# Optimal guidance and estimation of a 2D diffusion-advection process by a
team of mobile sensors

Sheng Cheng<EMAIL_ADDRESS>Derek A. Paley<EMAIL_ADDRESS>University of
Illinois Urbana-Champaign  University of Maryland, College Park

(01 November 2021)

###### Abstract

This paper describes an optimization framework to design guidance for a
possibly heterogeneous team of multiple mobile sensors to estimate a
spatiotemporal process modeled by a 2D diffusion-advection process. Owing to
the abstract linear system representation of the process, we apply the Kalman-
Bucy filter for estimation, where the sensors provide linear outputs. We
propose an optimization problem that minimizes the sum of the trace of the
covariance operator of the Kalman-Bucy filter and a generic mobility cost of
the mobile sensors, subject to the sensors’ motion modeled by linear dynamics.
We establish the existence of a solution to this problem. Moreover, we prove
convergence to the exact optimal solution of the approximate optimal solution.
That is, when evaluating these two solutions using the original cost function,
the difference becomes arbitrarily small as the approximation gets finer. To
compute the approximate solution, we use Pontryagin’s minimum principle after
approximating the infinite-dimensional terms originating from the diffusion-
advection process. The approximate solution is applied in simulation to
analyze how a single mobile sensor’s performance depends on two important
parameters: sensor noise variance and mobility penalty. We also illustrate the
application of the framework to multiple sensors, in particular the
performance of a heterogeneous team of sensors.

###### 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. +1 217 979 8496.

,

##  1 Introduction

The modern manufacturing industry has benefited from the advantages of mobile
robots for their reliability, economic efficiency, safety, and ease of use.
However, the monitoring and control of large-scale spatiotemporal processes,
e.g., oil spills, harmful algal blooms, and forest fires, have relied heavily
on human operators. These events can pose health threats, cause severe
environmental issues, and incur substantial financial costs. In such a
situation, mobile robots can replace the human operators and carry out the
tasks intelligently and cooperatively so long as they are informed about the
environment and guided with proper methods.

Spatiotemporal processes evolve in both space and time and, hence, they can be
treated as dynamical systems whose dynamics are modeled by partial
differential equations (PDEs). It is generally impossible to completely
identify the state of a PDE-modeled system, also known as a distributed
parameter system (DPS), with a finite number of sensors. Hence, an observer
for the DPS is necessary. For system-theoretical results on the observability
of parabolic PDEs, refer to [24, 25]. Early designs of the observer include
least-square filtering methods for systems governed by linear [28] and
nonlinear [26] partial differential equations.

For a DPS whose infinite-dimensional dynamics is linear with additive Gaussian
noise, an infinite-dimensional Kalman-Bucy filter (KF), which is an extension
to its finite-dimensional analog [22, 4], can be applied. The infinite-
dimensional version of the KF first appeared in [1]. The key to an infinite-
dimensional KF is the evolution of the estimation error’s covariance, which is
operator-valued and can be solved via Riccati equations whose properties have
been discussed in [6, 7]. For numerical approximation and computational
issues, approximation results are summarized in [8] for the infinite-
dimensional algebraic Riccati equations of a linear-quadratic regulator.
However, often the disturbance to a system may not be Gaussian, in which case
an $H_{2}$\- or $H_{\infty}$-observer design is favorable [41, 31].

We categorize the design of an observer for a PDE-modeled system by sensor
location, i.e., on the boundary or in the interior of the PDE’s domain. For
sensors placed on the boundary, one may design observers based on boundary
measurements. The challenge in using boundary measurements is that the output
operator is unbounded when characterized in an abstract linear system. A
tutorial paper [19] reviews the conditions for the well-posedness of the dual
control problem with boundary inputs. One approach to boundary observer design
is backstepping [37, 23], which stabilizes the observer using a Volterra
transformation that transforms the original system to a stable target system
and passes on the stability to the observer via its inverse transformation.
Optimization techniques, e.g., a linear-quadratic estimator [32], have been
proposed for boundary sensors using the method of variation.

For static in-domain sensors, a network can be deployed for estimating a PDE-
modeled system. The problem is how to place the sensors to yield effective
estimation, which is referred to as sensor placement. The solution is to
select the sensor placement that yields optimal performance for certain
criteria. The trace of the covariance operator of the KF, which quantifies the
uncertainty of the estimate, is a common choice of the cost function to be
minimized [2]. A similar problem is investigated in [45], which establishes
the well-posedness of the sensor placement problem and its approximation with
the cost function being the trace of the covariance operator. The same
criterion has been applied to sensor placement of the Boussinesq equation
[20]. In [34], a randomized observability constant is minimized by choosing
suitable shapes and locations of the sensors. Note that observability may not
be a useful criterion [45] based on the analysis of its dual, controllability.
Specifically, maximizing the minimum eigenvalue of the controllability gramian
is not useful (for actuator placement) [30], because the minimum eigenvalue
approaches zero as the dimension of approximation increases in computation.
Geometric approaches can also be applied to sensor placement, e.g., using the
centroidal Voronoi tessellation [16] and combining the transfer-function model
with geometric rules [40].

A variation of the sensor placement problem includes mobility of the sensors.
In this scenario, a guidance policy is necessary to take advantage of the
additional degree of freedom induced by mobility, which also makes the problem
more complicated by introducing the dynamics of the mobile sensors. One may
design sensor guidance using Lyapunov-based methods, where the guidance is
constructed to make the derivative of the Lyapunov function negative. The
Lyapunov function is usually designed to contain (quadratic) terms involving
the PDE state and the sensor guidance [13, 18]. The Lyapunov-based guidance
can further be used in monitoring a hazardous environment where the regions of
high information density reduce sensor longevity. Such guidance is combined
with a switching policy to balance the conflicting needs of information
collection and sensor life span [15].

Optimization can also be applied to design sensor guidance, where the guidance
(or the trajectory) of the sensor is selected to minimize a cost function. An
early work [9] proposes an optimization problem that minimizes the weighted
sum of the guidance effort for steering a sensor and the mean-square
estimation error at a terminal time. In [12], the sensors are guided to the
location that yields the maximum value of the estimation kernel. In [21],
receding horizon guidance is proposed to find the sensor path that maximizes
mutual information between sensor measurements and the predicted state of the
PDE.

This paper proposes an optimization framework that designs guidance for a team
of mobile sensors to estimate a 2D diffusion-advection process using a
centralized KF. The cost to be minimized is the sum of two terms: the trace of
the covariance operator of the KF, called the uncertainty cost, which
quantifies the uncertainty of the estimation error and the mobility cost
associated with the sensors’ motion. The covariance operator of the KF, which
is the solution of an operator-valued Riccati equation, has been studied in
[6]. Specifically, conditions for the existence of Bochner integrable
solutions (with values in the Schatten $p$-class) of an operator-valued
Riccati equation are established. The Bochner integrable solutions yield
simple numerical quadratures for computation of the covariance operator, which
is demonstrated in sensor placement [6] and sensor trajectory planning [7].
Both problems minimize the trace of a weighted covariance operator, whereas
the latter has the sensors’ dynamics as the constraint.

In our formulation, factors related to the mobile sensor platforms are
integrated in addition to reducing the estimation uncertainty only [7].
Specifically, the integration is reflected by the mobility cost, which can be
interpreted as the penalty associated with motion. Our formulation supports
various types of cost functions for evaluating the cost or penalty induced
from mobility. In prior work, the mobility cost is either limited to a
quadratic guidance effort [9] or cast as a general function of the guidance
without detailed discussion [5, 2]. Furthermore, the formulation permits
treating the proposed problem as an optimal control problem, where we show
both the uncertainty cost and the mobility cost are continuous mappings of the
sensors’ guidance. We use the techniques for the existence of an optimal
control [43, Theorem 6.1.4] to establish the existence of a solution to our
problem.

To compute an optimal solution, approximations of the infinite-dimensional
terms are necessary. Our treatment of the proposed problem (and its
approximation) permits the application of a two-point boundary value problem
derived using Pontryagin’s minimum principle. After restricting the admissible
guidance functions to a stringent set (with a reasonable physical
interpretation), we establish convergence to an exact optimal solution of an
approximate optimal solution, i.e., the cost difference between the original
and approximate solutions becomes arbitrarily small as the dimension of
approximation increases. The convergence result justifies the use of the
approximation and affirms that the performance of an approximate solution is
arbitrarily close to the performance of an exact solution.

We implement the solution method numerically in simulations to evaluate and
analyze the performance of a single sensor and multiple sensors. The flow
field that yields advection is set to drift the sensor platforms under
realistic conditions. It has been observed that the flow field is leveraged by
the optimal guidance to reduce the mobility cost in both cases of a single
sensor and a team of homogeneous sensors. Simulations with a team of
heterogeneous sensors suggest that such a team can reduce the cost of
investment with only a minor degradation in the overall performance.

