$ starting at
$(t,j)=(0,0)$ is given in (49) by
$\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\leq\rho_{\textup{C}}\left(\overline{\alpha_{1}}\left(\left\|\phi_{1}(0,0)\right\|_{\mathcal{A}}\right)\right)$,
where
$\rho_{\textup{C}}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+},\rho_{\textup{C}}\left(a\right)\coloneqq\underline{\alpha_{1}}^{-1}\left(\overline{\alpha_{1}}\left(a\right)+1\right)$.
By the outer semi-continuity and local boundness of the mapping $G$ in (5) for
all allowed sets of parameters, for each $\underline{r}>0$, there exists a
$\overline{r},r>0$, such that
$G_{1}\left((\mathcal{A}+\underline{r}\mathbb{B})\times\mathcal{X}^{\prime}_{2}\right)\subset
G_{1}(\mathcal{A}\times\mathcal{X}^{\prime}_{2})+\overline{r}\mathbb{B}\subset(\mathcal{A}+r\mathbb{B})\times\mathcal{X}^{\prime}_{2}$.
Via this property, we define the mapping
$\rho_{\textup{D}}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+},\rho_{\textup{C}}\left(\underline{r}\right)\coloneqq
r$. Thus, for any initial condition such that
$\left\|\phi(0,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\Delta$, the
maximal distance from the equilibrium set after $N$ irregular jumps, not
necessarily consecutive jumps, is given by
$\rho=\rho_{\textup{C}}\circ\rho_{\textup{D}}\circ\dots\circ\rho_{\textup{C}}(\Delta)$,
where $\rho_{\textup{D}}$ repeats $N$ times, and $\rho_{\textup{C}}$ repeats
$N+1$ times.

#### Semi-global stability for $\underline{\sigma}(\tau)$ trajectories

Let us consider trajectories after the $N$ irregular jumps. We show that for
any $R\geq\delta$, there exists $r>0$, such that
$\left\|\phi(l,i)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq
r\implies\left\|\phi(t,j)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq R$
for $l+i\leq t+j$. From (49) and
$\eta\leq\tfrac{1}{2}\underline{\alpha_{1}}\left(\delta\right)$, it follows
that

$\displaystyle\left\|\phi_{1}(l,i)\right\|_{\mathcal{A}}\leq\overline{\alpha_{1}}^{-1}\left(\underline{\alpha_{1}}\left(R\right)-\tfrac{1}{2}\underline{\alpha_{1}}\left(\delta\right)\right),$
$\displaystyle\text{ for
}\left\|\phi_{1}(t,j)\right\|_{\mathcal{A}}\geq\alpha_{\gamma}(\gamma^{*}).$
(50)

We note that the previous inequality holds up to the smallest radius of
interest
$r=\overline{\alpha_{1}}^{-1}\left(\tfrac{1}{2}\underline{\alpha_{1}}\left(\delta\right)\right)$,
because
$\gamma^{*}\leq\alpha_{\gamma}^{-1}(\overline{\alpha_{1}}^{-1}\left(\tfrac{1}{2}\underline{\alpha_{1}}\left(\delta\right)\right))$.
Then, for any $R\geq\delta$, and all $\gamma\in(0,\gamma^{*})$, we have
$r(R)\coloneqq\min\left\\{\overline{\alpha_{1}}^{-1}\left(\underline{\alpha_{1}}\left(R\right)-\eta\right),\Delta\right\\}$.

#### Practical attractivity

Without of loss of generality, we assume $R,r$ are given so that $\Delta\geq
R\geq r\geq\delta$. We have to show that there exists a period $T\geq 0$, such
that $\left\|\phi(0,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq
R\implies\left\|\phi(t,j)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq r$
for all $(t,j)$ such that $t+j\geq T$.
Let $(l,i)$ be the hybrid time instant after the $N$ irregular jumps. By
Assumption 6, it holds $l+i\leq T^{*}$. Then, from (47), for $\phi(0,0)$
replaced with $\phi(l,i)$, it follows

