# Optimal control of a 2D diffusion-advection process with a team of mobile
actuators under jointly optimal guidance

Sheng Cheng<EMAIL_ADDRESS>Derek A. Paley<EMAIL_ADDRESS>University of
Maryland, College Park  University of Maryland, College Park

(15 June 2021)

###### Abstract

This paper describes an optimization framework to control a distributed
parameter system (DPS) using a team of mobile actuators. The framework
simultaneously seeks optimal control of the DPS and optimal guidance of the
mobile actuators such that a cost function associated with both the DPS and
the mobile actuators is minimized subject to the dynamics of each. The cost
incurred from controlling the DPS is linear-quadratic, which is transformed
into an equivalent form as a quadratic term associated with an operator-valued
Riccati equation. This equivalent form reduces the problem to seeking for
guidance only because the optimal control can be recovered once the optimal
guidance is obtained. We establish conditions for the existence of a solution
to the proposed problem. Since computing an optimal solution requires
approximation, we also establish the conditions for convergence to the exact
optimal solution of the approximate optimal solution.  That is, when
evaluating these two solutions by the original cost function, the difference
becomes arbitrarily small as the approximation gets finer. Two numerical
examples demonstrate the performance of the optimal control and guidance
obtained from the proposed approach.

###### keywords:

Infinite-dimensional systems; Multi-agent systems; Modeling for control
optimization; Guidance navigation and control.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author Sheng Cheng (Tel. +1 301 335 2995).

,

##  1 Introduction

Recent development of mobile robots (unmanned aerial vehicles, terrestrial
robots, and underwater vehicles) has greatly extended the type of distributed
parameter system (DPS) over which mobile actuation and sensing can be
deployed. Such a system is often modeled by a partial differential equation
(PDE), which varies in both time and space. Exemplary applications of mobile
control and estimation of a DPS can be found in thermal manufacturing [13],
monitoring and neutralizing groundwater contamination [12], and wildfire
monitoring [19].

We propose an optimization framework that simultaneously solves for the
guidance of a team of mobile actuators and the control of a DPS. We consider a
2D diffusion-advection process as the DPS for its capability of modeling a
variety of processes governed by continuum mechanics and the convenience of
the state-space representation. The framework minimizes an integrated cost
function, evaluating both the control of the DPS and the guidance of the
actuators, subject to the dynamics of the DPS and the mobile actuators. The
problem addresses the mobile actuator and the DPS as a unified system, instead
of solely controlling the DPS. Furthermore, the additional degree of freedom
endowed by mobility yields improved control performance in comparison to using
stationary actuators.

The study of the control of a PDE-modeled DPS dates back to the 1960s (see the
survey [29]). For fundamental results, see the textbooks [7, 28, 1]. Although
it is possible to categorize prior work by whether the input operator is
bounded, our literature review is categorized by the location of the
actuation. When actuation acts on the boundary of the spatial domain, it is
called boundary control.  The main complexity in boundary control is that the
input operator, which associates with the PDE’s boundary condition, is
unbounded. This is addressed in the tutorial [15]. Recent developments on the
design of boundary control uses the backstepping method [33], where a Volterra
transformation determines a stabilizing control by transforming the system
into a stable target system. When actuation acts in the interior of the
spatial domain, it is called distributed control. For distributed control, the
DPS is actuated by in-domain actuators that are either stationary or mobile.

The problem of determining the location of stationary actuators is called the
actuator placement problem. Actuator placement has been studied for optimality
in the sense of linear-quadratic (LQ) [24], H2 [25], and H∞ [17]. The author
of [24] studies the actuator placement problem with the LQ performance
criterion. The actuators’ locations are chosen to minimize the operator norm
or trace norm of the Riccati operator solved from an algebraic Riccati
equation associated with an infinite-dimensional system. If the input operator
is compact and continuous with respect to actuator location in the operator
norm, then a solution exists for the problem minimizing the operator norm of
the Riccati operator [24, Theorem 2.6], under stabilizability and
detectability assumptions. When computing the optimal actuator locations, if
the approximated Riccati operator converges to the original Riccati operator
at each actuator location, then the approximate optimal actuator locations
converge to the exact optimal locations [24, Theorem 3.5]. For the above
results to hold when minimizing the trace-norm of the Riccati operator, the
input and output spaces have to be finite-dimensional [24, Theorems 2.10 and
3.9] in addition to the assumptions stated above.

The authors of [25] design optimal actuator placement by minimizing the
H2-control performance criterion, which minimizes the L2-norm of the linear
output of a linear system, subject state disturbances. Roughly speaking,
H2-control performance reduces the response to the disturbances while setting
a zero initial condition, whereas the LQ performance reduces the response to
the initial condition in a disturbance-free setting. For disturbances with
known or unknown spatial distribution, the trace of the Riccati solution
(scaled by the disturbance’s spatial distribution) or operator norm of the
Riccati solution are minimized, respectively, where the existence of a
solution and convergence to the exact optimal solution of the approximate
optimal solution are guaranteed. In [17], the H∞-performance criterion is
minimized for actuator placement. Specifically, the actuators are placed in
the locations that yield infimum of the attenuation bound (upper bound of the
infinity norm of the closed-loop disturbance-to-output transfer function). The
conditions for the existence of a solution and convergence to the exact
optimal placement of the approximate optimal placement are provided.

Geometric approaches have also been proposed for actuator placement. For
example, a modified centroidal Voronoi tessellation (mCVT) yields locations of
actuators and sensors that yields least-squares approximate control and
estimation kernels for a parabolic PDE [10]. The input operator is designed by
maximizing the H2-norm of the input-to-state transfer function, whereas the
kernel of the state feedback is obtained using the Riccati solution. Next,
mCVT determines the partition such that the actuator and sensor locations
achieve optimal performance (in the sense of least-squares) to the input
operator and state feedback kernel, respectively. A comparison of various
performance criteria for actuator placement has been conducted for controlling
a simply supported beam [26] and a diffusion process [27]. It has been
analyzed that maximizing the minimum eigenvalue of the controllability gramian
is not a useful criterion. Because the lower bound of the eigenvalue is zero,
the minimum eigenvalue approaches zero as the dimension of approximation
increases [26, 27].

The guidance of mobile actuators is designed to improve the control
performance in comparison to stationary actuators. Various performance
criteria have been proposed for guidance. In [13], a mobile heat source is
steered to maintain a spatially uniform temperature distribution in 1D using
the optimal control method. The formulation uses a finite-dimensional
approximation for modeling the process and evaluating performance.
Additionally, the admissible locations of the heat source are chosen within a
discrete set that yields approximate controllability requirements. Algorithms
are provided to solve the proposed problem with considerations on real-time
implementation and hardware constraints. Experimental results demonstrate the
performance of the proposed scheme. The authors of [14] propose an
optimization framework that steers mobile actuators to control a reaction-
diffusion process. A specific cost function, consisting of norms of control
input and measurement of the DPS and the steering force, is minimized subject
to dynamics of the actuator’s motion and the PDE, and bounds on the control
input and state of the DPS. The implementation of the framework is emphasized
by discrete mechanics and model predictive control which yield computationally
tractable solutions, in addition to an approximation of the PDE and a discrete
set of admissible actuator locations.

The problem of ultraviolet curing using a mobile radiant actuator is
investigated in [38], where the curing process is modeled by a 1D nonlinear
PDE. Both the radiant power and scanning velocity of the actuator are computed
for reaching a target curing state. A dual extended Kalman filter is applied
to estimate the state and parameters of the curing process for feedback
control, based on the phases of curing. In [11], a navigation problem is
studied in which a mobile agent moves through a diffusion process represented
by a hazardous field with given initial and terminal positions. Such a
formulation may be applied to emergency evacuation guidance from an indoor
environment with carbon monoxide. Both problems with minimum time and minimum
accumulated effects of hazards are formulated, and closed-form solutions are
derived using the Hamiltonian. A Lyapunov-based guidance strategy for
collocated sensors and actuators to estimate and control a diffusion process
is proposed in [8]. The decentralized guidance of the actuators for
controlling a diffusion process to a zero state is derived by constructing
suitable Lyapunov functions. The same methodology is applied to construct a
distributed consensus filter via the network among agents to improve state
estimation. A follow-up work [9] incorporates nonlinear vehicle dynamics in
addition.

The problem formulation in this paper includes a cost function that
simultaneously evaluates controlling the PDE-modeled DPS, referred to as the
PDE cost, and steering the mobile actuators, referred to as the mobility cost.
The PDE cost is a quadratic cost of the PDE state and control, whose optimal
value can be obtained by solving an operator-valued differential Riccati
equation. Our results are based on the related work [3], which establishes
Bochner integrable solutions of finite-horizon Riccati integral equations
(with values in Schatten p-classes) associated with infinite-dimensional
systems. The existence conditions for the solution of exact and approximate
Riccati integral equations are established in [3]. The significance of the
Bochner integrable solution is that it allows the implementation of simple
numerical quadratures for computing the approximated solution of Riccati
integral equations. In [3], the Riccati solution is applied in a sensor
placement problem, which computes optimal sensor locations that minimize the
trace of the covariance operator of the Kalman filter of a diffusion-advection
process. The same cost has been used in an optimization framework for mobile
sensor’s motion planning in [2]. The existence of a solution of the
optimization problem is established under the assumption that the integral
kernel of the output operator is continuous with respect to the location of
the sensor [2, Definition 4.5]. This assumption permits the construction of a
compact set of operators [2, Lemma 4.6] over which the cost function is
continuous, and hence establishes the existence of a solution. The assumption
is also made on the input operator in this paper, which allows the derivation
of a vital result on the Riccati operator’s continuity with respect to the
actuator trajectory (Lemma 3). The continuity property plays a crucial role in
establishing the existence of the proposed problem’s solution and the
convergence to the exact optimal solution of the approximate optimal solution.
The existence of a solution is established in using the fact that a weakly
sequentially lower semicontinuous function attains its minimum on a weakly
sequentially compact set over a normed linear space. In addition to the
assumptions made for the existence of a solution, a stringent (yet with
reasonable physical interpretation) requirement is placed on the admissible
set to yield compactness, which leads to convergence of the approximate
optimal solution.  The convergence is in the sense that when evaluating the
exact and approximate optimal solutions by the original cost function, the
difference becomes arbitrarily small as the dimension of approximation
increases.

The contributions of this paper are threefold. First, we propose an
optimization framework for controlling a PDE-modeled system using a team of
mobile actuators. The framework incorporates both controlling the process and
steering the mobile actuators. The former is handled by the linear-quadratic
regulator of the PDE. The latter is taken care of by designing generic cost
functions that address the constraints and limitations of the vehicles
carrying the actuators. Second, existence conditions of a solution of the
proposed problem are established. It turns out that the conditions are
generally satisfied in engineering problems, which allows the results to be
applied to a wide range of applications. Third, conditions are also
established under which the optimal solution computed using approximations
converges to the exact optimal solution.  The convergence is in the sense that
the cost function of the exact problem evaluated at these two solutions
becomes arbitrarily close as the dimensional of the approximation goes to
infinity. The convergence  is verified in numerical studies and confirms the
appropriateness of the optimal solution of the approximation.

The proposed framework is well-suited for the limited onboard resources of
mobile actuators in the following two aspects: (1) it adopts a finite-horizon
optimization scheme that characterizes the resource limitation more precisely
than the alternative approaches that do not specify a terminal time, such as
an infinite-horizon optimization or Lyapunov-based method; and (2) it provides
an intermediate step for the optimization problem that characterizes the
limited resources as inequality constraints, because the constraints can be
used to augment the cost function and turned into the proposed form using the
method of Lagrange multipliers. Potential applications of this work include
forest firefighting using unmanned aerial vehicles and oil spill removal or
harmful algae containment using autonomous skimmer boats. A preliminary
version of this paper [4] considered controlling a 1D diffusion process by a
team of mobile actuators. The results in this paper extends the controlled
process in [4] to a 2D diffusion-advection process and generalizes the
mobility cost therein. Furthermore, the proofs of the existence of a solution
and convergence of the approximated optimal solution are presented for the
first time in this paper. The results for a dual estimation framework can be
found in [5].