The contributions of this paper are threefold. First, we propose a guidance
policy for a team of mobile sensors to estimate a 2D diffusion-advection
process by solving a newly formulated optimization problem that minimizes the
trace of the covariance operator plus a generic cost of the sensors’ motion
subject to sensor platform dynamics. The resulting guidance minimizes the
estimation uncertainty while taking care of the sensor-oriented concerns. The
formulation with a generic mobility cost applies to a wide range of
applications, e.g., accumulated exposure to hazards, guidance effort, or
distance to terminal rendezvous locations. The mobility cost is incorporated
into the design process, unlike prior work that minimized the uncertainty cost
only [7]. Second, we establish realistic conditions for the existence of a
solution to the proposed problem. Third, we establish conditions for
convergence to an exact optimal solution of the approximate optimal solution.
When evaluating these two solutions using the original cost function, the
difference gets arbitrarily small as the approximation gets finer.

The problem studied in this paper is an extension of the optimal guidance
problem for a 1D diffusion process [10]. It is the dual problem of the one
studied in [11], which simultaneously designs guidance and actuation of a team
of mobile actuators to control a 2D diffusion-advection process.

The remainder of the paper is organized as follows. Section II introduces the
dynamics of the sensors, an abstract linear-system representation of the
diffusion-advection process, the measurement model, and the infinite-
dimensional Kalman-Bucy filter. Section III states the problem formulation and
establishes conditions for the existence of a solution to the problem. Section
IV proves the convergence to the exact optimal solution of the approximate
optimal solution and introduces a solution method to obtain optimal guidance.
Section V includes the simulation results of multiple parameter studies: a
single sensor, a team of homogeneous sensors, and a team of heterogeneous
sensors. Section VI summarizes the paper and discusses ongoing work.

##  2 Background

###  2.1 Notation and terminology

The paper adopts the following notation. The symbols $\mathbb{R}$ and
$\mathbb{R}^{+}$ denote the set of real numbers and the set of nonnegative
real numbers, respectively. The $n$-nary Cartesian power of a set $M$ is
denoted by $M^{n}$. The notation $X_{1}\hookrightarrow X_{2}$ means that the
space $X_{1}$ is densely and continuously embedded in $X_{2}$. The norm in a
finite- and infinite-dimensional space is denoted by $|\cdot|$ and
$\left\lVert\cdot\right\rVert$, respectively, with subscripts indicating its
type. The space of all bounded linear operators from space $X$ to space $Y$ is
denoted by $\mathcal{L}(X;Y)$ or $\mathcal{L}(X)$ if $Y=X$. We define the
space of continuous mappings by $C(I;X)=\\{F:I\rightarrow X\text{ such that
}t\mapsto F(t)\text{ is continuous in }\left\lVert\cdot\right\rVert_{X}\\}$
with the sup norm $\left\lVert F(\cdot)\right\rVert_{C(I;X)}=\sup_{t\in
I}\left\lVert F(t)\right\rVert_{X}$. For a Hilbert space $\mathcal{H}$
equipped with inner product $\langle\cdot,\cdot\rangle$ and
$\phi_{1},\phi_{2}\in\mathcal{H}$, define
$\phi_{1}\circ\phi_{2}\in\mathcal{L}(\mathcal{H})$ by
$(\phi_{1}\circ\phi_{2})\psi=\phi_{1}\langle\phi_{2},\psi\rangle$ for all
$\psi\in\mathcal{H}$. The superscript ∗ denotes an optimal variable, whereas ⋆
denotes the adjoint of a linear operator. The transpose of a matrix $A$ is
denoted by $A^{\top}$. An $n\times n$-dimensional diagonal matrix with
elements of vector $[a_{1},a_{2},\dots,a_{n}]$ on the main diagonal is denoted
by $\text{diag}(a_{1},a_{2},\dots,a_{n})$. The $i$th element of a vector $v$
is $[v]_{i}$. We follow the terminology of [11]: guidance refers to steering
the dynamics of the mobile sensor. For an optimization problem

$\displaystyle\underset{x}{\text{minimize}}$ $\displaystyle J(x)$ (P0) subject
to $\displaystyle x\in C,$

we use $J_{\eqref{prob: notation example problem}}(x)$ to denote the cost
function of (P0) evaluated at $x$. Specifically, $J^{*}_{\eqref{prob: notation
example problem}}(x^{*})$ indicates that the optimal value of (P0) is attained
when the cost function is evaluated at an optimal solution $x^{*}$.

###  2.2 Dynamics of the mobile sensors

Assume each of the $m_{s}$ mobile sensors has the linear dynamics

$\dot{\zeta}_{i}(t)=\alpha_{i}\zeta_{i}(t)+\beta_{i}p_{i}(t),\quad\zeta_{i}(0)=\zeta_{i,0},$
(1)

where $\zeta_{i}(t)\in\mathbb{R}^{n}$ and $p_{i}(t)\in
P_{i}\subset\mathbb{R}^{m}$ are the state and the guidance of sensor $i$ at
time $t$, respectively. The state $\zeta_{i}$ contains the 2D location of
sensor $i$ and hence $n\geq 2$. Assume that system (1) is controllable. One
special case of (1) would be a single integrator, where
$\zeta_{i}(t)\in\mathbb{R}^{2}$ is the location, $p_{i}(t)\in\mathbb{R}^{2}$
is the velocity command, and $\alpha_{i}$ and $\beta_{i}$ are the zero matrix
and identity matrix, respectively.

For conciseness, we concatenate the states and guidance of all $m_{s}$
sensors, respectively, and use one dynamical equation to describe the dynamics
of all agents:

$\dot{\zeta}(t)=\alpha\zeta(t)+\beta p(t),\quad\zeta(0)=\zeta_{0},$ (2)

where matrices $\alpha$ and $\beta$ are assembled from $\alpha_{i}$ and
$\beta_{i}$ for $i\in\\{1,2,\dots,m_{s}\\}$, respectively, and are consistent
with the concatenation for $\zeta$ and $p$. The controllability of the
concatenated system (2) inherits that of each individual system (1). With a
slight abuse of notation, we use $n$ for the dimension of $\zeta(t)$ and $m$
for the dimension of $p(t)$. Define the admissible set of guidance
$P:=P_{1}\times P_{2}\times\dots\times P_{m_{s}}$ such that $p(t)\in P$ for
$t\in[0,t_{f}]$.

###  2.3 Abstract linear system

Consider the following inhomogeneous diffusion-advection equation over a
smooth and bounded 2D spatial domain $\Omega$:

$\displaystyle\frac{\partial z(x,y,t)}{\partial t}=$ $\displaystyle\
a\nabla^{2}z(x,y,t)-\mathbf{v}\cdot\nabla z(x,y,t)$ $\displaystyle+w(x,y,t)$
(3) $\displaystyle z(x,y,0)=$ $\displaystyle\ \hat{z}(x,y,0)+w_{0}(x,y)$ (4)
$\displaystyle z(x,y,t)|_{(x,y)\in\partial\Omega}=$ $\displaystyle\ 0,$ (5)

where $(x,y)\in\Omega$, $t\in[0,t_{f}]$, $a>0$ is the diffusion coefficient,
and $\mathbf{v}\in\mathbb{R}^{2}$ is the flow that yields advection. The
initial condition $z(\cdot,\cdot,0)$ is perturbed around its nominal value
$\hat{z}(\cdot,\cdot,0)$ by an additive zero-mean Gaussian noise
$w_{0}(\cdot,\cdot)$. The dynamics (3) is subject to an additive zero-mean
Gaussian noise $w(\cdot,\cdot,t)$ with variance $Q(t)$, which is nonnegative
and self-adjoint. The state noise $w(\cdot,\cdot,t)$ and initial noise
$w_{0}(\cdot,\cdot)$ are mutually independent for all $t$.

For simplicity, represent the PDE state in (3) by an abstract linear system
whose state variable $\mathcal{Z}(t)$ represents $z(\cdot,\cdot,t)$ at time
$t$, such that

$\dot{\mathcal{Z}}(t)=\mathcal{A}\mathcal{Z}(t)+w(t),\quad\mathcal{Z}(0)=\hat{\mathcal{Z}}_{0}+w_{0},$
(6)

where $\mathcal{Z}$ belongs to a Hilbert space $\mathcal{H}$ with inner
product $\langle\cdot,\cdot\rangle$ and induced norm
$\left\lVert\cdot\right\rVert_{\mathcal{H}}$. Here, the variable $\mathcal{Z}$
is the state of the DPS and space $\mathcal{H}=L^{2}(\Omega)$ is the state
space. The operator $\mathcal{A}$ is defined as
$\mathcal{A}\psi=a\nabla^{2}\psi-\mathbf{v}\cdot\nabla\psi$ with
$\psi\in\text{Dom}(\mathcal{A})=\\{{\psi\in H_{0}^{1}(\Omega),}\
{\nabla^{2}\psi\in L^{2}(\Omega)}\\}=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$
[14]. Let $\mathcal{S}(\cdot)$ be the strongly continuous semigroup generated
by $\mathcal{A}$.