$\displaystyle\underline{\alpha_{1}}\left(r\right)\leq\overline{\alpha_{1}}\left(\rho\right)-\tfrac{1}{2}\alpha_{1}\left(\alpha_{\gamma}\left(\gamma^{*}\right)\right)[(\hat{\sigma}_{\tau}(\tau)(t-l)+j-i)],$
(51)

Then, when we have

$\displaystyle
t+j\leq\tfrac{2(\overline{\alpha_{1}}\left(\rho\right)-\underline{\alpha_{1}}\left(r\right))}{\min\\{\hat{\sigma}_{\tau},1\\}\alpha_{1}\left(\alpha_{\gamma}\left(\gamma^{*}\right)\right)}+T^{*}=T(R,r).$
(52)

#### Conclusion

Our restricted system renders the set $\mathcal{A}\times\mathcal{X}_{2}$
practically attractive. Finally, to show the equivalence between the solutions
of the original and restricted system, it is possible to use the same
procedure as in [16] after Equation (29). $\blacksquare$

##  Appendix B Proof of Theorem 2

Let $\Delta>\delta>0$ be the parameters of semi-global practical stability. We
denote with $\rho$ the maximum distance
$\left\|\phi(t,j)\right\|_{\mathcal{A}\times\mathcal{X}}$ for trajectories
starting in ${\mathcal{M}_{\mathcal{A}}}+\Delta\mathbb{B}$. On the other hand,
as it is possible to a priori bound the distance
$\left\|\phi(t,j)\right\|_{{\mathcal{M}_{\mathcal{A}}}}$ with $P>0$, see
Remark 3, we “redefine” the set $\mathcal{X}_{2}^{\prime}$ as a compact set
$\left(\\{x_{2}^{\prime}\mid x_{1}\in\mathcal{A},x_{2}^{\prime}\in
H_{1}(x_{1})\\}+P\mathbb{B}\right)\cap\mathcal{X}^{\prime}_{2}$. Now the same
procedure as in proof of Theorem 1 can be repeated. From Equations (35) and
(52) we have

$\displaystyle\left\|\phi(t,j)\right\|_{\mathcal{M}_{\rho}}\leq v\text{ for
all }t\in\mathrm{dom}(\phi(\cdot,j))$ $\displaystyle\text{ s.t.
}t\geq\underline{\sigma}(\tau^{*}(P,v)),$ (53)
$\displaystyle\left\|\phi(t,j)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq
r\text{ for all }(t,j)\in\mathrm{dom}(\phi)$ $\displaystyle\text{ s.t.
}t+j\geq T(\Delta,r),$ (54)

A key observation is that the intersection of sets
$\mathcal{A}\times\mathcal{X}_{2}$ and ${\mathcal{M}_{\rho}}$ gives us the set
${\mathcal{M}_{\mathcal{A}}}$, for which we want to prove stability. Distance
to the set $\mathcal{A}\times\mathcal{X}_{2}$ is given by (54), and the
distance to the set ${\mathcal{M}_{\rho}}$ is given by (53). Thus, it is
possible to quantify the distance to the equilibrium set using the distances
of the latter two sets using the the following results:

######  Lemma 4.

Let $\mathcal{A},\mathcal{B}$ be nonempty sets defined on a metric space,
where at least one is bounded. Let their intersection $\mathcal{S}$ be
nonempty. Then, for every $d>0$, there exists $\underline{d}>0$, such that
$\left\|x\right\|_{\mathcal{A}}\leq\underline{d}$ and
$\left\|x\right\|_{\mathcal{B}}\leq\underline{d}$ implies that
$\left\|x\right\|_{\mathcal{S}}\leq d$. ∎