The paper adopts the following notation. The symbols $\mathbb{R}$,
$\mathbb{R}^{+}$, and $\mathbb{N}$ denote the set of real numbers, nonnegative
real numbers, and nonnegative integers, respectively. The boundary of a set
$M$ is denoted by $\partial M$. The $n$-nary Cartesian power of a set $M$ is
denoted by $M^{n}$. A continuous embedding is denoted by $\hookrightarrow$. We
use $\left|\cdot\right|$ and $\left\lVert\cdot\right\rVert$ for the norm
defined on a finite- and infinite-dimensional space, respectively, with
subscript indicating type. The superscript ∗ denotes an optimal variable or an
optimal value, whereas ⋆ denotes the adjoint of a linear operator. The
transpose of a matrix $A$ is denoted by $A^{\top}$. An $n\times n$-dimensional
identity matrix is denoted by $I_{n}$. We denote by $0_{n\times m}$ and
$1_{n\times m}$ an $n\times m$-dimensional matrix with all entries being $0$
and $1$, respectively. The term guidance refers to the steering of the mobile
actuators, whereas the term control refers to the control input to the DPS.
For an optimization problem (P0) that minimizes cost function $J(\cdot)$ over
variable $x$ subject to constraints, we use $J_{\text{(P0)}}(x)$ to denote the
cost function of (P0) evaluated at $x$. Specifically, $J^{*}_{\text{(P0)}}(x)$
indicates that the optimal value of (P0) is attained when the cost function is
evaluated at $x$.

Section II introduces relevant mathematical background, including
representation of a diffusion-advection equation by an infinite-dimensional
system, the associated LQ optimal control, and its finite-dimensional
approximation. Section III introduces the proposed optimization problem and
its equivalent problem. Conditions for the existence of a solution are stated.
Section IV details the computation of an optimal solution using finite-
dimensional approximations. Conditions for the convergence to the exact
optimal solution of the approximate optimal solution are stated. A gradient-
based method is applied to find an optimal solution. Section V provides two
numerical examples to illustrate optimal guidance and control solved by the
proposed method. Section VI summarizes the paper and discusses ongoing work.

##  2 Background

This paper is motivated by the problem of controlling the following diffusion-
advection process on a two-dimensional spatial domain
$\Omega=[0,1]\times[0,1]$ with a team of $m_{a}$ mobile actuators:

$\displaystyle\frac{\partial z(x,y,t)}{\partial t}=$ $\displaystyle\
a\nabla^{2}z(x,y,t)-\mathbf{v}\cdot\nabla z(x,y,t)$ $\displaystyle\
+\sum_{i=1}^{m_{a}}(\mathcal{B}_{i}u_{i})(x,y,t),$ (1) $\displaystyle
z(\cdot,\cdot,t)|_{\partial\Omega}=$ $\displaystyle\ 0,$ (2) $\displaystyle
z(x,y,0)=$ $\displaystyle\ z_{0}(x,y),$ (3)

where $z(\cdot,\cdot,t)$ is the state at time $t$,
$\mathbf{v}\in\mathbb{R}^{2}$ is the velocity field for advection, and $u_{i}$
is the control implemented by actuator $i$, with the actuation characterized
spatially by $\mathcal{B}_{i}$. The state $z$ lives in the state space
$L^{2}(\Omega)$. A representative model of the actuation dispensed by each
actuator is Gaussian-shaped and centered at the actuator $i$’s location
$(x_{i},y_{i})$ with a bounded support such that

$\mathcal{B}_{i}(x,y)=\left\\{\begin{aligned}
\frac{1}{2\pi\sigma_{i}^{2}}&\text{exp}\left(-\frac{(x-x_{i})^{2}}{\sigma_{i}^{2}}-\frac{(y-y_{i})^{2}}{\sigma_{i}^{2}}\right)\\\
&\quad\text{if }|x-x_{i}|\leq\sigma_{i}\text{ and }|y-y_{i}|\leq\sigma_{i},\\\
0&\quad\text{otherwise},\end{aligned}\right.$ (4)

where the parameter $\sigma_{i}$ determines the spatial influence of the
actuation, which is concentrated mostly at the location of the actuator and
disperses to the surrounding with an exponential decay.

###  2.1 Dynamics of the mobile actuators

Assume the mobile actuators have linear dynamics, so that the dynamics of
actuator $i$ are

$\dot{\xi}_{i}(t)=\alpha_{i}\xi_{i}(t)+\beta_{i}p_{i}(t),\quad\xi_{i}(0)=\xi_{i,0},$
(5)

where $\xi_{i}(t)\in\mathbb{R}^{n}$ $(n\geq 2)$ and $p_{i}(t)\in
P_{i}\subset\mathbb{R}^{m}$ are the state and guidance at $t$, respectively.
Assume that system (5) is controllable. The first two elements of $\xi_{i}(t)$
are the horizontal and vertical position, $x_{i}(t)$ and $y_{i}(t)$, of the
actuator in the 2D domain. One special case would be a single integrator,
where $\xi_{i}(t)\in\mathbb{R}^{2}$ is the position,
$p_{i}(t)\in\mathbb{R}^{2}$ is the velocity commands, $\alpha_{i}=0_{2\times
2}$, and $\beta_{i}=I_{2}$.

For conciseness, we concatenate the states and guidance of all actuators,
respectively, and use one dynamical equation to characterize the dynamics of
all agents:

$\dot{\xi}(t)=\alpha\xi(t)+\beta p(t),\quad\xi(0)=\xi_{0},$ (6)

where matrices $\alpha$ and $\beta$ are assembled from $\alpha_{i}$ and
$\beta_{i}$ for $i\in\\{1,2,\dots,m_{a}\\}$, respectively and are consistent
with the concatenation for $\xi$ and $p$. With a slight abuse of notation, we
use $n$ for the dimension of $\xi(t)$ and $m$ for the dimension of $p(t)$.
Define the admissible set of guidance $P:=P_{1}\times P_{2}\times\dots\times
P_{m_{a}}$ such that $p(t)\in P$ for $t\in[0,t_{f}]$. Let
$M\in\mathbb{R}^{2m_{a}\times n}$ be a matrix such that $M\xi(t)$ is a vector
of locations of the actuators.

###  2.2 Abstract linear system and linear-quadratic regulation

To describe the dynamics of PDE (1)–(3), consider the following abstract
linear system:

$\dot{\mathcal{Z}}(t)=\mathcal{A}\mathcal{Z}(t)+\mathcal{B}(M\xi(t),t)u(t),\qquad\mathcal{Z}(0)=\mathcal{Z}_{0},$
(7)

where $\mathcal{Z}(\cdot)$ is the state within state space
$\mathcal{H}:=L^{2}(\Omega)$ and $u(\cdot)$ is the control within the control
space ${u(t)\in U\subseteq\mathbb{R}^{m_{a}}}$ for $t\in[0,t_{f}]$. In the
case of diffusion-advection process (1), for $\phi\in\mathcal{H}$,

$(\mathcal{A}\phi)(x,y)=a\nabla^{2}\phi(x,y)-\mathbf{v}\cdot\nabla\phi(x,y),$
(8)

where the operator $\mathcal{A}$ has domain
$\text{Dom}(\mathcal{A})=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$. The input
operator $\mathcal{B}(M\xi(t),t)\in\mathcal{L}(U,\mathcal{H})$ is a function
of the actuator locations such that
$\mathcal{B}(M\xi(t),t)=[\mathcal{B}_{1}(M\xi_{1}(t),t),\dots,\mathcal{B}_{m_{a}}(M\xi_{m_{a}}(t),t)]^{\top}$,
where $\mathcal{B}_{i}(\cdot,t)\in L^{2}(\Omega)$ for all $t\in[0,t_{f}]$ and
$i\in\\{1,2,\dots,m_{a}\\}$. A special case is the time-invariant input
operator in (4). Since the actuator state $\xi(t)$ is a function of time $t$,
we sometimes use $\mathcal{B}(t)$ for brevity.

The operator $\mathcal{A}:\text{Dom}(\mathcal{A})\rightarrow\mathcal{H}$ is an
infinitesimal generator of a strongly continuous semigroup $\mathcal{S}(t)$ on
$\mathcal{H}$. Subsequently, the dynamical system (7) has a unique mild
solution $\mathcal{Z}\in C([0,t_{f}];\mathcal{H})$ for any
$\mathcal{Z}_{0}\in\mathcal{H}$ and any $u\in L^{2}([0,t_{f}];U)$ such that
$\mathcal{Z}(t)=\mathcal{S}(t)\mathcal{Z}_{0}+\int_{0}^{t}\mathcal{S}(t-\tau)\mathcal{B}(\xi(\tau),\tau)u(\tau)\text{d}\tau$.

Similar to a finite-dimensional linear system, a linear-quadratic regulator
(LQR) problem can be formulated with respect to (7), which looks for a control
$u(\cdot)\in L^{2}([0,t_{f}];U)$ that minimizes the following quadratic cost:

$\displaystyle J(\mathcal{Z},u):=$
$\displaystyle\int_{0}^{t_{f}}\langle\mathcal{Z}(t),\mathcal{Q}(t)\mathcal{Z}(t)\rangle+u(t)^{\top}Ru(t)\text{d}t$
$\displaystyle+\langle\mathcal{Z}(t_{f}),\mathcal{Q}_{f}\mathcal{Z}(t_{f})\rangle,$
(9)

where $\mathcal{Q}(t)\in\mathcal{L}(\mathcal{H})$ and
$\mathcal{Q}_{f}\in\mathcal{L}(\mathcal{H})$ are self-adjoint and nonnegative,
which evaluates the running cost and terminal cost of the PDE state. The
coefficient $R$ is an $m_{a}\times m_{a}$-dimensional symmetric and positive
definite matrix that evaluates the control effort. We refers to
$J(\mathcal{Z},u)$ as the PDE cost.

Analogous to the finite-dimensional LQR, an optimal control $u^{*}$ that
minimizes the quadratic cost (9) is

$u^{*}(t)=-R^{-1}\mathcal{B}^{\star}(t)\Pi(t)\mathcal{Z}(t),$ (10)

where $\Pi$ is an operator that associates with the following backward
differential operator-valued Riccati equation:

$\dot{\Pi}(t)=-\mathcal{A}^{\star}\Pi(t)-\Pi(t)\mathcal{A}-\mathcal{Q}(t)\\\
+\Pi(t)\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(t)\Pi(t)$ (11)

with terminal condition $\Pi(t_{f})=\mathcal{Q}_{f}$, where
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(t)$ is short for
$\mathcal{B}(t)R^{-1}\mathcal{B}^{\star}(t)$. Before we proceed to state the
conditions for the existence of a unique solution of (11), we introduce the
$\mathcal{J}_{q}$-class as follows.

Denote the trace of a nonnegative operator $A\in\mathcal{L}(\mathcal{H})$ by
$\text{Tr}(A)$, where
$\text{Tr}(A):=\sum_{k=1}^{\infty}\langle\phi_{k},A\phi_{k}\rangle$ for any
orthonormal basis $\\{\phi_{k}\\}_{k=1}^{\infty}$ of $\mathcal{H}$ (the trace
is independent of the choice of basis functions). For $1\leq q<\infty$, let
$\mathcal{J}_{q}(\mathcal{H})$ denote the set of all bounded operators
$\mathcal{L}(\mathcal{H})$ such that
$\text{Tr}((\sqrt{A^{\star}A})^{q})<\infty$ [3]. If
$A\in\mathcal{J}_{q}(\mathcal{H})$, then the $\mathcal{J}_{q}$-norm of $A$ is
defined as $\left\lVert
A\right\rVert_{\mathcal{J}_{q}(\mathcal{H})}:=(\text{Tr}((\sqrt{A^{\star}A})^{q}))^{1/q}<\infty$.
The class $\mathcal{J}_{1}(\mathcal{H})$ and $\mathcal{J}_{2}(\mathcal{H})$
are known as the space of trace operators and the space of Hilbert-Schmidt
operators, respectively. Note that a continuous embedding
${\mathcal{J}_{q_{1}}(\mathcal{H})\hookrightarrow\mathcal{J}_{q_{2}}(\mathcal{H})}$
holds if $1\leq q_{1}<q_{2}\leq\infty$. In other words, if
$A\in\mathcal{J}_{q_{1}}(\mathcal{H})$, then
$A\in\mathcal{J}_{q_{2}}(\mathcal{H})$ and ${\left\lVert
A\right\rVert_{\mathcal{J}_{q_{2}}(\mathcal{H})}\leq\left\lVert
A\right\rVert_{\mathcal{J}_{q_{1}}(\mathcal{H})}}$.

The existence of a mild solution of (11) is established via Lemma 1. We omit
the proof of this lemma because it is a direct consequence of [3, Theorem
3.6].