The measurement by sensor $i$ depends on the sensor’s location such that

$y_{i}(t)=\mathcal{C}_{i}^{\star}(M_{0}\zeta_{i}(t),t)\mathcal{Z}(t)+v_{i}(t),$
(7)

where $M_{0}$ is a matrix with appropriate dimension such that
$M_{0}\zeta_{i}(t)\in\mathbb{R}^{2}$ is the location of sensor $i$ and
$\mathcal{C}_{i}^{\star}(M_{0}\zeta_{i}(t),t)\in\mathcal{L}(\mathcal{H};\mathbb{R})$
is the output operator that has an integral kernel
$\mathcal{C}_{i}(M_{0}\zeta_{i}(t),t)\in L^{2}(\Omega)$ such that

$\mathcal{C}_{i}^{\star}(M_{0}\zeta_{i}(t),t)\phi=\iint_{\Omega}\mathcal{C}_{i}(M_{0}\zeta_{i}(t),t)(x,y)\phi(x,y)\text{d}x\text{d}y$

for $\phi\in\mathcal{H}$. Additive zero-mean Gaussian noise $v_{i}(t)$ with
variance $\sigma_{i}^{2}$ is included in the measurement.

The measurement can have many types, e.g., pointwise [9, 25, 24], interval
integral [7, 18], interval average [17], and Gaussian-type kernel [7]. Later
in the simulation section, we will use a time-invariant kernel given by the
square-shaped average

$\mathcal{C}_{i}(M_{0}\zeta_{i}(t))(x,y)\\\ =\left\\{\begin{aligned}
\frac{1}{4r_{i}^{2}}&\ \text{ if }\begin{bmatrix}x\\\
y\end{bmatrix}-M_{0}\zeta_{i}(t)\in[-r_{i},r_{i}]\times[-r_{i},r_{i}]\\\ 0&\
\text{ otherwise,}\end{aligned}\right.$ (8)

where $2r_{i}$ is the length of the side of the square at time $t$.

The measurements $y(t)\in\mathbb{R}^{m_{s}}$ of all sensors are compactly
written as

$y(t)=\mathcal{C}^{\star}(M\zeta(t),t)\mathcal{Z}(t)+v(t),$ (9)

where $\mathcal{C}^{\star}(M\zeta(t),t)$ is an operator-valued vector
$\mathcal{C}^{\star}(M\zeta(t),t):=[\mathcal{C}_{1}^{\star}(M_{0}\zeta_{1}(t),t),\dots,\mathcal{C}_{m_{s}}^{\star}(M_{0}\zeta_{m_{s}}(t),t)]^{\top}$
and $M\in\mathbb{R}^{2m_{s}\times n}$ is a matrix such that $M\zeta(t)$ is a
vector of locations of the sensors, i.e.,
$(M\zeta(t))^{\top}=[(M_{0}\zeta_{1}(t))^{\top},\dots,(M_{0}\zeta_{m_{s}}(t))^{\top}]$.
We sometimes use $\mathcal{C}(t)$ for brevity instead of
$\mathcal{C}(\zeta(t),t)$, because the sensor state $\zeta(t)$ is a function
of $t$. The measurement noise $v(t)$ is a zero-mean Gaussian vector with
covariance
$R:=\text{diag}(\sigma_{1}^{2},\sigma_{2}^{2},\dots,\sigma_{m_{s}}^{2})$.
Assume the noise $w_{0}(\cdot,\cdot),\ w(\cdot,\cdot,t)$, and $v(t)$ are
mutually independent for all $t$.

###  2.4 Kalman-Bucy filter

Analogous to a finite-dimensional linear system, the infinite-dimensional
linear system (6) and (9) admits a Kalman-Bucy filter (KF). For the derivation
of the KF of an abstract linear system, one may refer to [28, 33]. The
estimation $\hat{\mathcal{Z}}(t)$ of the state $\mathcal{Z}(t)$ can be updated
from the measurement $y(t)$ by

$\displaystyle\dot{\hat{\mathcal{Z}}}(t)$
$\displaystyle=\mathcal{A}\hat{\mathcal{Z}}(t)+\Pi(t)\mathcal{C}(t)R^{-1}(y(t)-\hat{y}(t)),$
(10) $\displaystyle\hat{y}(t)$
$\displaystyle=\mathcal{C}^{\star}(t)\hat{\mathcal{Z}}(t),$ (11)

with initial condition $\hat{\mathcal{Z}}(0):=\hat{\mathcal{Z}}_{0}$. The
predicted observation of the estimated system is denoted by $\hat{y}(t)$. The
covariance operator of the estimation error
$\Pi(t):=\mathbb{E}[(\mathcal{Z}(t)-\hat{\mathcal{Z}}(t))\circ(\mathcal{Z}(t)-\hat{\mathcal{Z}}(t))]$
satisfies the following operator Riccati equation:

$\dot{\Pi}(t)=\mathcal{A}\Pi(t)+\Pi(t)\mathcal{A}^{\star}+Q(t)\\\
-\Pi(t)\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)\Pi(t),$ (12)

where $\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)$ is a compact
representation of $\mathcal{C}(t)R^{-1}\mathcal{C}^{\star}(t)$. The initial
condition $\Pi(0)$ is given as the covariance operator $\Pi_{0}$ of the
initial estimation error
$\Pi_{0}:=\mathbb{E}[(\mathcal{Z}(0)-\hat{\mathcal{Z}}(0))\circ(\mathcal{Z}(0)-\hat{\mathcal{Z}}(0))]$
[45], which is the variance of $w_{0}$ and is nonnegative and self-adjoint.

######  Definition 1 ([6, Definition 1.2]).

The trace operator
$\text{Tr}(\cdot):\mathcal{L}(\mathcal{H})\rightarrow\mathbb{R}$ is defined as
$\text{Tr}(\mathcal{G})=\sum_{i=1}^{\infty}\langle\phi_{i},\mathcal{G}\phi_{i}\rangle$
for nonnegative $\mathcal{G}\in\mathcal{L}(\mathcal{H})$, where
$\\{\phi_{i}\\}_{i=1}^{\infty}$ is an arbitrary orthonormal basis that spans
$\mathcal{H}$.

Note that $\text{Tr}(\mathcal{G})$ is independent of the choice of the
orthonormal basis.

######  Definition 2 ([7, Definition 3.2]).

Let $\mathbb{H}$ be a separable complex Hilbert space. For $1\leq q<\infty$,
let $\mathcal{J}_{q}(\mathbb{H})$ denote the set of all bounded operators
$\mathcal{L}(\mathbb{H})$ such that $\text{Tr}([A]^{q})<\infty$, where
$[A]:=\sqrt{A^{\star}A}$. If $A\in\mathcal{J}_{q}(\mathbb{H})$, then the
$\mathcal{J}_{q}$-norm of $A$ is defined as $\left\lVert
A\right\rVert_{\mathcal{J}_{q}(\mathbb{H})}:=(\text{Tr}([A]^{q}))^{1/q}$.

The classes $\mathcal{J}_{1}(\mathcal{H})$ and $\mathcal{J}_{2}(\mathcal{H})$
are known as the space of trace-class 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 $1\leq q_{1}<q_{2}\leq\infty$ [6]. In other words, if
$A\in\mathcal{J}_{q_{1}}(\mathcal{H})$, then
$A\in\mathcal{J}_{q_{2}}(\mathcal{H})$ and ${\left\lVert
A\right\rVert_{\mathcal{J}_{q_{2}}(\mathcal{H})}\leq\left\lVert
A\right\rVert_{\mathcal{J}_{q_{1}}(\mathcal{H})}}$.

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

  1. (A1)

$\Pi_{0}\in\mathcal{J}_{q}(\mathcal{H})$ and $\Pi_{0}$ is nonnegative.

  2. (A2)

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

  3. (A3)

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

The existence of a mild solution of (12) is established in Lemma 3. The proof
is omitted because the lemma follows directly from [6, Theorem 3.6].

######  Lemma 3.

[6, Theorem 3.6] Let $\mathcal{H}$ be a separable Hilbert space. Suppose
assumptions (A1)–(A3) hold. Then, the equation

$\Pi(t)=\mathcal{S}(t)\Pi_{0}\mathcal{S}^{\star}(t)+\int_{0}^{t}\mathcal{S}(t-\tau)\\\
\left(Q(\tau)-\Pi(\tau)\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(\tau)\Pi(\tau)\right)\mathcal{S}^{\star}(t-\tau)\text{d}\tau$
(13)

provides a unique mild solution to (12) in the space
$L^{2}([0,t_{f}];\mathcal{J}_{2q}(\mathcal{H}))$. The solution is in
$C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ and is pointwise self-adjoint and
nonnegative.

The covariance operator $\Pi(t)$ characterizes the uncertainty of the
estimation error. The expected value of the squared norm of the estimation
error is the trace of the covariance operator $\Pi(t)$ [7, 45]:
$\text{Tr}(\Pi(t))=\mathbb{E}[\lVert\mathcal{Z}(t)-\hat{\mathcal{Z}}(t)\rVert_{\mathcal{H}}^{2}]$.