Let us assume otherwise, i.e., there exists some $d>0$ such that for any
$\underline{d}>0$ there exists $x$ such that it holds
$\left\|x\right\|_{\mathcal{S}}>d$. Let us create a sequence of these points,
$\left(x_{i}\right)_{i\in\mathbb{N}}$, such that
$\left\|x_{i}\right\|_{\mathcal{A}}\leq\tfrac{1}{i}$ and
$\left\|x_{i}\right\|_{\mathcal{B}}\leq\tfrac{1}{i}$. Because the sequence is
bounded, there must exists a convergent subsequence. Let one such subsequence
converge to $x^{*}$. Because of the continuity of the metric, it holds that
$\left\|x^{*}\right\|_{\mathcal{A}}=0$ and
$\left\|x^{*}\right\|_{\mathcal{B}}=0$. Thus,
$x^{*}\in\operatorname{cl}\left(\mathcal{A}\right)$ and
$x^{*}\in\operatorname{cl}\left(\mathcal{B}\right)$, or in other words
$x^{*}\in\operatorname{cl}\left(\mathcal{A}\right)\cap\operatorname{cl}\left(\mathcal{B}\right)\equiv\operatorname{cl}\left(\mathcal{S}\right)$.
Then it holds $\left\|x^{*}\right\|_{\mathcal{S}}=0$, which is opposite of our
assumption. $\blacksquare$

######  Lemma 5.

Let $\mathcal{A},\mathcal{B}$ be nonempty sets defined on a metric space. Let
their intersection $\mathcal{S}$ be nonempty and bounded. Then, for every
$\overline{d}>0$, there exists ${d}>0$, such that
$\left\|x\right\|_{\mathcal{S}}\leq d$ implies
$\left\|x\right\|_{\mathcal{A}}\leq\overline{d}$ and
$\left\|x\right\|_{\mathcal{B}}\leq\overline{d}$. ∎

Let us assume otherwise, i.e there exists some $\overline{d}>0$ such that for
any $d>0$, there exists $x$ such that
$\left\|x\right\|_{\mathcal{A}}>\overline{d}$ and
$\left\|x\right\|_{\mathcal{B}}>\overline{d}$. Let us create a sequence of
these points, $\left(x_{i}\right)_{i\in\mathbb{N}}$, such that
$\left\|x\right\|_{\mathcal{A}}\leq\tfrac{1}{i}$. Because the sequence is
bounded there must exist a convergent subsequence. Let one such subsequence
converge to $x^{*}$. Because of the continuity of the metric, it holds that
$\left\|x^{*}\right\|_{\mathcal{S}}=0$. Thus,
$x^{*}\in\operatorname{cl}\left(\mathcal{S}\right)$ or in other words
$x^{*}\in\operatorname{cl}\left(\mathcal{S}\right)\equiv\operatorname{cl}\left(\mathcal{A}\right)\cap\operatorname{cl}\left(\mathcal{B}\right)$.
Then it holds $\left\|x^{*}\right\|_{\mathcal{A}}=0$ and
$\left\|x^{*}\right\|_{\mathcal{B}}=0$, which is opposite of our assumption.
$\blacksquare$

######  Reamrk 2.

Although we assume boundedness of some of the sets in Lemmas 4 and 5, it is
possible to prove the same results for the cases when $\mathcal{A}$ and
$\mathcal{B}$ are unbounded in the same dimensions, which is the case in our
setup.

#### Semi-global stability

To prove practical stability, we show that for any $R\geq\delta$ there exists
a neighborhood of the equilibrium, ${\mathcal{M}_{\mathcal{A}}}+r\mathbb{B}$,
such that any trajectory initiated in neighborhood will stay inside the set
${\mathcal{M}_{\mathcal{A}}}+R\mathbb{B}$, for properly chosen parameters. But
first, we prove a similar result for regular trajectories of the restricted
system.

######  Lemma 6 (Semi-global stability-like property).

Consider the hybrid system in (1) with restricted flow and jump sets in (29),
and let Assumptions 1, 2, 3, 5, 7 hold. Then, for every
$\overline{v}\geq\delta$, there exists a $\underline{v}>0$, and a set of
tuning parameters
$\varepsilon^{*},\tau^{*},\beta^{*}(\varepsilon),\gamma^{*}$, such that for
all regular trajectories $\phi$ with
$\left\|\phi(0,0)\right\|_{\mathcal{M}_{\mathcal{A}}}\leq\underline{v}$ and
$\varepsilon\in(0,\varepsilon^{*}),\tau\geq\tau^{*},\beta\in(0,\beta^{*}(\varepsilon)),\gamma\in(0,\gamma^{*})$,
it holds that
$\left\|\phi(t,j)\right\|_{\mathcal{M}_{\mathcal{A}}}\leq\overline{v}$ for all
$(t,j)\in\mathrm{dom}\left(\phi\right)$. ∎