Consider the following assumptions with $1\leq q<\infty$:

  1. (A1)

$\mathcal{Q}_{f}\in\mathcal{J}_{q}(\mathcal{H})$ and $\mathcal{Q}_{f}$ is
nonnegative.

  2. (A2)

$\mathcal{Q}(\cdot)\in L^{1}([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ and
$\mathcal{Q}(t)$ is nonnegative for all $t\in[0,t_{f}]$.

  3. (A3)

$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\cdot)\in
L^{\infty}([0,t_{f}];\mathcal{L}(\mathcal{H}))$ and
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(t)$ is nonnegative for
$t\in[0,t_{f}]$.

######  Lemma 1.

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathcal{S}(t)$ be a
strongly continuous semigroup on $\mathcal{H}$. Suppose assumptions (A1)–(A3)
hold. Then, the equation

$\Pi(t)=\mathcal{S}^{\star}(t_{f}-t)\mathcal{Q}_{f}\mathcal{S}(t_{f}-t)+\int_{t}^{t_{f}}\mathcal{S}^{\star}(\tau-t)\\\
\left(\mathcal{Q}(\tau)-\Pi(\tau)\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\tau)\Pi(\tau)\right)\mathcal{S}(\tau-t)\text{d}\tau$
(12)

provides a unique mild solution to (11) in the space
$L^{2}([0,t_{f}];\mathcal{J}_{2q}(\mathcal{H}))$. The solution also belongs to
$C([0,t_{f}];$ $\mathcal{J}_{q}(\mathcal{H}))$ and is pointwise self-adjoint
and nonnegative. Furthermore, if $\mathcal{Q}(\cdot)\in
C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ and
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\cdot)\in
C([0,t_{f}];\mathcal{L}(\mathcal{H}))$, then $\Pi$ is a weak solution to (11).

The equality introduced next in Lemma 2 allows for turning the optimal
quadratic PDE cost into a quadratic term associated with the initial condition
of the PDE and the Riccati operator. We state it without proof because it can
be established by integrating
$\text{d}\langle\mathcal{Z}(t),\Pi(t)\mathcal{Z}(t)\rangle/\text{d}t$ from $0$
to $t_{f}$; the differentiability of
$\langle\mathcal{Z}(t),\Pi(t)\mathcal{Z}(t)\rangle$ is proven in [7, Theorem
6.1.9].

######  Lemma 2.

Suppose $\Pi(t)$ is a mild solution to (11), given by (12). For every
$\mathcal{Z}_{0}\in\mathcal{H}$, the optimal PDE cost (9) satisfies the
equality
$J(\mathcal{Z}^{*},u^{*})=\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$,
where $\mathcal{Z}^{*}$ is the state that follows the dynamics (7) under
optimal control $u^{*}$ of (10), and $\Pi(0)$ is the solution (12) evaluated
at $t=0$.

The following assumption is vital to the main results in this paper.

  1. (A4)

The input operator $\mathcal{B}_{i}(x,t)$ is continuous with respect to
location $x\in\mathbb{R}^{2}$ [2, Definition 4.5], that is, there exists a
continuous function $l:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$l(0)=0$ and
$\left\lVert\mathcal{B}_{i}(x,t)-\mathcal{B}_{i}(y,t)\right\rVert_{L^{2}(\Omega)}\leq
l(|x-y|_{2})$ for all $t\in[0,t_{f}]$, all $x,y\in\mathbb{R}^{2}$, and all
$i\in\\{1,2,\dots,m_{a}\\}$.

The actuators’ locations determine where the input is actuated and,
furthermore, how $\Pi(\cdot)$ evolves through (12). Since the input operator
$\mathcal{B}(\cdot,t)$ is a mapping of the actuators’ locations at time $t$,
the composite input operator
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\cdot)$ is a mapping of the
actuator state in $[0,t_{f}]$ and so is $\Pi(0)$ by (12), although the
actuator state is not explicitly reflected in the notation of
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\cdot)$ or $\Pi(0)$. Hence, we can
define the optimal PDE cost
$\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$ as a mapping of the
actuator state. Let $K:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$
such that $K(\zeta):=\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$.
Assumption (A4) plays an important role in yielding the continuity of the
mapping $K(\cdot)$ stated below in Lemma 3, whose proof is in the
supplementary material.

######  Lemma 3.

Suppose $\mathcal{Z}_{0}\in\mathcal{H}$. Let assumptions (A1)–(A3) hold with
$q=1$ and $\Pi\in C([0,t_{f}];\mathcal{J}_{1}(\mathcal{H}))$ be defined as in
(12). If assumption (A4) holds, then the mapping
$K:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$ such that
$K(\xi):=\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$ is continuous.

Approximations to (7) and (12) permit numerical computation. Consider a
finite-dimensional subspace $\mathcal{H}_{N}\subset\mathcal{H}$ with dimension
$N$. The inner product and norm of $\mathcal{H}_{N}$ are inherited from that
of $\mathcal{H}$. Let $P_{N}:\mathcal{H}\to\mathcal{H}_{N}$ denote the
orthogonal projection of $\mathcal{H}$ onto $\mathcal{H}_{N}$. Let
$Z_{N}(t):=P_{N}\mathcal{Z}(t)$ and $S_{N}(t):=P_{N}\mathcal{S}(t)P_{N}$
denote the finite-dimensional approximation of $\mathcal{Z}(t)$ and
$\mathcal{S}(t)$, respectively. A finite-dimensional approximation of (7) is

$\displaystyle\dot{Z}_{N}(t)=$ $\displaystyle\
A_{N}Z_{N}(t)+B_{N}(M\xi(t),t)u(t),$ (13) $\displaystyle Z_{N}(0)=$
$\displaystyle\ Z_{0,N}:=P_{N}\mathcal{Z}_{0},$ (14)

where $A_{N}\in\mathcal{L}(\mathcal{H}_{N})$ and
$B_{N}(M\xi(t),t)\in\mathcal{L}(U,\mathcal{H}_{N})$ are approximations of
$\mathcal{A}$ and $\mathcal{B}(M\xi(t),t)$, respectively. Since the actuator
state $\xi(t)$ is a function of time $t$, we sometimes use $B_{N}(t)$ for
brevity. Correspondingly, the finite-dimensional approximation of (12) is

$\Pi_{N}(t)=S^{\star}_{N}(t_{f}-t)Q_{f,N}S_{N}(t_{f}-t)+\int_{t}^{t_{f}}S^{\star}_{N}(\tau-t)\\\
\left(Q_{N}(\tau)-\Pi_{N}(\tau)\bar{B}_{N}\bar{B}_{N}^{\star}(\tau)\Pi_{N}(\tau)\right)S_{N}(\tau-t)\text{d}\tau,$
(15)

where $Q_{N}=P_{N}\mathcal{Q}P_{N}$, $Q_{fN}=P_{N}\mathcal{Q}_{f}P_{N}$, and
$\bar{B}_{N}\bar{B}_{N}^{\star}(\tau)$ is short for
$B_{N}(\tau)R^{-1}B_{N}^{\star}(\tau)$.

The optimal control $u_{N}^{*}$ that minimizes the approximated PDE cost

$J_{N}(Z_{N},u_{N}):=\langle Z_{N}(t_{f}),Q_{f,N}Z_{N}(t_{f})\rangle\\\
+\int_{0}^{t_{f}}\langle
Z_{N}(t),Q_{N}(t)Z_{N}(t)\rangle+u_{N}^{\top}(t)Ru_{N}(t)\text{d}t$ (16)

is analogous to (10):

$u_{N}^{*}=-R^{-1}B_{N}^{\star}(t)\Pi_{N}(t)Z_{N}(t),$ (17)

where $\Pi_{N}(t)$ is a solution of (15).

The following assumptions are associated with the approximations:

  1. (A5)

Both $\mathcal{Q}_{f}$ and sequence $\\{Q_{f,N}\\}_{N=1}^{\infty}$ are
elements of $\mathcal{J}_{q}(\mathcal{H})$. Both $\mathcal{Q}_{f}$ and
$Q_{f,N}$ are nonnegative for all $N\in\mathbb{N}$ and
$\left\lVert\mathcal{Q}_{f}-Q_{f,N}\right\rVert_{\mathcal{J}_{q}(\mathcal{H})}\rightarrow
0$ as $N\rightarrow\infty$.

  2. (A6)

Both $\mathcal{Q}(\cdot)$ and sequence $\\{Q_{N}(\cdot)\\}_{N=1}^{\infty}$ are
elements of $L^{1}([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$. Both
$\mathcal{Q}(\tau)$ and $Q_{N}(\tau)$ are nonnegative for all
$\tau\in[0,t_{f}]$ and all $N\in\mathbb{N}$ and satisfy
$\int_{0}^{t}\left\lVert\mathcal{Q}(\tau)-Q_{N}(\tau)\right\rVert_{\mathcal{J}_{q}(\mathcal{H})}\text{d}\tau\rightarrow
0$ for all $t\in[0,t_{f}]$ as $N\rightarrow\infty$.

  3. (A7)

Both $\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(\cdot)$ and sequence
$\\{\bar{B}_{N}\bar{B}_{N}^{\star}(\cdot)\\}_{N=1}^{\infty}$ are elements of
$L^{\infty}([0,t_{f}];\mathcal{L}(\mathcal{H}))$. Both
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(t)$ and
$\bar{B}_{N}\bar{B}_{N}^{\star}(t)$ are nonnegative for all $t\in[0,t_{f}]$
and all $N\in\mathbb{N}$ and satisfy

$\underset{t\in[0,t_{f}]}{\operatorname*{ess\,sup}}\left\lVert\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}(t)-\bar{B}_{N}\bar{B}_{N}^{\star}(t)\right\rVert_{\text{op}}\rightarrow
0$ (18)

as $N\rightarrow\infty$ ($\left\lVert\cdot\right\rVert_{\text{op}}$ denotes
the operator norm).

Note that the assumptions (A1), (A2), and (A3) are contained in (A5), (A6),
and (A7), respectively.

The next theorem states the convergence of an approximate solution of the
Riccati equation, which is reproduced from [3, Theorem 3.5] and hence stated
without a proof.

######  Theorem 4.

Suppose $\mathcal{S}(t)$ is a strongly continuous semigroup of linear
operators over a Hilbert space $\mathcal{H}$ and that $\\{S_{N}(t)\\}$ is a
sequence of uniformly continuous semigroup over the same Hilbert space that
satisfy, for each $\phi\in\mathcal{H}$

$\left\lVert\mathcal{S}(t)\phi-S_{N}(t)\phi\right\rVert\rightarrow
0,\quad\left\lVert\mathcal{S}^{\star}(t)\phi-
S_{N}^{\star}(t)\phi\right\rVert\rightarrow 0$ (19)

as $N\rightarrow\infty$, uniformly in $[0,t_{f}]$. Suppose assumptions
(A5)–(A7) hold. If $\Pi(\cdot)\in C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$
is a solution of (12) and $\Pi_{N}(\cdot)\in
C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ is the sequence of solution of
(15), then

$\underset{t\in[0,t_{f}]}{\sup}\left\lVert\Pi(t)-\Pi_{N}(t)\right\rVert_{\mathcal{J}_{q}(\mathcal{H})}\rightarrow
0$ (20)

as $N\rightarrow\infty$.

The following assumption and lemma are analogous to (A4) and Lemma 3,
respectively:

  1. (A8)

The approximated input operator $B_{i,N}(x,t)$ is continuous with respect to
location $x\in\mathbb{R}^{2}$, that is, there exists a continuous function
$l_{N}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that $l_{N}(0)=0$ and
$\left\lVert B_{i,N}(x,t)-B_{i,N}(y,t)\right\rVert_{L^{2}(\Omega)}\leq
l_{N}(|x-y|_{2})$ for all $t\in[0,t_{f}]$, all $x,y\in\mathbb{R}^{2}$, and all
$i\in\\{1,2,\dots,m_{a}\\}$.

Similar to the mapping $K(\cdot)$ in Lemma 3, the optimal approximated PDE
cost can be characterized as a mapping of the actuator state through (15),
where the continuity is established in Lemma 5, whose proof is in the
supplementary material.

######  Lemma 5.

Suppose $Z_{0,N}\in\mathcal{H}_{N}$. Let assumptions (A5)–(A7) hold and
$\Pi_{N}(t)$ be defined as in (15). If assumption (A8) holds, then the mapping
$K_{N}:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$ such that
$K_{N}(\xi):=\langle Z_{0,N},\Pi_{N}(0)Z_{0,N}\rangle$ is continuous.