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

  1. (A4)

For each sensor $i\in\\{1,2,\dots,m_{s}\\}$, the kernel of the output operator
$\mathcal{C}_{i}(x,t)$ is continuous with respect to location
$x\in\mathbb{R}^{2}$ [7, Definition 4.5]. That is, there exists a continuous
function $l:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that $l(0)=0$ and
$\left\lVert\mathcal{C}_{i}(x_{1},t)-\mathcal{C}_{i}(x_{2},t)\right\rVert_{L^{2}(\Omega)}\leq
l(|x_{1}-x_{2}|_{2})$ for all $t\in[0,t_{f}]$ and all
$x_{1},x_{2}\in\mathbb{R}^{2}$.

Assumption (A4) is important in that, roughly speaking, it establishes the
continuity of the covariance operator with respect to sensor state (Lemma 5),
which further permits the existence of a solution to the optimization problem
proposed in this paper (Theorem 8), its finite-dimensional approximation
(Theorem 9), and the convergence to the exact optimal cost of the approximate
optimal cost (Theorem 10).

######  Remark 4.

The time invariant kernel in (8) is continuous with respect to location, where
$l(u)=(4r_{i}c_{0}u+u^{2})^{1/2}/(4r_{i}^{2})$ for sensor $i$ in assumption
(A4) for $c_{0}>0$.

The sensors’ locations determine where the output is measured and,
furthermore, how the covariance operator evolves through (13). We characterize
this relation by a composite mapping. Since the output operator
$\mathcal{C}^{\star}(\cdot,t)$ is a mapping of the sensors’ locations at time
$t$, the composite output operator
$\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(\cdot)$ is a mapping of the sensor
state in $[0,t_{f}]$ and so is $\Pi(\cdot)$ by (13), although the sensor state
is not explicitly reflected in the notation of the latter two mappings. Hence,
we can define the uncertainty cost
$\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t$ as a mapping of the sensor state.
Let $K:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$ such that
$K(\zeta):=\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t$. Lemma 5 below states
the continuity of the uncertainty cost with respect to the sensor state. Its
proof can be found in the supplementary material.

######  Lemma 5.

Let assumptions (A1)–(A3) hold with $q=1$ and $\Pi\in
C([0,t_{f}];\mathcal{J}_{1}(\mathcal{H}))$ be defined in (13). If assumption
(A4) holds, then the mapping $K(\cdot)$ is continuous.

###  2.5 Finite-dimensional approximation

A finite-dimensional approximation of the infinite-dimensional state estimate
$\hat{\mathcal{Z}}(t)$ and covariance operator $\Pi(t)$ is necessary for
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
$\hat{Z}_{N}(t):=P_{N}\hat{\mathcal{Z}}(t)$ and
$S_{N}(t):=P_{N}\mathcal{S}(t)P_{N}$ be the finite-dimensional approximations
of $\hat{\mathcal{Z}}(t)$ and $\mathcal{S}(t)$, respectively. The approximated
version of the estimation (10) is

$\displaystyle\dot{\hat{Z}}_{N}(t)$
$\displaystyle=A_{N}\hat{Z}_{N}(t)+\Pi_{N}(t)C_{N}(t)R^{-1}(y(t)-\hat{y}_{N}(t)),$
(14) $\displaystyle\hat{y}_{N}(t)$
$\displaystyle=C_{N}^{\star}(t)\hat{Z}_{N}(t),$ (15)

with initial condition $\hat{Z}_{N}(0)=P_{N}\hat{\mathcal{Z}}_{0}$. The
approximations $A_{N}\in\mathcal{L}(\mathcal{H}_{N})$ and
$C^{\star}_{N}(t)\in\mathcal{L}(\mathcal{H}_{N};\mathbb{R}^{m_{s}})$ are of
$\mathcal{A}$ and $\mathcal{C}^{\star}(t)$, respectively, and $\Pi_{N}(t)$ is
the finite-dimensional approximation of $\Pi(t)$ such that

$\displaystyle\Pi_{N}(t)=$ $\displaystyle
S_{N}(t)\Pi_{0,N}S_{N}^{\star}(t)+\int_{0}^{t}S_{N}(t-\tau)\big{(}Q_{N}(\tau)$
$\displaystyle-\Pi_{N}(\tau)\bar{C}_{N}\bar{C}_{N}^{\star}(\tau)\Pi_{N}(\tau)\big{)}S_{N}^{\star}(t-\tau)\text{d}\tau,$
(16)

where $\Pi_{0,N}:=P_{N}\Pi_{0}P_{N}$ and $Q_{N}(t):=P_{N}Q(t)P_{N}$ are
approximations of $\Pi_{0}$ and $Q(t)$, respectively, and
$\bar{C}_{N}\bar{C}_{N}^{\star}(\tau)$ is short for
$C_{N}R^{-1}C_{N}^{\star}(\tau)$.

If the subspace $\mathcal{H}_{N}$ is chosen such that it is spanned by the
first $N$ functions of the orthonormal basis $\\{\phi_{i}\\}_{i=1}^{\infty}$
that spans $\mathcal{H}$, then
$\text{Tr}(\Pi_{N}(t))=\text{Tr}(P_{N}\Pi(t)P_{N})=\sum_{i=1}^{N}\langle\phi_{i},\Pi(t)\phi_{i}\rangle.$
To establish convergence of the approximate covariance operator
$\Pi_{N}(\cdot)$ to the original operator $\Pi(\cdot)$, the following
assumptions are made:

  1. (A5)

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

  2. (A6)

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

  3. (A7)

Both $\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(\cdot)$ and sequence
$\\{\bar{C}_{N}\bar{C}_{N}^{\star}(\cdot)\\}_{N=1}^{\infty}$ are elements of
$L^{\infty}([0,t_{f}];\mathcal{L}(\mathcal{H}))$. Both
$\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)$ and
$\bar{C}_{N}\bar{C}_{N}^{\star}(t)$ are nonnegative for all $N$ and
$t\in[0,t_{f}]$. And
${\operatorname*{ess\,sup}}_{t\in[0,t_{f}]}\left\lVert\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)-\bar{C}_{N}\bar{C}_{N}^{\star}(t)\right\rVert_{\text{op}}\rightarrow
0$ holds as $N\rightarrow\infty$ ($\left\lVert\cdot\right\rVert_{\text{op}}$
denotes the operator norm).

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

The convergence of the approximate covariance operator $\Pi_{N}(\cdot)$ is
stated in the next theorem whose proof is omitted since the theorem is
reproduced from [6, Theorem 3.5].

######  Theorem 6 ([6, Theorem 3.5]).

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

$\left\lVert\mathcal{S}(t)\phi-S_{N}(t)\phi\right\rVert\rightarrow
0,\quad\left\lVert\mathcal{S}^{\star}(t)\phi-
S_{N}^{\star}(t)\phi\right\rVert\rightarrow 0$ (17)

as $N\rightarrow\infty$, uniformly in $[0,t_{f}]$ for each
$\phi\in\mathcal{H}$. Suppose assumptions (A5)–(A7) hold. If $\Pi(\cdot)\in
C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ is a solution of (13) and
$\Pi_{N}(\cdot)\in C([0,t_{f}];\mathcal{J}_{q}(\mathcal{H}))$ is the sequence
of solution of (2.5), then
$\sup_{t\in[0,t_{f}]}\left\lVert\Pi(t)-\Pi_{N}(t)\right\rVert_{\mathcal{J}_{q}(\mathcal{H})}\rightarrow
0$ as $N\rightarrow\infty$.

The following assumption and lemma are related to the continuity with respect
to location of the approximate output kernel and the continuity with respect
to sensor state of the trace of the approximate covariance operator, which are
analogous to assumption (A4) and Lemma 5, respectively.

  1. (A8)

The approximated input operator $C_{i,N}(x,t)$ 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
$\left\lVert C_{i,N}(x_{1},t)-C_{i,N}(x_{2},t)\right\rVert_{L^{2}(\Omega)}\leq
l_{N}(|x_{1}-x_{2}|_{2})$ for all $t\in[0,t_{f}]$, all
$x_{1},x_{2}\in\mathbb{R}^{2}$, and all $i\in\\{1,2,\dots,m_{s}\\}$.

Similar to the mapping $K(\cdot)$ in Lemma 5, we can characterize the
approximate uncertainty cost as a mapping of the sensor state $\zeta$, where
continuity is established in Lemma 7, whose proof is in the supplementary
material.

######  Lemma 7.

Let assumptions (A5)–(A7) hold and $\Pi_{N}(t)$ be defined as in (2.5). If
assumption (A8) holds, then the mapping
$K_{N}:C([0,t_{f}];\mathbb{R}^{n})\rightarrow\mathbb{R}^{+}$ such that
$K_{N}(\zeta):=\int_{0}^{t_{f}}\text{Tr}(\Pi_{N}(t))\text{d}t$ is continuous.