_Sketch of the proof_
First, we find $\hat{v}>0$ such that any trajectory initiated in
${\mathcal{M}_{\mathcal{A}}}+\hat{v}\mathbb{B}$, stays in
${\mathcal{M}_{\mathcal{A}}}+\overline{v}\mathbb{B}$ during flows. Then we
find $\tilde{v}>0$ such that jumps from
${\mathcal{M}_{\mathcal{A}}}+\tilde{v}\mathbb{B}$ will end in
${\mathcal{M}_{\mathcal{A}}}+\hat{v}\mathbb{B}$. Next, we find $\underline{v}$
such that any trajectory initiated in
${\mathcal{M}_{\mathcal{A}}}+\underline{v}\mathbb{B}$, stays in
${\mathcal{M}_{\mathcal{A}}}+\tilde{v}\mathbb{B}$ during flows. Finally, we
choose $\varepsilon$, $\tfrac{1}{\tau}$, $\beta$, $\gamma$ small enough such
that all trajectories end up in
${\mathcal{M}_{\mathcal{A}}}+\underline{v}\mathbb{B}$ before jumps.

Consider the following system of implications:

$\displaystyle\quad\,\,\,\left\|\phi(0,0)\right\|_{{\mathcal{M}_{\mathcal{A}}}}{\leq}{\underline{v}}\overset{(1)}{\Rightarrow}\begin{array}[]{l}\left\|\phi(0,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\underline{u}\\\
\left\|\phi(0,0)\right\|_{{\mathcal{M}_{\rho}}}\leq\underline{u}\end{array}$
$\displaystyle\overset{(2)}{\Rightarrow}\begin{array}[]{l}\left\|\phi(t,0)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\tilde{u}\\\
\left\|\phi(t,0)\right\|_{{\mathcal{M}_{\rho}}}\leq\tilde{u}\end{array}\overset{(3)}{\Rightarrow}\left\|\phi(t,0)\right\|_{{\mathcal{M}_{\mathcal{A}}}}\leq\tilde{v}$
$\displaystyle\overset{(4)}{\Rightarrow}\left\|\phi(t,1)\right\|_{{\mathcal{M}_{\mathcal{A}}}}\leq\hat{v}\overset{(5)}{\Rightarrow}\begin{array}[]{l}\left\|\phi(t,1)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\hat{u}\\\
\left\|\phi(t,1)\right\|_{{\mathcal{M}_{\rho}}}\leq\hat{u}\end{array}$
$\displaystyle\overset{(6)}{\Rightarrow}\begin{array}[]{l}\left\|\phi(l,1)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\overline{u}\\\
\left\|\phi(l,1)\right\|_{{\mathcal{M}_{\rho}}}\leq\overline{u}\end{array}\overset{(7)}{\Rightarrow}\left\|\phi(l,1)\right\|_{{\mathcal{M}_{\mathcal{A}}}}\leq\overline{v}.$