##  3 Problem formulation

This paper seeks to derive the guidance and control input of each actuator
such that the state $\mathcal{Z}$ of the abstract linear system (7) can be
driven to zero. Specifically, consider the following problem:

$\displaystyle\underset{\begin{subarray}{c}u\in L^{2}([0,t_{f}];U)\\\ p\in
L^{2}([0,t_{f}];P)\end{subarray}}{\text{minimize}}$ $\displaystyle
J(\mathcal{Z},u)+J_{\text{m}}(\xi,p)$ (P) subject to
$\displaystyle\dot{\mathcal{Z}}(t)=\mathcal{A}\mathcal{Z}(t)+\mathcal{B}(t)u(t),\quad\mathcal{Z}(0)=\mathcal{Z}_{0},$
$\displaystyle\dot{\xi}(t)=\alpha\xi(t)+\beta p(t),\quad\xi(0)=\xi_{0},$

where
$J_{\text{m}}(\xi,p):=\int_{0}^{t_{f}}h(\xi(t),t)+g(p(t),t)\text{d}t+h_{f}(\xi(t_{f}))$
is the cost associated with the motion of the actuators, named the mobility
cost, such that the mappings
$h:\mathbb{R}^{n}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$ and
$g:\mathbb{R}^{m}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$ evaluate the
running state cost and running guidance cost, respectively, and the mapping
$h_{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ evaluates the terminal state
cost.

The running state cost $h(\cdot,\cdot)$ may characterize restrictions to
actuator state. For example, a Gaussian-type function with its peak in the
center of the spatial domain, i.e.,

$h(\left[\begin{smallmatrix}x\\\ y\end{smallmatrix}\right],t)=\\\
\frac{1}{2\pi\sigma_{x}(t)\sigma_{y}(t)}\text{exp}\left(-\frac{(x-0.5)^{2}}{\sigma_{x}^{2}(t)}-\frac{(y-0.5)^{2}}{\sigma_{y}^{2}(t)}\right),$
(21)

where $\sigma_{x}(t),\sigma_{y}(t)>0$ and $x,y\in[0,1]$, can model a hazardous
field that may shorten the life span of an actuator. The integral of this
function in the interval $[0,t_{f}]$ evaluates the accumulated exposure of the
mobile actuator along its trajectory, which may need to be contained as small
as possible (see [11]). Another example is the artificial potential field
[16], cast as a soft constraint, that penalizes the trajectory when it passes
an inaccessible region such as an obstacle. The running guidance cost
$g(\cdot,\cdot)$ may be the absolute value or a quadratic function of the
guidance, which characterizes the total amount (of fuel) or energy for
steering, respectively. And the terminal state cost $h_{f}(\cdot)$ may
characterize restrictions of the terminal state of the mobile actuators. For
example, if an application specifies terminal positions, then $h_{f}(\cdot)$
may be a quadratic function that penalizes the deviation of the actual
terminal positions.

The formulation in (P) provides an intermediate step for minimizing the PDE
cost subject to mobility constraints, in addition to the dynamics constraints.
The mobility constraints are characterized by inequalities of $h_{f}(\cdot)$
and the integrals of $h(\cdot,\cdot)$ and $g(\cdot,\cdot)$, because these
constraints can be used to augment the cost function and turned into the form
of (P) using the method of Lagrange multipliers.

An equivalent problem of (P) can be derived using Lemma 2. For an arbitrary
admissible guidance $p$, the actuator trajectory $\xi$ is determined following
the dynamics (6), which also determines the input operator
$\mathcal{B}(\xi(\cdot),\cdot)$. By Lemma 2, the control $u$ that minimizes
the cost function of (P)—specifically, the PDE cost $J(\mathcal{Z},u)$—is
given by (10), and the minimum PDE cost is
$\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$, where $\Pi(0)$ is the
mild solution of (12) with actuator trajectory steered by guidance $p$. Hence,
we derive the following problem equivalent to (P):

$\displaystyle\underset{p\in L^{2}([0,t_{f}];P)}{\text{minimize}}$
$\displaystyle\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle+J_{\text{m}}(\xi,p)$
(P1) subject to $\displaystyle\dot{\xi}(t)=\alpha\xi(t)+\beta
p(t),\quad\xi(0)=\xi_{0},$

where $\Pi(0)$ is defined in (12) with $t=0$.

To prove the existence of a solution to (P1), we make the following
assumptions on the admissible set of guidance and the functions composing the
mobility cost:

  1. (A9)

The set of admissible guidance $P\subset\mathbb{R}^{m}$ is closed and convex.

  2. (A10)

The mappings $h:\mathbb{R}^{n}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$,
$g:\mathbb{R}^{m}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$, and
$h_{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ are continuous. For every
$t\in[0,t_{f}]$, the function $g(\cdot,t)$ is convex.

  3. (A11)

There exists a constant $d_{1}>0$ with $g(p,t)\geq d_{1}|p|_{2}^{2}$ for all
$(p,t)\in P\times[0,t_{f}]$.

Assumptions (A9)–(A11) are generally satisfied in applications with vehicles
carrying the actuators. Assumption (A9) is a physically reasonable
characterization of the steering of a vehicle, where the admissible steering
is generally a continuum with attainable limits within its range. Assumption
(A10) places a general continuity requirement on the cost functions and a
convexity requirement on the steering cost function. Assumption (A11) requires
the function $g(p,t)$ to be bounded below by a quadratic function of the
guidance $p$ for all $t$, which is generally satisfied, e.g., with $g$ itself
being a quadratic function of $p$. These assumptions are applied in Theorem 6
below regarding the existence of a solution of (P1), whose proof is in
Appendix A. Subsequently, the solution to (P1) can be used to reconstruct the
solutions to (P), which is stated in Theorem 7 with its proof in Appendix B.

######  Theorem 6.

Consider problem (P1) and let assumptions (A1)–(A4) and (A9)–(A11) hold. Then
(P1) has a solution.

######  Theorem 7.

Consider problems (P) and (P1). Let assumptions (A4) and (A9)–(A11) hold. Let
$p^{*}$ be the optimal solution of (P1) and $u^{*}$ be the optimal control
obtained from (10) with actuator trajectory steered by $p^{*}$. Then $u^{*}$
and $p^{*}$ minimize problem (P).

The equivalent problem (P1) allows us to search for an optimal guidance $p$
such that the mobility cost plus the optimal PDE cost is minimized. The
control is no longer an optimization variable, because it is determined by the
LQR of the abstract linear system for arbitrary trajectories of the mobile
actuators.

##  4 Computation of optimal control and guidance

Approximation of the infinite-dimensional terms in problem (P) is necessary
when computing the optimal control and guidance. Hence, we replace the PDE
cost and dynamics of (P) by (16) and (13), respectively, and obtain the
following approximate problem (AP):

$\displaystyle\underset{\begin{subarray}{c}u\in L^{2}([0,t_{f}];U)\\\ p\in
L^{2}([0,t_{f}];P)\end{subarray}}{\text{minimize}}$ $\displaystyle
J_{N}(Z_{N},u)+J_{\text{m}}(\xi,p)$ (AP) subject to
$\displaystyle\dot{Z}_{N}(t)=A_{N}Z_{N}(t)+B_{N}(M\xi(t),t)u(t)$
$\displaystyle Z_{N}(0)=Z_{0,N},$
$\displaystyle\dot{\xi}(t)=\alpha\xi(t)+\beta p(t),$
$\displaystyle\xi(0)=\xi_{0}.$

Similar to (P), problem (AP) can be turned into an equivalent form using LQR
results for a finite-dimensional system:

$\displaystyle\underset{p\in L^{2}([0,t_{f}];P)}{\text{minimize}}$
$\displaystyle\langle Z_{0,N},\Pi_{N}(0)Z_{0,N}\rangle+J_{\text{m}}(\xi,p)$
(AP1) subject to $\displaystyle\dot{\xi}(t)=\alpha\xi(t)+\beta
p(t),\quad\xi(0)=\xi_{0},$

where $\Pi_{N}(0)$ is defined in (15) with $t=0$. Analogous to Theorems 6 and
7, the existence of a solution of (AP1) and how to use its solution to
reconstruct a solution for (AP) are stated in Theorem 8 below, whose proof is
presented in Appendix C.

######  Theorem 8.

Consider problem (AP1) and let assumptions (A5)–(A8) and (A9)–(A11) hold. Then
(AP1) has a solution, denoted by $p_{N}^{*}$. Let $u_{N}^{*}$ be the optimal
control obtained from (17) with actuator trajectory steered by $p_{N}^{*}$.
Then $u_{N}^{*}$ and $p_{N}^{*}$ minimize problem (AP).

An extension to Theorem 8 is that an optimal feedback control can be obtained
from (17) whenever the optimal guidance is solved from (AP) or (AP1).
Basically, when the trajectory is determined via the optimal guidance, a
feedback control can be implemented.

To establish convergence to the solution of (P1) of (AP1)’s solution, we need
to restrict the set of admissible guidance to a smaller set as introduced
below in assumption (A12).

  1. (A12)

There exist $p_{\max}>0$ and $a_{\max}>0$ such that the set of admissible
guidance is $\mathcal{P}(p_{\max},a_{\max}):=\\{p\in C([0,t_{f}];P):|p(t)|$ is
uniformly bounded by $p_{\max}$ and $|p(t_{1})-p(t_{2})|\leq
a_{\max}|t_{1}-t_{2}|,\ \forall t_{1},t_{2}\in[0,t_{f}]\\}$.

There are two perspectives to interpreting the assumption (A12).
Mathematically, (A12) requires the admissible guidance to be a continuous
function that is uniformly bounded and uniformly equicontinuous. These two
properties yield the sequential compactness of the set
$\mathcal{P}(p_{\max},a_{\max})$ by the Arzelà-Ascoli Theorem [30].
Practically, (A12) requires the input signal to be continuous and have bounds
$p_{\max}$ and $a_{\max}$ on the magnitude and the rate of change,
respectively. This requirement is reasonable and checkable because a
continuous signal is commonly used for smooth operation, and the bounds on
magnitude and changing rate are due to the physical limits of the motion of a
platform. For example, in the case of single integrator dynamics where $p$ is
the velocity command, $p_{\max}$ and $a_{\max}$ refer to the maximum speed and
maximum acceleration, respectively.  Moreover, since time discretization of
the signal is applied when computing the optimal guidance, as long as the
bound $p_{\max}$ on the magnitude of the signal is determined, then the
changing rate is bounded by $a_{\max}:=2p_{\max}/\Delta t_{\min}$ for the
smallest discrete interval length $\Delta t_{\min}$. Theorem 9 below states
the convergence of the approximate optimal solution with its proof in Appendix
D.

######  Theorem 9.

Consider problem (P1) and its finite-dimensional approximation (AP1). Let
assumptions (A4)–(A12) hold and let $p^{*}$ and $p_{N}^{*}$ denote the optimal
guidance of (P1) and (AP1), respectively. Then

$\lim_{N\rightarrow\infty}|J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: new
equivalent IOCA}}(p^{*})|=0.$ (22)

Furthermore, the cost function of (P1) evaluated at the guidance $p_{N}^{*}$
converges to the optimal cost of (P1)

$\lim_{N\rightarrow\infty}|J_{\eqref{prob: new equivalent
IOCA}}(p_{N}^{*})-J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})|=0.$ (23)

######  Remark 10.

Two implications of Theorem 9 follow. First, (22) implies that the optimal
cost of the approximated problem (AP1) converges to that of the exact problem
(P1), which justifies the approximation in (AP). Second, (23) implies that the
approximate optimal guidance $p_{N}^{*}$, when evaluated by the cost function
of (P1), yields a cost that is arbitrarily close to the exact optimal cost of
(P1). Since $p_{N}^{*}$ is computable and $p^{*}$ is not, the convergence in
(23) qualifies $p_{N}^{*}$ as an appropriate optimal guidance.

The convergence stated in Theorem 9 is established based on several earlier
stated results, including

  1. 1.

the input operator’s continuity with respect to location (assumption (A4)),
which leads to the continuity of the PDE cost with respect to actuator
trajectory (Lemma 3);

  2. 2.

existence of the Riccati operator (Lemma 1) and convergence of its
approximation (Theorem 4); and

  3. 3.

sequential compactness of the set of admissible guidance (assumption (A12)),
which leads to the continuity of the cost function with respect to guidance
(Lemma 12).