##  3 Problem formulation

We now introduce the formulation of the optimization problem. Given the
dynamics and initial condition (2) of the sensors, the dynamics of the
diffusion-advection process (6), and the second moment of the initial state
noise $w_{0}(\cdot,\cdot)$, the process noise $w(\cdot,\cdot,t)$, and
measurement noise $v(t)$, the problem below yields the optimal guidance for a
team of mobile sensors to estimate a 2D diffusion-advection process.

The cost function consists of two parts: one part accounts for reducing the
estimation uncertainty (uncertainty cost), and the other accounts for the
motion of the sensors (mobility cost). The uncertainty cost is the integral of
the trace of the covariance operator $\Pi(\cdot)$ over the horizon
$[0,t_{f}]$, i.e., $\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t$. The mobility
cost $J_{\text{m}}(\zeta,p)$ is defined as

$J_{\text{m}}(\zeta,p)=\int_{0}^{t_{f}}h(\zeta(t),t)+g(p(t),t)\text{d}t+h_{f}(\zeta(t_{f})).$
(18)

Here, $h:\mathbb{R}^{n}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$ is a
continuous function that characterizes the cost associated with the state of
the mobile sensors. For example, a hazardous field can be modeled by $h$,
where $h(\zeta(t),t)$ evaluates the exposure of the mobile sensors, which can
shorten the sensor’s life span. The cost of the guidance is characterized by
$g:\mathbb{R}^{m}\times[0,t_{f}]\rightarrow\mathbb{R}^{+}$. For example, a
quadratic guidance effort is $g(p(t),t)=p^{\top}(t)\gamma p(t)$, where
$\gamma\in\mathbb{R}^{m\times m}$ is symmetric and positive definite. The
guidance cost can address limited onboard resources, like fuel or batteries,
by treating $\gamma$ as the penalty coefficient. The terminal state cost
$h_{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ evaluates the cost associated
with the terminal state. An exemplary scenario is when the sensors are expect
to come close to a set of pre-assigned terminal locations
$x_{f}\in\Omega^{m_{s}}$, where
$h_{f}(\zeta(t_{f})):=|M\zeta(t_{f})-x_{f}|_{2}^{2}$.

The motion of the sensors follow from the dynamics (2), which constrain the
optimization. Denote the admissible set of guidance functions as
$\mathcal{P}:=L^{2}([0,t_{f}];P)$, where $P$ is the set of admissible guidance
(values) defined at the end of Section 2.2.

The optimization problem is formulated as follows:

$\displaystyle\underset{p\in\mathcal{P}}{\text{minimize}}$
$\displaystyle\int_{0}^{t_{f}}\text{Tr}(\Pi(t))+h(\zeta(t),t)+g(p(t),t)\text{d}t$
(P) $\displaystyle+h_{f}(\zeta(t_{f}))$ subject to
$\displaystyle\dot{\zeta}(t)=a\zeta(t)+bp(t),\ \zeta(0)=\zeta_{0},$

where $\Pi(t)$ is given by (13) with a given initial condition
$\Pi(0)=\Pi_{0}$. It suffices to search for guidance $p$ that minimizes the
cost of (P), because the sensor state $\zeta$ is entirely determined by
guidance $p$ via the sensor dynamics and the given initial condition
$\zeta_{0}$, which further determines $\Pi(\cdot)$ through (13) with a given
initial covariance $\Pi(0)$.

A special case is considered in [10] where only a quadratic guidance effort is
considered in the mobility cost. Such a formulation applies to the case of
limited onboard resources of each mobile sensor when $\gamma$ is diagonal. It
minimizes the Lagrangian function of the optimization problem that minimizes
the uncertainty cost subject to the constraints of bounded guidance effort and
linear dynamics of the mobile sensors.

The following three assumptions are necessary for the existence of a solution
to problem (P).

  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 $g(p,t)$ is convex about $p$.

  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 met in applications with real vehicles.
Assumption (A9) is satisfied when the values of admissible guidance vary along
a continuum. The continuity requirement in assumption (A10) on the cost
functions $h$, $g$, and $h_{f}$ is typically satisfied. And the convexity
requirement in assumption (A10) and quadratic boundedness from below in
assumption (A11) can be met if $g$ is quadratic in $p$, e.g.,
$g(p,t)=p^{\top}(t)\gamma(t)p(t)$ for a symmetric and positive definite matrix
$\gamma(t)$ (which is continuous with respect to $t$) such that $d_{1}$ can be
chosen as the minimum eigenvalue of $\gamma(t)$ for $t\in[0,t_{f}]$ (see [3,
Corollary VI.1.6]).

Theorem 8 below states the existence of a solution to problem (P), whose proof
can be found in Appendix A.

######  Theorem 8.

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

We use Pontryagin’s minimum principle to characterize an optimal solution of
(P). Consider the Hamiltonian

$H(\zeta(t),p(t),\lambda(t),t)=\text{Tr}(\Pi(t))\\\
+h(\zeta(t),t)+g(p(t),t)+\lambda^{\top}(t)(a\zeta(t)+bp(t)),$ (19)

where $\lambda(t)\in\mathbb{R}^{n}$ is the costate associated with $\zeta(t)$.
The necessary conditions of (local) optimality are as follows:

$\displaystyle\dot{\zeta}^{*}(t)=$ $\displaystyle\ a\zeta^{*}(t)+bp^{*}(t),$
(20a) $\displaystyle\zeta^{*}(0)=$ $\displaystyle\ \zeta_{0},$ (20b)
$\displaystyle\dot{\lambda}^{*}(t)=$ $\displaystyle\
-a^{\top}\lambda^{*}(t)-\nabla_{\zeta}h(\zeta^{*}(t),t)$ $\displaystyle\
-\nabla_{\zeta}\text{Tr}(\Pi^{*}(t)),$ (20c)
$\displaystyle\lambda^{*}(t_{f})=$ $\displaystyle\
\nabla_{\zeta}h_{f}(\zeta^{*}(t_{f})),$ (20d) $\displaystyle 0=$
$\displaystyle\ \nabla_{p}g(p^{*}(t),t)+b^{\top}\lambda^{*}(t),$ (20e)

where $\Pi^{*}(\cdot)$ is evaluated along the optimal system state
$\zeta^{*}(\cdot)$ and we use the first-order necessary condition
$\nabla_{p}H(\zeta^{*}(t),p^{*}(t),\lambda^{*}(t),t)=0$ in (20e) for $H$ to
attain its minimum at $p^{*}(t)$. The necessary condition (20) essentially
requires the solution to a two-point boundary value problem, which further
requires the derivation of $\nabla_{\zeta}\text{Tr}(\Pi^{*}(t))$. We refer to
a similar derivation in [7], where the gradient of the covariance operator’s
trace with respect to sensor’s guidance is taken. The $i$th row of
$\nabla_{\zeta}\text{Tr}(\Pi^{*}(t))$, denoted by
$[\nabla_{\zeta}\text{Tr}(\Pi^{*}(t))]_{i}$, is the partial derivative of
$\text{Tr}(\Pi^{*}(t))$ with respect to the $i$th element of the state
$\zeta^{*}(t)$ for $i\in\\{1,2,\dots,n\\}$. Since trace is a linear operator,
we have

$[\nabla_{\zeta}\text{Tr}(\Pi^{*}(t))]_{i}=\frac{\partial\text{Tr}(\Pi^{*}(t))}{\partial[\zeta(t)]_{i}}=\text{Tr}(\frac{\partial\Pi^{*}(t)}{\partial[\zeta(t)]_{i}}).$
(21)

By the chain rule, (21) becomes

$\text{Tr}(\frac{\partial\Pi^{*}(t)}{\partial[\zeta(t)]_{i}})=\text{Tr}(D_{\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)}\Pi^{*}(t)\circ
D_{[\zeta(t)]_{i}}\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)),$ (22)

where $D_{\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)}\Pi(t)$ is the Fréchet
derivative of the Riccati operator with respect to the composite output
operator $\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)$ and
$D_{[\zeta(t)]_{i}}\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)$ is the
Fréchet derivative of $\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)$ with
respect to $[\zeta(t)]_{i}$. Denote
$D_{\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(t)}\Pi^{*}(t)$ by $\Lambda(t)$
and, by [7, Theorem 5.5], $\Lambda(t)$ is the unique solution to

$\displaystyle\Lambda h(t)=$ $\displaystyle\
-\int_{0}^{t}\mathcal{S}(t-s)\big{(}(\Lambda
h)(s)\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(s)\Pi(s)$
$\displaystyle+\Pi(s)\bar{\mathcal{C}}\bar{\mathcal{C}}^{\star}(s)(\Lambda
h)(s)$
$\displaystyle+\Pi(s)h(s)\Pi(s)\big{)}\mathcal{S}^{\star}(t-s)\text{d}s$ (23)

with $\Lambda(0)=0$ for all $h\in C([0,t_{f}];\mathcal{J}_{1}(\mathcal{H}))$
and all $t\in[0,t_{f}]$. The approximated version of problem (P) and the two-
point boundary value problem (20) will be applied to solve for optimal
guidance in Section 4.