Implication (7) follows from Lemma 4, while implication (6) follows from
Equations (50) with
$\hat{u}^{1}\leq\overline{\alpha_{1}}^{-1}\left(\underline{\alpha_{1}}\left(\tfrac{1}{2}\overline{u}\right)\right)$,
Equation (34) and
$\hat{u}^{1}=\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(\overline{u}\right)\right)$,
and $\hat{u}=\min\\{\hat{u}^{1},\hat{u}^{2}\\}$; Implication (5) follows from
Lemma 5; Implication (4) proceeds from [3, Lemma 5.15], outer semicontinuity,
local boundedness of the mapping $G$, Assumption 8, thus for every
$\hat{v}>0$, there exists a $\tilde{v}\leq\hat{v}$ such that
$G({\mathcal{M}_{\mathcal{A}}}+\tilde{v}\mathbb{B})\subset{\mathcal{M}_{\mathcal{A}}}+\hat{v}\mathbb{B}$;
Implication (3) follows from Lemma 4, while implication (2) follows from
Equations (50) with
$\underline{u}^{1}=\overline{\alpha_{1}}^{-1}\left(\underline{\alpha_{1}}\left(\tfrac{1}{2}\tilde{u}\right)\right)$,
Equation (34) and
$\underline{u}^{2}=\overline{\alpha_{2,\rho}}^{-1}\left(\underline{\alpha_{2,\rho}}\left(\tilde{u}\right)\right)$
and $\underline{u}=\min\\{\underline{u}^{1},\underline{u}^{2}\\}$; Implication
(1) follows from Lemma 5. To satisfy the inequalities in Equations (50), (34),
let
$\eta=\tfrac{1}{2}\min\\{\underline{\alpha_{1}}\left(\underline{u}\right),\alpha_{1}\left(\overline{\alpha_{1}}^{-1}\left(\tfrac{1}{2}\underline{\alpha_{1}}\left(\underline{u}\right)\right)\right),2\\}$,
$\gamma^{*}\coloneqq$
$\alpha_{\gamma}^{-1}\left(\overline{\alpha_{1}}^{-1}\left(\tfrac{1}{2}\underline{\alpha_{1}}\left(\underline{u}\right)\right)\right)$,
$\gamma\in(0,\gamma^{*})$; Via Equation (33) and Lemmas 1 and 2, for
$e=\hat{\alpha}_{\gamma}(\gamma)\eta$, we have $\tau^{*}$,
$\varepsilon_{1}^{*}$, $\beta_{1}^{*}$. Next, we choose
$\tau\in(0,\min(\tau^{*},\overline{\tau}))$. From Equation (41) and Lemma 3
for $e=\hat{\sigma}(\tau)\hat{\alpha}_{\gamma}(\gamma)\eta$, we have
$\varepsilon_{2}^{*}$, $\beta_{2}^{*}(\varepsilon)$. Finally, let
$\varepsilon_{3}^{*}$ be defined as in (45). We define
$\varepsilon^{*}\coloneqq\min\\{\varepsilon^{*}_{1},\varepsilon^{*}_{2}\,\varepsilon^{*}_{3},\overline{\varepsilon}\\}$,
$\beta^{*}(\varepsilon)\coloneqq\min\\{\beta^{*}_{1},\beta^{*}_{2}(\varepsilon),$
$\overline{\beta}\\}$, and set the parameters as follows:
$\varepsilon\in(0,\varepsilon^{*})$, $\beta\in(0,\beta^{*}(\varepsilon))$.
Furthermore, as Equation (50) holds for jumps and flows, it follows that
$\left\|\phi(l,1)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\leq\tilde{u}$,
and due to Lemma 2, it holds that
$\left\|\phi(l,1)\right\|_{{\mathcal{M}_{\rho}}}\leq\tilde{u}$ for
$l\geq\min_{t}\mathrm{dom}\left(\phi(\cdot,1)\right)+\underline{\sigma}(\tau^{*})$.
Thus, it is possible to follow the same reasoning with implications (3) to (7)
for the next, and all the following regular jumps, which proves our Lemma.
$\blacksquare$

######  Reamrk 3.

We can “reverse” Lemma 6 so that we claim that for every $\underline{v}>0$,
there exists a $\overline{v}>0$ that satisfies the same inequality. Then, by
doing an inverse procedure of the proof of stability, we can derive $P>0$ such
that $\left\|\phi(0,0)\right\|_{\mathcal{M}_{\mathcal{A}}}\leq\Delta$ implies
that $\left\|\phi(t,j)\right\|_{\mathcal{M}_{\mathcal{A}}}\leq P$. These
bounds depend on the proprieties of mapping $G$ and the lower and upper bounds
of the Lyapunov functions, thus can be computed a priori. ∎