Notice that these key results, in an analogous manner, are also required in
[24] when establishing the convergence to the exact optimal actuator locations
of the approximate optimal locations [24, Theorem 3.5], i.e.,

  1. 1.

continuity with respect to location and compactness of the input operator [24,
Theorem 2.6], which lead to continuity of the Riccati operator with respect to
actuator locations [24, Theorem 2.6];

  2. 2.

existence of the Riccati operator [24, Theorem 2.3] and the convergence of its
approximation [24, Theorem 3.1]; and

  3. 3.

sequential compactness of the set of admissible locations, which is inherited
from the setting that the spatial domain is closed and bounded in a finite-
dimensional space.

Although the establishment of convergence is similar to the one in [24], the
cost function and type of Riccati equation are different: we have the
quadratic PDE cost plus generic mobility cost and differential Riccati
equation in this paper for control and actuator guidance versus the Riccati
operator’s norm as cost function and algebraic Riccati equation in [24] for
actuator placement.  The similarity comes from the infinite-dimensional nature
of PDEs such that approximation is necessary for computation, and convergence
in approximation qualifies the approximate optimal solutions.

###  4.1 Checking assumptions (A4)–(A12)

For the approximated optimal guidance to be a good proxy of the exact optimal
guidance, by Theorem 9, assumption (A4)–(A12) have to be checked to ensure the
convergence. We summarize methods for checking these assumptions here. (A4):
examine the explicit form of $\mathcal{B}$; (A5)–(A7): examine the explicit
form of the operators $\mathcal{Q}$, $\mathcal{Q}_{f}$, and
$\bar{\mathcal{B}}\bar{\mathcal{B}}^{\star}$ and their approximations; (A8):
examine the explicit form of $B_{N}$; (A9)–(A11): examine the explicit form of
$J_{\text{m}}$; (A12): examining the bounds on the magnitude and changing rate
of the admissible guidance.

###  4.2 Gradient-descent for solving problem (AP)

Define the costates $\lambda(t)\in\mathcal{H}_{N}$ and
${\mu(t)\in\mathbb{R}^{n}}$ associated with $Z_{N}(t)$ and $\xi(t)$,
respectively, for $t\in[0,t_{f}]$ and define the Hamiltonian:

$\displaystyle H(Z_{N}(t),\xi(t),u(t),p(t),\lambda(t),\mu(t))$
$\displaystyle=$ $\displaystyle\ \langle
Z_{N}(t),Q_{N}(t)Z_{N}(t)\rangle+u^{\top}(t)Ru(t)+h(\xi(t),t)$ $\displaystyle\
+g(p(t),t)+\lambda^{\top}(t)\left(A_{N}Z_{N}(t)+B_{N}(M\xi(t),t)u(t)\right)$
$\displaystyle\ +\mu^{\top}(t)\left(\alpha\xi(t)+\beta p(t)\right).$ (24)

By Pontryagin’s minimum principle [22], we can solve a two-point boundary
value problem originated from (24) to find a local minimum of (AP). The
iterative procedure for solving the two-point boundary value problem can be
implemented in a gradient-descent manner [18, 21].

##  5 Numerical examples

We demonstrate the performance of the optimal guidance and control in two
numerical examples. The first example uses the diffusion-advection process
with zero Dirichlet boundary condition (1)–(3). The second example uses the
same process but with zero Neumann boundary condition.

The examples are motivated by and simplified from practical applications,
e.g., removal of harmful algal blooms (HAB). In this case, the distribution of
the HAB’s concentration on the water surface can be modeled by a 2D diffusion-
advection process. The cases of zero Dirichlet and Neumann boundary conditions
correspond to the scenarios where the surface is circumvented by absorbent and
nonabsorbent materials, respectively. The control to the process is
implemented by the surface vehicles that use physical methods (e.g., emitting
ultrasonic waves or hauling algae filters) or chemical methods (by releasing
algal treatment) [32], whose impact on the process can be characterized by the
input operator (4). The magnitude of the control determines how fast the
concentration is reduced at the location of the actuator. The optimal control
and guidance minimize the cost such that the HAB concentration is reduced
while the vehicles do not exercise too much control nor conduct aggressive
maneuvers. And vehicles’ low-level control can track the optimal trajectories
despite the model mismatch between the dynamics of the vehicles and those
applied in the optimization problem (P).

We apply the following values in the numerical examples:
$\Omega=[0,1]\times[0,1]$, $z_{0}(x,y)=320(x-x^{2})(y-y^{2})$, $N=13$,
$m_{a}=4$, $t_{f}=1$, $\mathbf{v}=[0.1,-0.1]^{\top}$, $a=0.05$,
$U=\mathbb{R}^{4}$, $P_{i}=[-100,100]$, $p_{\max}=a_{\max}=100$, $R=0.1I_{4}$,
$\mathcal{Q}=\mathcal{Q}_{f}=\chi(x,y)$, $h(\xi(t),t)=h_{f}(\xi(t_{f}))=0$,
$g(p(t),t)=0.1p^{\top}(t)p(t)$, $\xi_{1}(0)=[0.1,0.1]^{\top}$,
$\xi_{2}(0)=[0.125,0.1]^{\top}$, $\xi_{3}(0)=[0.125,0.125]^{\top}$,
$\xi_{4}(0)=[0.1,0.125]^{\top}$, $\sigma_{i}=0.05$, $\alpha_{i}=0_{2\times
2}$, and $\beta_{i}=I_{2}$ for $i\in\\{1,2,3,4\\}$, where the indicator
function $\chi(x,y)=1$ if $x=y$, and $\chi(x,y)=0$ if $x\neq y$. We use (4)
for the input operator of each actuator.  The Péclect number of the process is
$|\mathbf{v}|_{2}/a\approx 2.83$, which implies neither the diffusion or the
advection dominates the process.

###  5.1 Diffusion-advection process with Dirichlet boundary condition

We use the dynamics in (1)–(3) with the Dirichlet boundary condition. We use
the Galerkin scheme to approximate the infinite-dimensional variables. The
orthonormal set of eigenfunctions of the Laplacian operator $\nabla^{2}$ (with
zero Dirichlet boundary condition) over the spatial domain
$\Omega=[0,1]\times[0,1]$ is $\phi_{i,j}(x,y)=2\sin(\pi ix)\sin(\pi jy)$. We
introduce a single index $k:=(i-1)N+j$ such that $\phi_{k}:=\phi_{i,j}$. For
brevity, we use $\mathcal{H}_{N}$ to denote the $N^{2}$-dimensional space
spanned by the basis functions $\\{\phi_{k}\\}_{k=1}^{N^{2}}$. Recall the
orthogonal projection $P_{N}:\mathcal{H}\rightarrow\mathcal{H}_{N}$. It
follows that $P_{N}^{\star}=P_{N}$ and $P_{N}^{\star}P_{N}\rightarrow I$
strongly [3]. Let $\Phi_{N}:=[\phi_{1}\ \phi_{2}\ \dots\
\phi_{N^{2}}]^{\top}$.  We choose $N=13$ because it is the smallest dimension
such that the resulting optimal cost is within the 1% of the optimal cost
evaluated with the maximum dimension $N=20$ in the numerical studies (see Fig.
7).

Assumption (A4) holds for the choice of input operator. With the Galerkin
approximation using the orthonormal eigenfunctions of the Laplacian operator
$\nabla^{2}$ with zero Dirichlet boundary condition, it can be shown that
assumption (A8) holds for $l_{N}(\cdot)=N^{2}l(\cdot)$. Assumptions (A5)–(A7)
hold with $q=1$ under the Galerkin approximation with aforementioned basis
functions $\Phi_{N}$ [3]. Assumptions (A9)–(A11) and (A12) hold for the choice
of functions in the mobility cost and parameters of the set of admissible
guidance, respectively.

We use the forward-backward sweeping method [23] to solve the two-point
boundary value problem originated from the Hamiltonian (24). The forward
propagation of $Z_{N}$ and $\xi$ and backward propagation of $\lambda$ and
$\mu$ are computed using the Runge-Kutta method. The same method is also
applied to propagate the approximate Riccati solution $\Pi(t)$. Spatial
integrals are computed using Legendre-Gauss quadrature.  To verify the
convergence of the approximate optimal cost $J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization problem}}(p_{N}^{*})$ stated in
(22), we compute $J^{*}_{\eqref{prob: equivalent finite dim approx integrated
optimization problem}}(p_{N}^{*})$ for $N\in\\{6,7,\dots,20\\}$. Note that the
total number of basis functions is $N^{2}$. The result is shown in Fig. 7,
where exponential convergence can be observed.

Figure 1: Evolution of the diffusion-advection process with Dirichlet boundary
condition under the optimal feedback control $\bar{u}^{*}$. The actuators are
steered by the optimal guidance $p^{*}$. Snapshots at $t=0.05$ and $0.2$ s
show the transient stage, whereas the one at $t=1$ s shows the relatively
steady stage. The mobile disturbance is shown by the gray circle. Figure 2:
Optimal feedback control $\bar{u}^{*}$ of each actuator in the case of
Dirichlet boundary condition. The circles along the horizontal axis correspond
to the snapshots in Fig. 1. Figure 3: Norm of the PDE state in the case of
Dirichlet boundary condition with pairs of control and guidance in Table 1.
The circles along the horizontal axis correspond to the snapshots in Fig. 1.

In the simulation, a mobile disturbance $0.5\mathcal{B}(x_{d}(t),t)$, whose
trajectory is $x_{d}(t)=[0.5+0.3\sin(2\pi t),0.5+0.3\cos(2\pi t)]^{\top}$ is
added to the right-hand side of the dynamics (1).

Denote the optimal open-loop control and optimal guidance solved using the
gradient-descent method in Section 4.2 by $u^{*}$ and $p^{*}$, respectively.
The trajectory steered by $p^{*}$ is denoted by $\xi^{*}$. Recall that an
optimal feedback control, denoted by $\bar{u}^{*}$, can be synthesized using
(17) based on the optimal trajectory $\xi^{*}$ of the actuators.

Fig. 1 shows the evolution of the process controlled by the optimal feedback
control and the optimal trajectories of the actuators. The actuation
concentrates in the first $0.2$ s, which is shown in Fig. 2. Meanwhile, the
actuators quickly pass the peak of the initial PDE at the center of the
spatial domain and spread evenly in space. Subsequently, the actuators 2–4
cease active steering and dispensing actuation. The flow field causes the
actuators to drift until the terminal time.

To demonstrate the performance of the optimal feedback control $\bar{u}^{*}$,
we compare it with semi-naive control $u_{\text{sn}}$ and naive control
$u_{\text{n}}$ defined as local feedback controls:
$u_{\text{sn}}(t)=-0.1z_{\text{sn}}(\xi^{*}(t),t)$ and
$u_{\text{n}}(t)=-0.1z_{\text{n}}(\xi_{\text{n}}(t),t)$. The semi-naive
actuators follow the optimal trajectory $\xi^{*}$, whereas the naive actuators
follow the trajectory $\xi_{\text{n}}$, which moves at a constant speed from
$\xi_{0}$ to $1_{n\times 1}-\xi_{0}$. Table 1 compares the cost breakdown of
all the control and guidance strategies. The optimal feedback control yields a
smaller cost than the optimal open-loop control due to the capability of
feedback control in rejecting disturbances. Simulations with a disturbance-
free model (not shown) yield identical total cost for optimal open-loop
control and optimal feedback control, which justifies the correctness of the
synthesis. Fig. 3 compares the norm of the PDE state controlled by pairs of
control and guidance listed in Table 1. As can be seen, the PDE is effectively
regulated using optimal feedback control. As a comparison, the norm associated
with optimal open-loop control grows slowly after $0.3$ s due the influence of
the disturbance, although its reduction in the beginning is indistinguishable
from that of the optimal feedback control.