##  4 Solving optimal guidance using approximation

Since the infinite-dimensional terms in (P) have to be approximated for
computation as introduced in Section 2.5, we arrive at the approximated
problem:

$\displaystyle\underset{p_{N}\in\mathcal{P}}{\text{minimize}}$
$\displaystyle\int_{0}^{t_{f}}\text{Tr}(\Pi_{N}(t))+h(\zeta(t),t)+g(p_{N}(t),t)\text{d}t$
(AP) $\displaystyle+h_{f}(\zeta(t_{f}))$ subject to
$\displaystyle\dot{\zeta}(t)=a\zeta(t)+bp_{N}(t),\ \zeta(0)=\zeta_{0},$

where $\Pi_{N}(t)$ is obtained through (2.5). It suffices to search for
guidance $p_{N}$, because both the sensor state $\zeta$ and the approximated
estimation covariance $\Pi_{N}$ are fully determined by the guidance and
initial conditions. The existence of a solution of problem (AP) is guaranteed
in Theorem 9 with the proof in Appendix B.

######  Theorem 9.

Consider problem (AP) and let assumptions (A5)–(A11) hold. Then problem (AP)
has a solution.

Solving problem (AP) provides a candidate solution, denoted by $p_{N}^{*}$,
where $N$ is the dimension of the approximation. The candidate $p_{N}^{*}$ may
not equal the exact optimal solution, denoted by $p^{*}$, of the original
problem (P). However, as we show in the following theorem, the candidate
$p_{N}^{*}$ yields the optimal value of (AP) arbitrarily close to the one of
(P), as the dimension $N$ goes to infinity. Moreover, when $p_{N}^{*}$ is
evaluated in the original problem (P), the resulting cost is arbitrarily close
to the optimal cost of (P).

Before we state this convergence result, we introduce an assumption on the set
of admissible guidance functions.

  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 $|p(t_{1})-p(t_{2})|\leq
a_{\max}|t_{1}-t_{2}|,\ \forall t_{1},t_{2}\in[0,t_{f}]\\}$.

Notice that the set $\mathcal{P}(p_{\max},a_{\max})$ is sequentially compact,
due to the Arzelà-Ascoli Theorem [35], since the guidance functions in
$\mathcal{P}(p_{\max},a_{\max})$ are uniformly equicontinuous and uniformly
bounded. The parameters $p_{\max}$ and $a_{\max}$ may be determined by the
vehicles carrying sensors. For example, $p_{\max}$ and $a_{\max}$ refer to the
maximum speed and maximum acceleration, respectively, in the case of single
integrator dynamics where $p$ is the velocity command.

Theorem 10 below states the convergence of the approximation with the proof in
Appendix C.

######  Theorem 10.

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

$\lim_{N\rightarrow\infty}|J^{*}_{\eqref{prob: approximated min covariance
sensor guidance problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: min covariance sensor
guidance problem}}(p^{*})|=0.$ (24)

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

$\lim_{N\rightarrow\infty}|J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})|=0.$ (25)

######  Remark 11.

Two implications follow from the convergence result in Theorem 10. First, the
convergence in (24) affirms the usage of approximation since the optimal cost
of the approximate problem (AP) gets arbitrarily close to that of the exact
problem (P) as the approximation gets finer. Second, the convergence in (25)
affirms the optimal guidance computed using approximation. When the
approximate optimal guidance is evaluated by the cost function of the original
problem (P), the resulting value is arbitrarily close to the optimal cost of
(P) as the approximation gets finer. In other words, the approximate optimal
guidance is a sufficiently accurate proxy for the exact optimal guidance.

######  Remark 12.

Assumptions are made for the existence of a solution to problem (P)
((A9)–(A11)), well-posedness of the Riccati operators ((A1)–(A4)), and the
convergence of the approximated solution ((A5)–(A8) and (A12)). Assumptions
(A9)–(A11) are regarding the mobility cost and the set of admissible guidance,
which are generally satisfied in engineering applications (see the discussion
in [11] before Theorem 3.1). The rest of the assumptions are typically
satisfied with the diffusion-advection equation and Galerkin approximation
(using eigenfunctions of the Laplacian operator). Details of how to check
similar assumptions for the dual control problem can be found in [11, Section
4.1].

The convergence stated in Theorem 10 is established based on several earlier
stated results, including

  1. 1.

the output operator’s continuity with respect to location (assumption (A4)),
which leads to the continuity of the uncertainty cost with respect to sensor
state (Lemma 5);

  2. 2.

existence of the Riccati operator (Lemma 3) and convergence of its
approximation (Theorem 6); and

  3. 3.

sequential compactness of the set of admissible guidance functions (assumption
(A12)), which leads to the continuity of the cost function with respect to
guidance (Lemma 14 in the Appendix).

Notice that these key results, in an analogous manner, are also required in
[45] when establishing the convergence to the exact optimal sensor locations
of the approximate optimal locations [45, Theorem 4.3], i.e.,

  1. 1.

continuity with respect to location and compactness of the output operator
(dual of [29, Theorem 2.10]), which lead to continuity of the Riccati operator
with respect to sensor locations [45, Theorem 4.1];

  2. 2.

existence of the Riccati operator [45, Theorem 2.5] and the convergence of its
approximation [45, Theorem 4.2]; 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.

To compute an optimal solution to problem (AP), we use Pontryagin’s minimum
principle, which introduces a Hamiltonian function that has the same form as
(19) except that the covariance operator $\Pi(t)$ in (19) is replaced by its
approximation $\Pi_{N}(t)$. And correspondingly, the resulting two-point
boundary value problem has the same form as (20) except for $\Pi(t)$ being
replaced by $\Pi_{N}(t)$.

######  Remark 13.

The optimal sensor trajectory $\zeta^{*}$ steered by the optimal guidance
$p^{*}$ may be used as the reference trajectory tracked by the vehicle’s
lower-level control. Although collision avoidance among the sensors is not
discussed in this paper, it can be incorporated into the lower-level control
using numerous methods in the existing literature, e.g., [42] and the
references therein.

##  5 Simulation results

This section shows the simulation results obtained using the solution method
proposed in Section 4. Comparison and analysis are made regarding the
performance of the mobile sensor(s) under optimal guidance for the case of a
single sensor, a team of homogeneous sensors, and a team of heterogeneous
sensors.

We use the Galerkin scheme to approximate the infinite-dimensional terms. 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 $\phi_{i,j}(x,y)=2\sin(\pi ix)\sin(\pi jy)$. With
a single index $k:=(i-1)N+j$ such that $\phi_{k}:=\phi_{i,j}$, the set of
eigenfunctions $\\{\phi_{k}\\}_{k=1}^{N^{2}}$ spans an $N^{2}$-dimensional
space that is previously denoted by $\mathcal{H}_{N}$. For 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. And the
assumption (17) in Theorem 6 holds uniformly for all $t\in[0,t_{f}]$ as
$N\rightarrow\infty$ [6]. We set $p_{\max}$ and $a_{\max}$ to be sufficiently
large so that the solution is in the set $\mathcal{P}(p_{\max},a_{\max})$. We
plot the optimal cost $J^{*}_{\eqref{prob: approximated min covariance sensor
guidance problem}}(p_{N}^{*})$ for $N$ from 7 to 20, as shown in Fig. 1. The
optimal cost shows a tendency of exponential convergence as we increase the
number of basis functions. And we choose $N=12$ in the rest of the simulations
since it is the smallest dimension with the optimal cost within 1% of the
optimal cost evaluated with the maximum dimension $N=20$ in the trials.

The parameters in the simulation are $t_{f}=2$, and $a=0.01$. We use single
integrator dynamics for each sensor. The state $\zeta$ is the 2D location of
the sensors and guidance $p$ is the 2D velocity command. In some applications,
the flow field $\mathbf{v}$ of the diffusion-advection process can affect the
mobile sensors. For example, surface vehicles that measure the concentration
of certain chemical substances or biological entities in a water body are
subject to the movement of the water. Considering this realistic condition, we
append the flow field $\mathbf{v}=[0.1,-0.1]^{\top}$ of advection to the
right-hand side of the single integrator dynamics, which means the sensors
will drift along the flow when zero guidance is implemented. The previous
statements and results on the existence of solution and convergence of the
approximate solution still hold within this setting. The optimization will
find optimal guidance subject to (or possibly taking advantage of) this flow
field. The sensor has the square-shaped average output kernel (see (8)) with
$r_{i}=0.05$, in which case its footprint covers only $1\%$ of the domain in
area.

We set $g(p(t))=\gamma p^{\top}(t)p(t)/2$ and
$h(\zeta(t),t)=h_{f}(\zeta(t_{f}))=0$ as the mobility cost, which is simply
the quadratic guidance effort for $\gamma>0$.