Let $N$ be the number of irregular jumps for the given $\Delta$. Via Lemma 6,
for $\overline{v}=R$, we find $\overline{r}_{N}=\underline{v}$ and parameters
$\varepsilon^{*}_{N},\tau^{*}_{N},\beta^{*}_{N}(\varepsilon),\gamma^{*}_{N}$.
Then, from [3, Lemma 5.15], outer semicontinuity, local boundedness of the
mapping $G$, Assumption 8, we can find $\underline{r}_{N}$ such that
$G({\mathcal{M}_{\mathcal{A}}}+\underline{r}_{N}\mathbb{B})\subset{\mathcal{M}_{\mathcal{A}}}+\overline{r}_{N}\mathbb{B}$.
Then again we use Lemma 6, with $\overline{v}=\underline{r}_{N}$, to find
$\underline{v}=\overline{r}_{N-1}$ and parameters
$\varepsilon^{*}_{N-1},\tau^{*}_{N-1},\beta^{*}_{N-1}(\varepsilon),\gamma^{*}_{N-1}$.
These steps are repeated until we reach the first jump. Then, we use Lemma 6,
for $\overline{v}=\underline{r}_{1}$ to find $r=\underline{v}$ and parameters
$\varepsilon^{*}_{0},\tau^{*}_{0},\beta^{*}_{0}(\varepsilon),\gamma^{*}_{0}$.
We note that for
$\varepsilon^{*}\coloneqq\min\\{\varepsilon^{*}_{0},\dots,\varepsilon^{*}_{N}\\}$,
$\tau^{*}\coloneqq\max\\{\tau^{*}_{0},\dots,\tau^{*}_{N}\\}$,
$\beta^{*}(\varepsilon)\coloneqq\min\\{\beta^{*}_{0}(\varepsilon),\dots,\beta^{*}_{N}(\varepsilon)\\}$
and $\gamma^{*}\coloneqq\min\\{\gamma^{*}_{0},\dots,\gamma^{*}_{N}\\}$, all
the inequalities hold._Practical attractivity_

Without of loss of generality, we assume $R,r$ are given so that $\Delta\geq
R\geq r\geq\delta$. Let $(l,i)$ be a hybrid time instant after the $N$
irregular jumps. By Assumption 6, it holds $l+i\leq T^{*}$. Furthermore, Lemma
6 gives us $\underline{r}=\underline{v}$ for $\overline{v}=\delta$, and the
corresponding tuning parameters
$\varepsilon^{*},\tau^{*},\beta^{*}(\varepsilon),\gamma^{*}$. From the
definition of parameters in Lemma 6, it follows that the Lyapunov derivatives
and differences for functions in Equation (30) and Assumption 5, are defined
for
$\left\|\phi(t,j)\right\|_{\mathcal{A}\times\mathcal{X}_{2}}\geq\underline{u},\left\|\phi(t,j)\right\|_{{\mathcal{M}_{\rho}}}\geq\underline{u}$,
where $\underline{u}$ is given in the the system of implications in (B). Thus
Equations (53) and (54) guarantee that the trajectories eventually enter and
stay in $\tilde{u}$ neighborhoods before jumps, for $v=r=\underline{u}$. And
from our practical-stability result, it follows that the trajectory stays in
the $\overline{r}$ neighborhood $\blacksquare$

##  Appendix C Proof of Theorem 3

Similarly to [10, Equ. 13], let the Lyapunov function candidate be given by

$\displaystyle
V_{i}(q_{i})\coloneqq\tfrac{1}{2}\left({x}_{i}^{e}-c_{3}{\omega}_{i}y_{i}^{e}\right)^{2}+\tfrac{1}{2}{y^{e}_{i}}^{2}+\tfrac{1}{2}{\theta^{e}_{i}}^{2},$
(55)

where
$q_{i}=\operatorname{col}\left({{x}_{i},{y}_{i},\theta_{i}^{e},{\tau}_{i},\theta_{i},\hat{v}_{i},\hat{\omega}_{i}}\right)$.
First, we characterize the upper and lower bounds of the Lyapunov function
candidate. It holds