Table 1: Cost comparison of control and guidance strategies in the case of Dirichlet boundary condition. All costs are normalized with respect to the total cost of the case with no control. Control (C) and Guidance (G) |  | Cost
---|---|---
| C | G |  | $J_{N}$ | $J_{\text{m}}$ | Total
opt. feedback | $\bar{u}^{*}$ | $\xi^{*}$ |  | 13.7% | 3.0% | 16.7%
opt. open-loop | $u^{*}$ | $\xi^{*}$ |  | 17.5% | 3.0% | 20.5%
semi-naive | $u_{\text{sn}}$ | $\xi^{*}$ |  | 42.5% | 3.0% | 45.5%
naive | $u_{\text{n}}$ | $\xi_{\text{n}}$ |  | 78.8% | 0.5% | 79.3%
no control | - | - |  | 100.0% | 0.0% | 100.0%

###  5.2 Diffusion-advection process with Neumann boundary condition

The results derived in this paper also apply to the operator $\mathcal{A}$
defined in (8) with a Neumann boundary condition (BC), because a general
second-order and uniformly elliptic operator with Neumann BC yields a strongly
continuous analytic semigroup on $L^{2}(\Omega)$ [20]. In this example, we
consider the diffusion-advection process (1) with initial condition (3) and
zero Neumann BC: ${\partial z(x,y,t)}/{\partial\mathbf{n}}=0$, where
$\mathbf{n}$ is the normal to the boundary $\partial\Omega$ and
$(x,y)\in\partial\Omega$. Notice that the basis functions applied for Galerkin
approximation in this case are the eigenfunctions of the Laplacian with zero
Neumann BC, $\phi_{i,j}(x,y)=2\cos(\pi ix)\cos(\pi jy)$ for
$i,j\in\\{0,1,\dots\\}$. All the parameters, disturbance, and pairs of control
and guidance for comparison applied in this example are identical to those in
Section 5.1.  Exponential convergence in the approximate optimal cost can be
observed in Fig. 7.

Fig. 4 shows the evolution of the process and the optimal trajectory of the
actuators. Similar to the case of Dirichlet BC, the actuators spread out to
cover most of the domain in the initial $0.2$ s, with most of the actuation
implemented during the same interval, seen in Fig. 5. However, the actuators
span a slightly larger area (Fig. 4) and the maximum amplitude of actuation is
bigger (Fig. 5), compared to the case of Dirichlet BC in Fig. 1 and Fig. 2,
respectively. The difference is a consequence of the fact that the zero
Neumann BC does not contribute to the regulation of the process because it
insulates the process from the outside. Contrarily, the zero Dirichlet BC acts
as a passive control that can essentially regulate the process to a zero state
when there is no inhomoegeneous term in the dynamics (1). This difference can
be observed when comparing the norm of the PDE state in Fig. 6 with Fig. 3.
The norm of the uncontrolled state reduces slightly in the case of Neumann BC
(Fig. 6) compared to the almost linear reduction in the case of Dirichlet BC
(Fig. 3). Fig. 6 also shows the difference of norm reduction between the
optimal feedback control and optimal open-loop control. Once again, the former
yields a smaller terminal norm than the latter due to the feedback’s
capability of disturbance rejection. The cost breakdown of the pairs of
control and guidance in comparison is shown in Table. 2.

Figure 4: Evolution of the process with Neumann boundary condition under the optimal feedback control $\bar{u}^{*}$. The actuators are steered by the optimal guidance $p^{*}$. Snapshots at $t=0.05$ and $0.2$ s show the transient stage, whereas the one at $t=1$ s shows the relatively steady stage. The mobile disturbance is shown by the gray circle. Figure 5: Optimal feedback control $\bar{u}^{*}$ of each actuator in the case of Neumann boundary condition. The circles along the horizontal axis correspond to the snapshots in Fig. 4. Figure 6: Norm of the PDE state in the case of Neumann boundary condition with pairs of control and guidance in Table 2. The circles along the horizontal axis correspond to the snapshots in Fig. 4. Table 2: Cost comparison of control and guidance strategies in the case of Neumann boundary condition. All costs are normalized with respect to the total cost of the case with no control. Control (C) and Guidance (G) |  | Cost
---|---|---
| C | G |  | $J_{N}$ | $J_{\text{m}}$ | Total
opt. feedback | $\bar{u}^{*}$ | $\xi^{*}$ |  | 6.4% | 1.6% | 8.0%
opt. open-loop | $u^{*}$ | $\xi^{*}$ |  | 7.1% | 1.6% | 8.7%
semi-naive | $u_{\text{sn}}$ | $\xi^{*}$ |  | 63.7% | 1.6% | 65.3%
naive | $u_{\text{n}}$ | $\xi_{\text{n}}$ |  | 65.9% | 0.2% | 66.1%
no control | - | - |  | 100.0% | 0.0% | 100.0%
Figure 7: Approximate optimal costs $J^{*}_{\eqref{prob: equivalent finite dim
approx integrated optimization problem}}(p_{N}^{*})$ normalized with respect
to the optimal cost for $N^{2}=400$.

##  6 Conclusion

This paper proposes an optimization framework that steers a team of mobile
actuators to control a DPS modeled by a 2D diffusion-advection process.
Specifically, jointly optimal control of the DPS and guidance of the mobile
actuators are solved such that the sum of a quadratic PDE cost and a generic
mobility cost is minimized subject to the dynamics of the DPS and of the
mobile actuators. We obtain an equivalent problem using LQR of an abstract
linear system, which reduces the problem to search for optimal guidance only.
The optimal control can be synthesized once the optimal guidance is obtained.
Conditions on the existence of a solution are established based on the
equivalent problem. We use the Galerkin approximation scheme to reduce the
problem to a finite-dimensional one and apply a gradient-descent method to
compute optimal guidance and control numerically. We prove conditions under
which the approximate optimal guidance converges to that of the exact optimal
guidance  in the sense that when evaluating these two solutions by the
original cost function, the difference becomes arbitrarily small as the
dimension of approximation increases. The convergence justifies the
appropriateness of both the approximate problem and its solution. The
performance of the proposed optimal control and guidance is illustrated with
two numerical examples , where exponential convergence of the approximate
optimal cost is observed.

Ongoing and future work includes  establishing the convergence rate of the
approximate optimal cost and studying problems with other types of PDE cost,
such as the operator norm of the Riccati operator [24, 6] to characterize
unknown initial conditions and H2\- or H∞-performance criteria for different
types of perturbation [25, 17]. On the actuator side, decentralized guidance
design may be incorporated in future work to enable more autonomy of the team
than a centralized implementation.  Actuators that travel along the boundary
may be considered as well which would result in boundary controller design.

##  Appendix A Proof of Theorem 6

The proof uses Theorem 11 (stated below) to establish the existence of an
optimal solution of (P1).

######  Theorem 11.

[36, Theorem 6.1.4] Suppose $(X,\left\lVert\cdot\right\rVert)$ is a normed
linear space, $M_{0}\subset X$ is weakly sequentially compact and
$f:M_{0}\rightarrow\mathbb{R}$ is weakly sequentially lower semicontinuous on
$M_{0}$. Then there exists an $\bar{x}\in M_{0}$ such that
$f(\bar{X})=\inf\\{f(x):x\in M_{0}\\}$.

Proof of Theorem 6 Without loss of generality, we consider the case of one
mobile actuator, i.e., $m_{a}=1$. The case of $m_{a}\geq 2$ follows naturally.

We want to apply Theorem 11 to prove that the minimum of the cost function of
(P1) is achieved on a subset $\mathcal{P}_{0}$ (defined below) of the
admissible set in which the cost of guidance is upper bounded. Consider
problem (P1)’s admissible set of guidance functions $\mathcal{P}:=\\{p\in
L^{2}([0,t_{f}];\mathbb{R}^{m}):p(t)\in P,t\in[0,t_{f}]\\}$. Assume there
exists $p_{0}\in\mathcal{P}$ such that $J_{\eqref{prob: new equivalent
IOCA}}(p_{0})<\infty$ and let $\mathcal{P}_{0}:=\\{p\in\mathcal{P}:\
J_{\eqref{prob: new equivalent IOCA}}(p)\leq J_{\eqref{prob: new equivalent
IOCA}}(p_{0})\\}$. We wish to prove Condition-1, Condition-2, and Condition-3
stated below:

Condition-1: The set $\mathcal{P}_{0}$ is bounded.

Condition-2: The set $\mathcal{P}_{0}$ is weakly sequentially closed.

Condition-3: The mapping $J_{\eqref{prob: new equivalent
IOCA}}(\cdot):\mathcal{P}\rightarrow\mathbb{R}$ is weakly sequentially lower
semicontinuous on $\mathcal{P}_{0}$.

Condition-1 and Condition-2 imply that $\mathcal{P}_{0}$ is weakly
sequentially compact. By Theorem 11, problem (P1) has a solution when
Condition-1–Condition-3 hold.

Before proving these three conditions, we define a mapping
$T:L^{2}([0,t_{f}];\mathbb{R}^{m})\rightarrow C([0,t_{f}];\mathbb{R}^{n})$ by
$(Tp)(t):=\xi(t)=e^{\alpha t}\xi_{0}+\int_{0}^{t}e^{\alpha(t-\tau)}\beta
p(\tau)\text{d}\tau$ for $t\in[0,t_{f}]$. For $p_{1},p_{2}\in
L^{2}([0,t_{f}];\mathbb{R}^{m})$ and $t\in[0,t_{f}]$, we have

$\displaystyle|Tp_{1}(t)-Tp_{2}(t)|_{1}$ $\displaystyle\leq$
$\displaystyle\textstyle\int_{0}^{t}|e^{\alpha(t-\tau)}\beta|_{1}|p_{1}(\tau)-p_{2}(\tau)|_{1}\text{d}\tau$
$\displaystyle\leq$ $\displaystyle
c_{5}\textstyle\int_{0}^{t}|e^{\alpha(t-\tau)}\beta|_{1}|p_{1}(\tau)-p_{2}(\tau)|_{2}\text{d}\tau$
$\displaystyle=$ $\displaystyle
c_{5}(\textstyle\int_{0}^{t}|e^{\alpha(t-\tau)}\beta|_{1}^{2}\text{d}\tau)^{1/2}\left\lVert
p_{1}-p_{2}\right\rVert_{L^{2}([0,t];\mathbb{R}^{m})}$ $\displaystyle\leq$
$\displaystyle c_{5}c_{6}\left\lVert
p_{1}-p_{2}\right\rVert_{L^{2}([0,t];\mathbb{R}^{m})}$ (25)

for $c_{5}$ and $c_{6}>0$. Hence, $\left\lVert
Tp_{1}-Tp_{2}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}=\sup_{t\in[0,t_{f}]}|Tp_{1}(t)-Tp_{2}(t)|_{1}\leq
c_{5}c_{6}\left\lVert
p_{1}-p_{2}\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$, which also shows
that $T$ is a continuous mapping, i.e., $\left\lVert
Tp\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}\leq c_{5}c_{6}\left\lVert
p\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$ for all $p\in
L^{2}([0,t_{f}];\mathbb{R}^{m})$.

Proof of Condition-1: Suppose $p\in\mathcal{P}_{0}$, then

$\displaystyle J_{\eqref{prob: new equivalent IOCA}}(p_{0})\geq$
$\displaystyle\ J_{\eqref{prob: new equivalent IOCA}}(p)$ $\displaystyle=$
$\displaystyle\
h_{f}(Tp(t_{f}))+\textstyle\int_{0}^{t_{f}}h(Tp(t),t)+g(p(t),t)\text{d}t$
$\displaystyle+\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$
$\displaystyle\geq$ $\displaystyle\
\textstyle\int_{0}^{t_{f}}d_{1}|p(t)|^{2}_{2}\text{d}t$ $\displaystyle=$
$\displaystyle\ d_{1}\left\lVert
p\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}^{2},$ (26)

where the second inequality follows from the nonnegativity of
$h_{f}(\cdot),h(\cdot,\cdot)$, and
$\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$. Since $d_{1}>0$, the
boundedness of $\mathcal{P}_{0}$ follows.

Proof of Condition-2: Suppose $\\{p_{k}\\}\subset\mathcal{P}_{0}$ and
$\\{p_{k}\\}$ converges weakly to $p$ (denoted by $p_{k}\rightharpoonup p$).
We want to show $p\in\mathcal{P}_{0}$. We start with proving that
$\mathcal{P}$ is weakly sequentially closed and, hence, $p\in\mathcal{P}$.
Subsequently, we show $J_{\eqref{prob: new equivalent IOCA}}(p)\leq
J_{\eqref{prob: new equivalent IOCA}}(p_{0})$ to conclude Condition-2.