Assumption (A4) holds for the choice of output operator (see Remark 4). With
the Galerkin approximation using the orthonormal eigenfunctions
$\\{\phi_{k}\\}_{k=1}^{N^{2}}$, 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
$\\{\phi_{k}\\}_{k=1}^{N^{2}}$ [6]. Assumptions (A9)–(A11) and (A12) hold for
the choice of functions in the mobility cost and parameters of the set of
admissible guidance functions, respectively.

Figure 1: Approximate optimal costs $J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}}(p_{N}^{*})$ normalized with respect to
the optimal cost for $N^{2}=400$.

To evaluate the performance of the optimal guidance in simulation, we set the
deterministic portion of the initial condition of the PDE to zero, i.e.,
$\hat{Z}_{0}=0$, which excludes the bias from choosing a particular non-zero
one. The stochastic portion of the initial condition $w_{0}$ is chosen as a
zero-mean Gaussian process with non-stationary kernel function
$k_{0}:\Omega\times\Omega\rightarrow\mathbb{R}^{+}$ such that
$k_{0}(x_{1},x_{2}):=9\text{exp}(-|x_{12}|_{2}^{2}/200-|x_{10}|_{2}^{2}/10-|x_{20}|_{2}^{2}/10)$,
where $x_{ij}:=x_{i}-x_{j}$ for $i,j\in\\{0,1,2\\}$ and $x_{0}\in\Omega$
represents the peak of the uncertainty in the domain and we set
$x_{0}=[0.75,0.25]^{\top}$. For the state noise $w(t)$, we use a zero-mean
Gaussian process with a homogeneous kernel function
$k:\Omega\times\Omega\rightarrow\mathbb{R}^{+}$ such that
$k(x_{1},x_{2})=\text{exp}(-|x_{12}|_{2}^{2}/2000)$.

We use the forward-backward sweep method [27] to solve the two-point boundary
problem (20) (with $\Pi$ replaced by $\Pi_{N}$) and subsequently compute the
optimal guidance. A fixed-step length of 0.01 and a relative tolerance of
$1\times 10^{-6}$ are applied in the iterative procedure. The forward
propagation of (20a) and (2.5) and the backward propagation of (20c) are
computed via the Runge-Kutta method.

###  5.1 Single sensor results

Two important parameters in the problem setting are the sensor noise variance
$R$ and mobility penalty $\gamma$. Smaller $R$ yields higher sensor quality,
whereas smaller $\gamma$ yields better mobility of the vehicle. For example,
if $\gamma$ is the mass of the vehicle, then the guidance effort is the
kinetic energy of the vehicle. These parameters affect the performance of the
estimation as shown next. In this simulation, the sensor is initiated at
$\zeta_{0}=[0.3,0.1]^{\top}$. Monte Carlo simulations of 100 trials compare
the optimal guidance with three naive guidance policies as follows: naive 1
crosses the domain by reaching the opposite of the initial location within
domain at $[0.7,0.9]^{\top}$ at a constant speed; naive 2 reaches the peak of
the initial uncertainty $x_{0}$ at a constant speed; and naive 3 circulates
the domain in the clockwise direction with center $[0.5,0.5]^{\top}$, radius
$1/\sqrt{5}$, and angular speed $\pi$ rad/s. A stationary sensor is also
included for comparison whose null guidance merely compensates for the flow
field $\mathbf{v}$.

(a) Optimal trajectories for various values of the sensor noise variance $R$

(b) Norm of the terminal estimation error for various values of sensor noise
variance $R$. The color bar shows the mean value; the error bar shows the
standard deviation.

Figure 2: The value of mobility penalty $\gamma$ is fixed at 0.5, whereas the
sensor noise’s variance $R$ takes values in the set $\\{0.2,0.4,0.6,0.8,1\\}$.

First, hold either $R$ or $\gamma$ fixed and vary the other to observe the
variation of the optimal trajectory. Fig. 2(a) displays the trajectories when
$\gamma=0.5$ and $R$ varies from $0.2$ to $1$. The sensor maneuvers less as
$R$ increases, which indicates the optimal guidance’s compensation for
deteriorating sensor quality by moving it closer to the peak of the initial
uncertainty at $x_{0}=[0.75,0.25]^{\top}$.

For the Monte Carlo trials, the mean and standard deviation of the terminal
estimation error’s norm are shown in Fig. 2(b). The optimal guidance exhibits
smaller mean and variance of terminal estimation error over the naive guidance
policies and the null guidance at each evaluated $R$. Notice that the
advantage is preserved when the sensor quality deteriorates as $R$ increases.

(a) Optimal trajectories for various values of the mobility penalty $\gamma$

(b) Norm of the terminal estimation error for various values of the mobility
penalty $\gamma$. The color bar shows the mean value; the error bar shows the
standard deviation.

Figure 3: The value of sensor noise’s variance $R$ is fixed at $0.2$, whereas
the mobility penalty $\gamma$ takes values in the set $\\{0.5,1,1.5,2,2.5\\}$.

Fig. 3(a) displays the trajectories when $R=0.2$ and $\gamma$ varies from
$0.5$ to $2.5$. A bigger value of $\gamma$ suppresses the guidance effort and
hence reduces the range of the sensor’s motion. The results of Monte Carlo
trials when $\gamma$ varies are shown in Fig. 3(b). Notice that the mean and
standard deviation are invariant for each naive guidance policy and the null
guidance among the varying $\gamma$ since the sensor trajectory steered by
each of these guidance policies is independent of $\gamma$. The optimal
guidance shows its advantage over the other guidance policies at a relatively
smaller values of $\gamma$, e.g., at $0.5$ and $1$. This advantage is
gradually lost as $\gamma$ takes relatively bigger values, e.g., $2$ and
$2.5$. This comparison suggests that the optimal guidance may not be the best
option for vehicles with large $\gamma$ (e.g., when the vehicle is heavy),
despite the fact that the guidance is still optimal for the chosen cost
function in simulation.

Figure 4: Trajectory of the sensor following the optimal guidance. The
contours represent the level curves of pointwise variance of the estimation
error on a uniform grid within the domain, computed from the Monte Carlo
simulations.

Fig. 4 shows the snapshots of the sensor trajectory under the optimal guidance
and contour plot of the pointwise variance of the estimation error among the
Monte Carlo trials with $\gamma=1$ and $R=0.2$. The pointwise variance is
computed at each point in a uniform grid of $144\times 144$ sampling points in
the domain $\Omega$. The sensor is steered quickly towards the area with
higher uncertainty near $[0.75,0.25]^{\top}$ (see the snapshot at $t=0.8$ s).
Eventually, the sensor starts to drift along the flow field
$\mathbf{v}=[0.1,-0.1]^{\top}$ (see the snapshot at $t=2$ s). That sensor can
effectively reduce the uncertainty of the estimation error as can be observed
from the drop in the pointwise variance in the sensor’s footprint. Note that
the zero Dirichlet boundary condition also contributes to reducing the
uncertainty of the estimation error via diffusion and advection.

###  5.2 Team of homogeneous sensors

Figure 5: Trajectories of multiple homogeneous sensors following the optimal
guidance. The contours represent the level curves of pointwise variance of the
estimation error on a uniform grid within the domain.

To demonstrate the framework’s capability in guiding a team of multiple
sensors, we simulate four homogeneous sensors ($R=0.2$, $\gamma=0.5$). To
adapt to a total of four sensors, the kernel functions for $w_{0}$ and $w$ are
set to be $4k_{0}(x_{1},x_{2})$ and $4k(x_{1},x_{2})$, respectively, and the
peak of the initial uncertainty is set to the center $[0.5,0.5]^{\top}$. The
other settings are identical to those in Section 5.1. Fig. 5 shows the
snapshots of the sensors’ trajectories under the optimal guidance and contour
plot of the pointwise variance of the estimation error among the Monte Carlo
trials. Similar to the case of a single sensor in Fig. 4, the sensors quickly
sweep the peak of the initial uncertainty in the center and expand to cover
the domain (see the snapshot at $t=0.6$ s). The pointwise variance drops along
sensors’ footprints. The sensors essentially drift along the flow to reduce
the guidance effort (see the snapshot at $t=2$ s).

###  5.3 Team of heterogeneous sensors

The parameters $R$ and $\gamma$ essentially relate to operational planning:
one may invest more for better sensor quality or a swifter vehicle.
Consequently, one would necessarily invest more for a team of superior mobile
sensors (e.g., $R=0.2$ and $\gamma=0.5$) than a team of poor mobile sensors
(e.g., $R=1$ and $\gamma=2.5$). The latter has five times as much sensor noise
(in terms of standard deviation) and five times the mobility penalty as the
former. One may balance the conflicting needs of performance and investment by
deploying a team of heterogeneous sensors, i.e., a mixed team of superior and
poor sensors.