$\displaystyle
V_{i}(q_{i})=\tfrac{1}{2}{{x}_{i}^{e}}^{2}-c_{3}{\omega}_{i}x_{i}^{e}y_{i}^{e}+\tfrac{1}{2}c_{3}^{2}{\omega}_{i}^{2}{y_{i}^{e}}^{2}+\tfrac{1}{2}{y^{e}_{i}}^{2}+\tfrac{1}{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\geq\tfrac{1}{2}\,{{x}_{i}^{e}}^{2}\left(1-\gamma\right)+\tfrac{1}{2}c_{3}^{2}{\omega}_{i}^{2}{y_{i}^{e}}^{2}\left(1-\tfrac{1}{\gamma}\right)+\tfrac{1}{2}{y^{e}_{i}}^{2}+\tfrac{1}{2}{\theta^{e}_{i}}^{2}$
$\displaystyle{\geq}\tfrac{1}{4}{{x}_{i}^{e}}^{2}+\tfrac{1}{2}\left(1-c_{3}^{2}{\omega}_{i}^{2}\right){y^{e}_{i}}^{2}+\tfrac{1}{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\geq\tfrac{1}{4}\left(x_{i}-u_{i}^{1}\right)^{2}+\tfrac{1}{4}\left(y_{i}-u_{i}^{2}\right)^{2}+\tfrac{1}{4}{\theta^{e}_{i}}^{2}=\tfrac{1}{4}\left\|r_{i}\right\|^{2},$

where the second line follows from
$ab\leq\tfrac{\gamma}{2}a^{2}+\tfrac{1}{2\gamma}b^{2}$, third line follows
from $\gamma=\tfrac{1}{2}$, in the forth line we assume that
$c_{3}\leq\tfrac{\sqrt{2}}{2{\omega}_{i}}$, and
$r_{i}\coloneqq\operatorname{col}\left({x_{i}-u_{i}^{1},y_{i}-u_{i}^{2},\theta_{i}-\theta_{\textup{r}}}\right)$.
Furthermore, for the upper bound we have

$\displaystyle V_{i}(q_{i})$
$\displaystyle\leq{{x}_{i}^{e}}^{2}+c_{3}^{2}{\omega}^{2}{y_{i}^{e}}^{2}+\tfrac{1}{2}{y^{e}_{i}}^{2}+\tfrac{1}{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\leq{{x}_{i}^{e}}^{2}+{y^{e}_{i}}^{2}+{\theta^{e}_{i}}^{2}=\left\|r_{i}\right\|^{2},$

where the second line follows from the former assumption on constant $c_{3}$.
Thus, the bound on the Lyapunov function are given by.

$\displaystyle\tfrac{1}{4}\left\|r_{i}\right\|^{2}\leq
V_{i}(q_{i})\leq\left\|r_{i}\right\|^{2}.$

The Lyapunov derivative is bounded similarly to [10, Equ. 14]:

$\displaystyle\left\langle\nabla V_{i}(q_{i})\ \middle|\
f_{i}(q_{i})\right\rangle=-\Sigma_{i}(q_{i})+\Lambda_{i}(q_{i}),$

where $\Sigma_{i}(q_{i})\coloneqq
c_{1}\left({x}^{e}_{i}-c_{3}{\omega}_{i}y^{e}_{i}\right)^{2}+c_{2}{\theta^{e}_{i}}^{2}+c_{3}\omega_{i}^{2}{y^{e}_{i}}^{2}$
and
$\Lambda_{i}(q_{i})\coloneqq\left({x}^{e}_{i}-c_{3}{\omega}_{i}y^{e}_{i}\right)\left(e^{w}_{i}y^{e}_{i}-e^{v}_{i}-c_{3}y^{e}_{i}e^{w}_{i}c_{2}\right.$
$\left.+c_{3}we^{\omega}_{i}x^{e}_{i}\right)-y^{e}_{i}e^{w}_{i}x^{e}_{i}-\theta^{e}_{i}e^{w}_{i}$,
with $e^{w}_{i}:=$ $\hat{\omega}_{i}-\omega_{i}$,
$e^{v}_{i}=\hat{v}_{i}-v_{i}$, and $v_{i}\coloneqq
c_{1}(x_{i}^{e}-c_{3}\omega_{i}y_{i}^{e})-c_{3}c_{2}(\omega_{\textup{r}}-\omega_{i})y_{i}^{e}+c_{3}\omega_{i}^{2}x_{i}^{e}$.
To characterize the convergence rate, we upper bound $\Sigma(q_{i})_{i}$ as
follows:

$\displaystyle\Sigma_{i}(q_{i})=c_{1}\left({x}^{e}_{i}-c_{3}{\omega}_{i}y^{e}_{i}\right)^{2}+c_{2}{\theta^{e}_{i}}^{2}+c_{3}\omega_{i}^{2}{y^{e}_{i}}^{2}$
$\displaystyle\geq
c_{1}{{x}^{e}_{i}}^{2}\left(1-\gamma\right)+c_{1}c_{3}^{2}\omega_{i}^{2}{y^{e}_{i}}^{2}\left(1-\tfrac{1}{\gamma}\right)+c_{3}\omega_{i}^{2}{y^{e}_{i}}^{2}+c_{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\geq\tfrac{1}{2}c_{1}{{x}^{e}_{i}}^{2}+\left(c_{3}\omega_{i}^{2}-c_{1}c_{3}^{2}\omega_{i}^{2}\right){y^{e}_{i}}^{2}+c_{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\geq\tfrac{1}{2}c_{1}{{x}^{e}_{i}}^{2}+\tfrac{1}{2}c_{3}\omega_{i}^{2}{y^{e}_{i}}^{2}+c_{2}{\theta^{e}_{i}}^{2}$
$\displaystyle\geq\underline{c}\left\|r_{i}\right\|^{2},$

where the third line follows for $\gamma=\tfrac{1}{2}$, in fourth line we
assume $c_{1}\leq\tfrac{1}{2c_{3}}$,
$\underline{c}\coloneqq\min\\{\tfrac{1}{2}c_{1},c_{2},\tfrac{1}{2}c_{3}\omega_{i}^{2}\\}$
Now, we write the Lyapunov derivative as

$\displaystyle\left\langle\nabla V_{i}(q_{i})\ \middle|\
f_{i}(q_{i})\right\rangle\leq-\underline{c}V_{i}(q_{i})+\Lambda_{i}(q_{i}).$

As the jumps restart $\Lambda_{i}(q_{i})$ to $0$, the jumps of the Lyapunov
given by

$\displaystyle V(q_{i}^{+})-V(q_{i})\leq 0.$

Let $\Delta>\delta>0$ be the parameters of the semi-global practical
stability. If our Lyapunov derivative is negative on the desired domain, it
follows that

$\displaystyle\tfrac{1}{4}\left\|r_{i}(t,j)\right\|^{2}\leq
V_{i}(q_{i}(t,j))\leq V_{i}(q_{i}(0,0))\leq\left\|r_{i}(0,0)\right\|^{2},$

Thus for any initial condition with $\left\|r_{i}(0,0)\right\|\leq\Delta$, it
holds that $\left\|r_{i}(t,j)\right\|\leq 2\Delta$. Using the previous bound,
we can estimate the minimal and maximal value of $\omega_{i}$ as

$\displaystyle\underline{\omega}_{i}$
$\displaystyle\coloneqq\min\omega_{i}=\omega_{\textup{r}}-2c_{2}\Delta$ (56)
$\displaystyle\overline{\omega}_{i}$
$\displaystyle\coloneqq\max\omega_{i}=\omega_{\textup{r}}+2c_{2}\Delta.$ (57)

Hence, we choose $c_{2}=\tfrac{\omega_{\textup{r}}}{4\Delta}$ to ensure
$\underline{\omega}_{i}$ is positive,
$c_{3}=\tfrac{1}{3{{\omega}}_{\textup{r}}}\leq\tfrac{\sqrt{2}}{3{{\omega}}_{\textup{r}}}$,
and $c_{1}=\tfrac{1}{2c_{3}}$.
As $\Lambda_{i}(q_{i})$ is differentiable and all of its variables and their
derivatives are bounded, we can approximate it with a constant $M$ and write
the derivative as

$\displaystyle\left\langle\nabla V_{i}(q_{i})\ \middle|\
f_{i}(q_{i})\right\rangle$
$\displaystyle\leq-\underline{c}V_{i}(q_{i})+M\tilde{\tau}_{i}$
$\displaystyle\leq-\underline{c}V_{i}(q_{i})+M\sigma_{i}$

Parameter $\sigma_{i}$ can be made arbitrarily small, thus enabling
arbitrarily close convergence to the equilibrium point. As there is a constant
time between jumps, semi-global practical stability follows for
$c_{1}=\sigma_{i}\rightarrow 0$ [16, Cor. 8.7].