To show that the set $\mathcal{P}$ is weakly sequentially closed, by [35,
Theorem 2.11], it suffices to show that $\mathcal{P}$ is closed and convex.
Let $\\{q_{k}\\}\subset\mathcal{P}$ and $q_{k}\rightarrow q$. We want to show
$q\in\mathcal{P}$, i.e., $q\in L^{2}([0,t_{f}],\mathbb{R}^{m})$ and $q(t)\in
P$ for $t\in[0,t_{f}]$. Since $L^{2}([0,t_{f}],\mathbb{R}^{m})$ is complete,
we can choose a subsequence $\\{q_{k_{j}}\\}\subset\mathcal{P}$ that converges
to $q$ pointwise almost everywhere on $[0,t_{f}]$ [37, p. 53]. Since $P$ is
closed (assumption (A9)), $q(t)\in P$ for almost all $t\in[0,t_{f}]$. Hence,
$\mathcal{P}$ is closed. The convexity of $\mathcal{P}$ follows from that of
$P$ (assumption (A9)), i.e., if $p_{1},p_{2}\in\mathcal{P}$, then $\lambda
p_{1}+(1-\lambda)p_{2}\in L^{2}([0,t_{f}];\mathbb{R}^{m})$ and $\lambda
p_{1}(t)+(1-\lambda)p_{2}(t)\in P$ for $t\in[0,t_{f}]$ and $\lambda\in[0,1]$.

What remain to be shown is $J_{\eqref{prob: new equivalent IOCA}}(p)\leq
J_{\eqref{prob: new equivalent IOCA}}(p_{0})$. Since $p_{k}\rightharpoonup p$,
by definition, we have $Tp_{k}\rightarrow Tp$. We now show that the sequence
$\\{Tp_{k}\\}$ contains a uniformly convergent subsequence in
$C([0,t_{f}];\mathbb{R}^{n})$. The sequence $\\{Tp_{k}\\}\subset
C([0,t_{f}];\mathbb{R}^{n})$ is uniformly bounded and uniformly equicontinuous
for the following reasons: Since $\left\lVert
Tp_{k}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}\leq c_{5}c_{6}\left\lVert
p_{k}\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$, it follows that
$\left\lVert Tp_{k}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}$ is uniformly
bounded, because $\\{p_{k}\\}\subset\mathcal{P}_{0}$ which is a bounded set.
For $s,t\in[0,t_{f}]$, we have

$\displaystyle\ |Tp_{k}(t)-Tp_{k}(s)|_{1}$ $\displaystyle=$ $\displaystyle\
\left|\textstyle\int_{s}^{t}\alpha Tp_{k}(\tau)+\beta
p_{k}(\tau)\text{d}\tau\right|_{1}$ $\displaystyle\leq$ $\displaystyle\
|t-s||\alpha|_{1}\left\lVert Tp_{k}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}$
$\displaystyle\ +|t-s|^{1/2}|\beta|_{2}\left\lVert
p_{k}\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}.$

Since $\\{\left\lVert p_{k}\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}\\}$
and $\\{\left\lVert Tp_{k}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}\\}$ both
are uniformly bounded for all $p_{k}\in\mathcal{P}_{0}$, $\\{Tp_{k}\\}$ is
uniformly equicontinuous. By the Arzelà-Ascoli Theorem [30], there is a
uniformly convergent subsequence $\\{Tp_{k_{j}}\\}\subset\\{Tp_{k}\\}$.

Without loss of generality, we assume $p_{k}\rightharpoonup p$ and
$Tp_{k}\rightarrow Tp$ uniformly on $[0,t_{f}]$, and $J_{\eqref{prob: new
equivalent IOCA}}(p_{k})\leq J_{\eqref{prob: new equivalent IOCA}}(p_{0})$. We
have $J_{\eqref{prob: new equivalent IOCA}}(p_{0})-J_{\eqref{prob: new
equivalent IOCA}}(p)=J_{\eqref{prob: new equivalent
IOCA}}(p_{0})-J_{\eqref{prob: new equivalent IOCA}}(p_{k})+J_{\eqref{prob: new
equivalent IOCA}}(p_{k})-J_{\eqref{prob: new equivalent IOCA}}(p)\geq
J_{\eqref{prob: new equivalent IOCA}}(p_{k})-J_{\eqref{prob: new equivalent
IOCA}}(p)$, by which, to show $J_{\eqref{prob: new equivalent IOCA}}(p)\leq
J_{\eqref{prob: new equivalent IOCA}}(p_{0})$, it suffices to show
$J_{\eqref{prob: new equivalent
IOCA}}(p)\leq\liminf_{k\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{k})$, which is to show

$\displaystyle
h_{f}(Tp(t_{f}))+\textstyle\int_{0}^{t_{f}}h(Tp(t),t)+g(p(t),t)\text{d}t$
$\displaystyle+\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$
$\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}h_{f}(Tp_{k}(t_{f}))+\textstyle\int_{0}^{t_{f}}h(Tp_{k}(t),t)+g(p_{k}(t),t)\text{d}t$
$\displaystyle+\langle\mathcal{Z}_{0},\Pi^{k}(0)\mathcal{Z}_{0}\rangle,$ (27)

where $\Pi^{k}(0)$ is the solution of (12) associated with actuator state
$Tp_{k}$. Since $\\{Tp_{k}\\}$ converges to $Tp$ uniformly on $[0,t_{f}]$, the
continuity of $h_{f}(\cdot)$ implies

$h_{f}(Tp(t_{f}))=\liminf_{k\rightarrow\infty}h_{f}(Tp_{k}(t_{f}));$ (28)

Fatou’s lemma [30] implies

$\textstyle\int_{0}^{t_{f}}h(Tp(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}h(Tp_{k}(t),t)\text{d}t;$
(29)

and Lemma 3 implies

$\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle=\liminf_{k\rightarrow\infty}\langle\mathcal{Z}_{0},\Pi^{k}(0)\mathcal{Z}_{0}\rangle.$
(30)

To prove (27), based on (28)–(30), it suffices to show
$\textstyle\int_{0}^{t_{f}}g(p(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}g(p_{k}(t),t)\text{d}t$.
By contradiction, assume there is $\lambda>0$ such that

$\liminf_{k\rightarrow\infty}\textstyle\int_{o}^{t_{f}}g(p_{k}(t),t)\text{d}t<\lambda<\textstyle\int_{0}^{t_{f}}g(p(t),t)\text{d}t.$
(31)

There exists a subsequence $\\{p_{k_{j}}\\}\subset\\{p_{k}\\}$ such that
$O_{\lambda}:=\\{q\in
L^{2}([0,t_{f}];\mathbb{R}^{m}):\textstyle\int_{0}^{t_{f}}g(q(t),t)\text{d}t\leq\lambda\\}$
and $\\{p_{k_{j}}\\}\subset O_{\lambda}$. We wish to show that $O_{\lambda}$
is weakly sequentially closed. By [35, Theorem 2.11], it suffices to show that
$O_{\lambda}$ is convex and closed. Since
$g(\cdot,t):\mathbb{R}^{m}\rightarrow\mathbb{R}$ is convex for all
$t\in[0,t_{f}]$, it follows that $O_{\lambda}$ is convex. Let
$\\{q_{k}\\}\subset O_{\lambda}$ and $\left\lVert
q_{k}-q\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$ converges to $0$ as
$k\rightarrow\infty$. We can choose a subsequence
$\\{q_{k_{j}}\\}\subset\\{q_{k}\\}$ such that $q_{k_{j}}$ converges to $q$
pointwise almost everywhere on $[0,t_{f}]$ [37, p. 53]. Now we have
(1) $g(q_{k_{j}}(t),t)\geq 0$ for all $t\in[0,t_{f}]$ (assumption (A11));
(2) $\lim_{j\rightarrow\infty}g(q_{k_{j}}(t),t)=g(q(t),t)$ almost everywhere
on $[0,t_{f}]$.

By Fatou’s lemma [30],

$\textstyle\int_{0}^{t_{f}}g(q(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}g(q_{k_{j}}(t),t)\text{d}t\leq\lambda,$

where the last inequality holds due to $\\{q_{k_{j}}\\}\subset O_{\lambda}$.
Hence, $q\in O_{\lambda}$ and $O_{\lambda}$ is closed.

Since $O_{\lambda}$ is weakly sequentially closed, $p_{k_{j}}\rightharpoonup
p$ implies that $p\in O_{\lambda}$, which contradicts (31). Hence,
$J_{\eqref{prob: new equivalent IOCA}}(p)\leq J_{\eqref{prob: new equivalent
IOCA}}(p_{0})$ is proved, and we conclude Condition-2.

Proof of Condition-3: We now show that the mapping $J_{\eqref{prob: new
equivalent IOCA}}(\cdot):\mathcal{P}\rightarrow\mathbb{R}$ is weakly
sequentially lower semicontinuous on $\mathcal{P}_{0}$. Suppose
$\\{p_{k}\\}\subset\mathcal{P}_{0}$ and $p_{k}\rightharpoonup
p\in\mathcal{P}_{0}$. We wish to show $J_{\eqref{prob: new equivalent
IOCA}}(p)\leq\liminf_{k\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{k})$, which has been established when we proved $J_{\eqref{prob: new
equivalent IOCA}}(p)\leq J_{\eqref{prob: new equivalent IOCA}}(p_{0})$ in
Condition-2 (starting from (27)).

So we conclude that the existence of a solution of problem (P1). ∎

##  Appendix B Proof of Theorem 7

Proof By contradiction, assume there are $p_{0}^{*}$ and $u_{0}^{*}$
minimizing (P) and $p_{0}^{*}\neq p^{*}$ and $u_{0}^{*}\neq u^{*}$ such that
$J^{*}_{\eqref{prob: new IOCA}}(u_{0}^{*},p_{0}^{*})<J_{\eqref{prob: new
IOCA}}(u^{*},p^{*})=J_{\eqref{prob: new equivalent IOCA}}(p^{*})$. Denote
$\bar{u}_{0}^{*}$ the optimal control (10) associated with actuator trajectory
steered by $p_{0}^{*}$. It follows that $J^{*}_{\eqref{prob: new
IOCA}}(u_{0}^{*},p_{0}^{*})=J_{\eqref{prob: new
IOCA}}(\bar{u}_{0}^{*},p_{0}^{*})$, because $J^{*}_{\eqref{prob: new
IOCA}}(u_{0}^{*},p_{0}^{*})>J_{\eqref{prob: new
IOCA}}(\bar{u}_{0}^{*},p_{0}^{*})$ violates the optimality of $u_{0}^{*}$ and
$J^{*}_{\eqref{prob: new IOCA}}(u_{0}^{*},p_{0}^{*})<J_{\eqref{prob: new
IOCA}}(\bar{u}_{0}^{*},p_{0}^{*})$ contradicts the fact that $\bar{u}_{0}^{*}$
minimizes the quadratic cost $J(\mathcal{Z},u)$ (see Lemma 2). Since
$J_{\eqref{prob: new
IOCA}}(\bar{u}_{0}^{*},p_{0}^{*})=\langle\mathcal{Z}_{0},\Pi_{0}^{*}(0)\mathcal{Z}_{0}\rangle+J_{\text{m}}(\xi_{0}^{*},p_{0}^{*})=J_{\eqref{prob:
new equivalent IOCA}}(p_{0}^{*})<J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})$, where $\Pi_{0}^{*}(0)$ associates with trajectory
$\xi_{0}^{*}$ steered by $p_{0}^{*}$, it follows that $p^{*}$ is not an
optimal solution of (P1), which contradicts the optimality of $p^{*}$ for
(P1).∎

##  Appendix C Proof of Theorem 8

Proof Since $\langle Z_{0,N},\Pi_{N}(0)Z_{0,N}\rangle\geq 0$ and the mapping
$K_{N}:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$ is continuous
(see Lemma 5), the proof is analogous to that of Theorem 6, where we use
$\langle Z_{0,N},\Pi_{N}(0)Z_{0,N}\rangle$ to substitute
$\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle$. The proof that
$u_{N}^{*}$ and $p_{N}^{*}$ minimize problem (AP) follows from the same logic
as the proof of Theorem 7. ∎

##  Appendix D Proof of Theorem 9

Before we prove Theorem 9, we first establish two intermediate results in
Lemma 12, whose proof is in the supplementary material.

######  Lemma 12.

Consider problem (P1) and its approximation (AP1). If assumptions (A4)–(A7)
and (A9)–(A12) hold, then the following two implications hold:
1\. For $p\in C([0,t_{f}];P)$, $\lim_{N\rightarrow\infty}|J_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}}(p)-J_{\eqref{prob: new equivalent IOCA}}(p)|=0$, where $N$ is the
dimension of approximation applied in (AP).
2\. The mapping $J_{\eqref{prob: new equivalent
IOCA}}:C([0,t_{f}];P)\rightarrow\mathbb{R}^{+}$ is continuous, where
$J_{\eqref{prob: new equivalent
IOCA}}(p)=\langle\mathcal{Z}_{0},\Pi(0)\mathcal{Z}_{0}\rangle+J_{\text{m}}(\xi,p)$.
Here, the actuator state $\xi$ follows the dynamics (6) steered by the
guidance $p$, and $\Pi(0)$ follows (11) with the actuator state $\xi$.

Proof of Theorem 9 In the notation $J_{\eqref{prob: equivalent finite dim
approx integrated optimization problem}}(p_{N}^{*})$, the dimension of
approximation in (AP1), which is $N$ in this case, is indicated by its
solution $p_{N}^{*}$. We append a subscript to indicate the dimension when it
is not explicitly reflected by the argument, e.g., $J_{\eqref{prob: equivalent
finite dim approx integrated optimization problem}_{N}}(p)$.

We first show (22), i.e., $|J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: new
equivalent IOCA}}(p^{*})|\rightarrow 0$ as $N\rightarrow\infty$. First,

$\displaystyle J^{*}_{\eqref{prob: equivalent finite dim approx integrated
optimization problem}}(p_{N}^{*})=$ $\displaystyle\
\underset{p\in\mathcal{P}(p_{\max},a_{\max})}{\min}J_{\eqref{prob: equivalent
finite dim approx integrated optimization problem}}(p)$ $\displaystyle\leq$
$\displaystyle\ J_{\eqref{prob: equivalent finite dim approx integrated
optimization problem}}(p^{*})$ $\displaystyle\leq$ $\displaystyle\
|J_{\eqref{prob: equivalent finite dim approx integrated optimization
problem}}(p^{*})-J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})|+J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*}).$

Since $|J_{\eqref{prob: equivalent finite dim approx integrated optimization
problem}}(p^{*})-J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})|\rightarrow
0$ as $N\rightarrow 0$ (see Lemma 12-1), it follows that

$\displaystyle\limsup_{N\rightarrow\infty}J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization problem}}(p_{N}^{*})\leq$
$\displaystyle\ J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*}).$ (32)

To proceed with proving (22), in addition to (32), we shall show
$\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}}(p_{N}^{*})\geq J^{*}_{\eqref{prob: new
equivalent IOCA}}(p^{*})$. Choose a convergent subsequence
$\\{J^{*}_{\eqref{prob: equivalent finite dim approx integrated optimization
problem}}(p_{N_{k}}^{*})\\}_{k=1}^{\infty}$ such that
$\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization
problem}}(p_{N_{k}}^{*})=\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization problem}}(p_{N}^{*})$.
Since the guidance functions defined in the set
$\mathcal{P}(p_{\max},a_{\max})$ are uniformly equicontinuous and uniformly
bounded, by the Arzelà-Ascoli Theorem [30], there is a uniformly convergent
subsequence of $\\{p_{N_{k}}^{*}\\}_{k=1}^{\infty}$ which we use the same
index $\\{N_{k}\\}_{k=1}^{\infty}$ to simplify notation and let the limit of
$\\{p_{N_{k}}^{*}\\}_{k=1}^{\infty}$ be $p_{\inf}^{*}$, i.e.,

$\lim_{k\rightarrow\infty}\left\lVert
p_{N_{k}}^{*}-p_{\inf}^{*}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}=0.$ (33)

Now, $|J^{*}_{\eqref{prob: equivalent finite dim approx integrated
optimization problem}}(p_{N_{k}}^{*})-J_{\eqref{prob: new equivalent
IOCA}}(p_{\inf}^{*})|\leq|J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}}(p_{N_{k}}^{*})-J_{\eqref{prob: new
equivalent IOCA}}(p_{N_{k}}^{*})|+|J_{\eqref{prob: new equivalent
IOCA}}(p_{N_{k}}^{*})-J_{\eqref{prob: new equivalent IOCA}}(p_{\inf}^{*})|$,
which implies

$\displaystyle\ \limsup_{k\rightarrow\infty}|J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization
problem}}(p_{N_{k}}^{*})-J_{\eqref{prob: new equivalent IOCA}}(p_{\inf}^{*})|$
$\displaystyle\leq$ $\displaystyle\
\lim_{k\rightarrow\infty}|J^{*}_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}}(p_{N_{k}}^{*})-J_{\eqref{prob: new
equivalent IOCA}}(p_{N_{k}}^{*})|$ $\displaystyle\
+\lim_{k\rightarrow\infty}|J_{\eqref{prob: new equivalent
IOCA}}(p_{N_{k}}^{*})-J_{\eqref{prob: new equivalent IOCA}}(p_{\inf}^{*})|.$
(34)

The first limit on the right-hand side of (34) is zero for the following
reason. For all $p\in\mathcal{P}(p_{\max},a_{\max})$, $J_{\eqref{prob:
equivalent finite dim approx integrated optimization problem}_{N}}(p)$
converges to $J_{\eqref{prob: new equivalent IOCA}}(p)$ pointwise as the
dimension of approximation $N\rightarrow\infty$ (see Lemma 12-1). Furthermore,
since the sequence of approximated PDE cost $\\{\langle
Z_{N}(0),\Pi_{N}(0)Z_{N}(0)\rangle\\}_{N=1}^{\infty}$ is a monotonically
increasing sequence, the sequence $\\{J_{\eqref{prob: equivalent finite dim
approx integrated optimization problem}_{N}}(p)\\}_{N=1}^{\infty}$ is a
monotonically increasing sequence for each $p$ on the compact set
$\mathcal{P}(p_{\max},a_{\max})$. By Dini’s Theorem [31, Theorem 7.13],
$|J_{\eqref{prob: equivalent finite dim approx integrated optimization
problem}_{N}}(p)-J_{\eqref{prob: new equivalent IOCA}}(p)|\rightarrow 0$
uniformly on $\mathcal{P}(p_{\max},a_{\max})$ as $N\rightarrow\infty$. By
Moore-Osgood Theorem [31, Theorem 7.11], this uniform convergence and the
convergence $p_{N_{k}}^{*}\rightarrow p_{\inf}^{*}$ as $k\rightarrow\infty$
(see (33)) imply that $\lim_{k\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N_{k}}^{*})=\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}_{j}}(p_{N_{k}}^{*})$, in which the iterated limit equals the double
limit [34, p. 140], i.e.,

$\displaystyle\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}_{j}}(p_{N_{k}}^{*})=$ $\displaystyle\
\lim_{\begin{subarray}{c}j\rightarrow\infty\\\
k\rightarrow\infty\end{subarray}}J^{*}_{\eqref{prob: equivalent finite dim
approx integrated optimization problem}_{j}}(p_{N_{k}}^{*})$ $\displaystyle=$
$\displaystyle\ \lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: equivalent finite
dim approx integrated optimization problem}}(p_{N_{k}}^{*}).$

The second limit on the right-hand side of (34) is zero due to Lemma 12-2.
Hence, it follows from (34) that $\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}}(p_{N_{k}}^{*})=J_{\eqref{prob: new equivalent IOCA}}(p_{\inf}^{*})$,
which implies

$\displaystyle\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization problem}}(p_{N}^{*})=$
$\displaystyle\ \lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: equivalent finite
dim approx integrated optimization problem}}(p_{N_{k}}^{*})$ $\displaystyle=$
$\displaystyle\ J_{\eqref{prob: new equivalent IOCA}}(p_{\inf}^{*})$
$\displaystyle\geq$ $\displaystyle\ J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*}).$ (35)

Therefore, we conclude $\lim_{N\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}}(p_{N}^{*})=J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})$ from
(32) and (35).

Next, we show (23), i.e., $|J_{\eqref{prob: new equivalent
IOCA}}(p_{N}^{*})-J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})|\rightarrow
0$ as $N\rightarrow\infty$.  We start with $J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})\leq J_{\eqref{prob: new equivalent IOCA}}(p_{N}^{*})$ for all
$N$, which implies that

$J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})\leq\liminf_{N\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N}^{*}).$ (36)

To prove (23), what remains to be shown is $J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})\geq\limsup_{N\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N}^{*})$. Choose a convergent subsequence $\\{J_{\eqref{prob: new
equivalent IOCA}}(p_{N_{j}}^{*})\\}_{j=1}^{\infty}$ such that
$\lim_{j\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N_{j}}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: new
equivalent IOCA}}(p_{N}^{*})$. Since
$\\{p_{N_{j}}^{*}\\}_{j=1}^{\infty}\subset\mathcal{P}(p_{\max},a_{\max})$ is
uniformly equicontinuous and uniformly bounded, by Arzelà-Ascoli Theorem [30],
the sequence has a (uniformly) convergent subsequence which we denote with the
same indices $N_{j}$ to simplify notation. Denote the limit of
$\\{p_{N_{j}}^{*}\\}_{j=1}^{\infty}$ by $p_{\sup}^{*}$ such that

$\lim_{j\rightarrow\infty}\left\lVert
p_{N_{j}}^{*}-p_{\sup}^{*}\right\rVert_{C([0,t_{f}];\mathbb{R}^{m})}=0.$ (37)

Due to the continuity of $J_{\eqref{prob: new equivalent IOCA}}(\cdot)$ (see
Lemma 12-1), we have

$J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})=\lim_{j\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N_{j}}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: new
equivalent IOCA}}(p_{N}^{*}).$

It follows that

$\displaystyle J_{\eqref{prob: new equivalent IOCA}}(p_{\sup}^{*})$
$\displaystyle\leq$ $\displaystyle|J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})-J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})|+J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})$
$\displaystyle=$ $\displaystyle|J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})-\lim_{N\rightarrow\infty}J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization
problem}}(p_{N}^{*})|+J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})$
$\displaystyle=$ $\displaystyle|J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})-\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob: equivalent
finite dim approx integrated optimization
problem}}(p_{N_{j}}^{*})|+J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*}).$
(38)

Since the sequence of approximated PDE cost $\\{\langle{Z_{N}(0)},$
${\Pi_{N}(0)Z_{N}(0)}\rangle\\}_{N=1}^{\infty}$ is monotonically increasing,
the sequence $\\{J_{\eqref{prob: equivalent finite dim approx integrated
optimization problem}_{N}}(p)\\}_{N=1}^{\infty}$ is a monotonically increasing
sequence for each $p$ on the compact set $\mathcal{P}(p_{\max},a_{\max})$.
Since $\lim_{N\rightarrow\infty}J_{\eqref{prob: equivalent finite dim approx
integrated optimization problem}_{N}}(p)=J_{\eqref{prob: new equivalent
IOCA}}(p)$ for all $p\in\mathcal{P}(p_{\max},a_{\max})$ (see Lemma 12-1), by
Dini’s Theorem [31, Theorem 7.13], the limit holds uniformly on
$\mathcal{P}(p_{\max},a_{\max})$ as $N\rightarrow\infty$. By Moore-Osgood
Theorem [31, Theorem 7.11], this uniform convergence and the convergence
$p_{N_{j}}^{*}\rightarrow p_{\sup}^{*}$ as $j\rightarrow\infty$ (see (37))
imply that

$\displaystyle J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})=\lim_{k\rightarrow\infty}\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}_{k}}(p_{N_{j}}^{*}).$ (39)

Furthermore, the iterated limit equals the double limit [34, p. 140], i.e.,

$\displaystyle\lim_{k\rightarrow\infty}\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
equivalent finite dim approx integrated optimization
problem}_{k}}(p_{N_{j}}^{*})=$
$\displaystyle\lim_{\begin{subarray}{c}j\rightarrow\infty\\\
k\rightarrow\infty\end{subarray}}J^{*}_{\eqref{prob: equivalent finite dim
approx integrated optimization problem}_{k}}(p_{N_{j}}^{*})$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob: equivalent finite
dim approx integrated optimization problem}}(p_{N_{j}}^{*}).$ (40)

Hence, combining (38)–(40), we have $J^{*}_{\eqref{prob: new equivalent
IOCA}}(p^{*})\geq J_{\eqref{prob: new equivalent
IOCA}}(p_{\sup}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: new
equivalent IOCA}}(p_{N}^{*})$, from which and (36) we conclude the desired
convergence $\lim_{N\rightarrow\infty}J_{\eqref{prob: new equivalent
IOCA}}(p_{N}^{*})=J^{*}_{\eqref{prob: new equivalent IOCA}}(p^{*})$. ∎

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