The following simulation compares the performance of a team of heterogeneous
sensors (including $m_{p}$ poor sensors and $8-m_{p}$ superior sensors, for
$m_{p}$ in the range of $\\{1,2,\dots,7\\}$) with that of homogeneous teams
($m_{p}=0$ for superior sensors only and $m_{p}=8$ for poor sensors only). The
sensors are initiated in the lower left corner of the domain. To adapt to a
total of eight sensors, the kernel functions for $w_{0}$ and $w$ are set to be
$8k_{0}(x_{1},x_{2})$ and $8k(x_{1},x_{2})$, respectively, and the peak of the
initial uncertainty is set to the center of the domain at $[0.5,0.5]^{\top}$.
Fig. 6 shows the normalized optimal total cost and uncertainty cost for the
heterogeneous team compared with homogeneous teams of superior and poor
sensors. The performance deteriorates judged by the rising costs as the number
of poor sensors increases in team. However, the degradation of the
heterogeneous team (when $m_{p}\leq 5$) is maintained within $20\%$ of the
superior team in both the total cost and uncertainty cost, which indicates the
cost effectiveness of the heterogeneous team since the investment reduces
linearly as $m_{p}$ increases.

Figure 6: Normalized optimal total cost and uncertainty cost of a
heterogeneous team with $m_{p}$ poor sensors and $8-m_{p}$ superior sensors.

##  6 Conclusion

This paper proposes a guidance design method for a team of mobile sensors to
estimate a spatiotemporal process modeled by a 2D diffusion-advection process.
We formulate an optimization problem that minimizes the sum of the trace of
the covariance operator of the Kalman-Bucy filter and a generic mobility cost
of the sensors subject to the dynamics of the sensor platforms. Conditions for
the existence of a solution to the proposed problem are established, where a
fundamental assumption is that the output kernel is continuous with respect to
location. Approximation of the infinite-dimensional terms permits the
computation of the optimal guidance, and we prove that the approximated
problem’s optimal cost converges to that of the exact problem. Moreover, we
prove that the optimal guidance obtained from an approximate problem yields
the cost evaluated by the exact problem’s cost function arbitrarily close to
the exact optimal cost. Exponential convergence in the optimal cost can be
observed in simulations when the approximation dimension increases.

We use Pontryagin’s minimum principle to numerically compute optimal guidance.
The numerical solutions are evaluated in simulations. We study how the optimal
trajectory and terminal estimation error change subject to the varying values
of sensor noise variance and the mobility penalty for a single sensor.
Trajectories of the sensors and the evolution of the pointwise variance are
shown for a single sensor and a homogeneous team of sensors in a flow field.
We also study the cost-effectiveness of a heterogeneous team of mixed sensors
with superior and poor qualities by comparing the performance degradation to
that of the homogeneous team of superior sensors.

Ongoing and future work includes establishing a convergence rate in Theorem
10, incorporating constraints to the problem formulation (e.g., constraints on
sensor states), extension to more complicated sensor dynamics (e.g., control-
affine), and demonstrating the framework experimentally using a swarm of
quadrotor helicopters in an outdoor testbed.

## Acknowledgments

The authors would like to thank the reviewers for their helpful suggestions.
Work reported in this article was sponsored under a seed grant from Northrop
Grumman.

##  Appendix A Proof of Theorem 8

Since we have proved that the uncertainty cost
$\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t$ is a continuous mapping
$K(\cdot)$ of the sensor state $\zeta$ (see Lemma 5), the existence of optimal
guidance of (P) can be proved using the techniques of proving existence of
solution to an optimal control (guidance) problem.

Proof of Theorem 8 Without loss of generality, we consider the case of one
mobile sensor, i.e., $m_{s}=1$. The case of $m_{s}\geq 2$ follows naturally.

Our proof follows the proof in [43, Chapter 6.2] which proves existence of a
solution to an optimal control problem based on functional analytic theorems.
Consider problem (P)’s admissible set of guidance functions
$\mathcal{P}:=\\{p\in L^{2}([0,t_{f}];\mathbb{R}^{m}):p(t)\in
P,t\in[0,t_{f}]\\}$. Since there exists $p_{0}\in\mathcal{P}$ such that
$J_{\eqref{prob: min covariance sensor guidance problem}}(p_{0})<\infty$
(e.g., $p_{0}=0$ that yields a stationary sensor at $\zeta_{0}$), let
$\mathcal{P}_{0}:=\\{p\in\mathcal{P}:\ J_{\eqref{prob: min covariance sensor
guidance problem}}(p)\leq J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})\\}$.

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

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

Condition-3: The mapping $J_{\eqref{prob: min covariance sensor guidance
problem}}(\cdot):\mathcal{P}\rightarrow\mathbb{R}^{+}$ is weakly sequentially
lower semicontinuous on $\mathcal{P}_{0}$.

Condition-1 and Condition-2 imply that $\mathcal{P}_{0}$ is weakly
sequentially compact. By [43, Theorem 6.1.4], problem (P) has a solution when
Condition-1–Condition-3 hold.

Before showing these three conditions, we first define a the solution map
$T:L^{2}([0,t_{f}];\mathbb{R}^{m})\rightarrow C([0,t_{f}];\mathbb{R}^{n})$ of
the sensor dynamics (2) by $(Tp)(t):=\zeta(t)=e^{\alpha
t}\zeta_{0}+\int_{0}^{t}e^{\alpha(t-\tau)}\beta p(\tau)\text{d}\tau$ for
$t\in[0,t_{f}]$. The continuity of the map $T$ is straightforward [43], i.e.,
there exists $c_{4}>0$ such that $\left\lVert
Tp\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}\leq c_{4}\left\lVert
p\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$.

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

$\displaystyle J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})\geq$ $\displaystyle\ J_{\eqref{prob: min covariance sensor
guidance problem}}(p)$ $\displaystyle=$
$\displaystyle\textstyle\int_{0}^{t_{f}}h(Tp(t),t)+g(p(t),t)+\text{Tr}(\Pi(t))\text{d}t$
$\displaystyle+h_{f}(Tp(t_{f}))$ $\displaystyle\geq$ $\displaystyle\
\textstyle\int_{0}^{t_{f}}d_{1}|p(t)|^{2}_{2}\text{d}t$ $\displaystyle=$
$\displaystyle\ d_{1}\left\lVert
p\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}^{2}.$ (26)

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 to $p$ weakly (denoted by $p_{k}\rightharpoonup p$).
We wish to show $p\in\mathcal{P}_{0}$. We start with establishing that
$\mathcal{P}$ is weakly sequentially closed and, hence, $p\in\mathcal{P}$.
Subsequently, we show $J_{\eqref{prob: min covariance sensor guidance
problem}}(p)\leq J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})$ to conclude Condition-2.

To show that the set $\mathcal{P}_{0}$ is weakly sequentially closed, by [39,
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 $[0,t_{f}]$ [44, 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 $J_{\eqref{prob: min covariance sensor guidance
problem}}(p)\leq J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})$. 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
$\left\lVert Tp_{k}\right\rVert_{C([0,t_{f}];\mathbb{R}^{n})}\leq
c_{4}\left\lVert p_{k}\right\rVert_{L^{2}([0,t_{f}];\mathbb{R}^{m})}$, 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

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

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 [35], 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 $J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{k})\leq J_{\eqref{prob: min covariance
sensor guidance problem}}(p_{0})$. We have

$\displaystyle J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})-J_{\eqref{prob: min covariance sensor guidance problem}}(p)$
$\displaystyle=$ $\displaystyle J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})-J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{k})+J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{k})-J_{\eqref{prob: min covariance sensor guidance problem}}(p)$
$\displaystyle\geq$ $\displaystyle J_{\eqref{prob: min covariance sensor
guidance problem}}(p_{k})-J_{\eqref{prob: min covariance sensor guidance
problem}}(p).$ (27)

Hence, to show $J_{\eqref{prob: min covariance sensor guidance
problem}}(p)\leq J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{0})$, it suffices to show $J_{\eqref{prob: min covariance sensor
guidance problem}}(p)\leq\liminf_{k\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{k})$, which is to show

$\displaystyle
h_{f}(Tp(t_{f}))+\textstyle\int_{0}^{t_{f}}h(Tp(t),t)+g(p(t),t)+\text{Tr}(\Pi(t))\text{d}t$
$\displaystyle\leq$
$\displaystyle\liminf_{k\rightarrow\infty}h_{f}(Tp_{k}(t_{f}))+\textstyle\int_{0}^{t_{f}}h(Tp_{k}(t),t)+g(p_{k}(t),t)$
$\displaystyle+\text{Tr}(\Pi^{k}(t))\text{d}t,$ (28)

where $\Pi^{k}(t)$ is the solution of (13) associated with sensor state
$Tp_{k}$. Since $\\{Tp_{k}\\}$ converges to $Tp$ uniformly on $[0,t_{f}]$, the
continuity of $h_{f}(\cdot)$ implies

$h_{f}(Tp(t_{f}))=\liminf_{k\rightarrow\infty}h_{f}(Tp_{k}(t_{f}));$ (29)

Fatou’s lemma [35] implies

$\textstyle\int_{0}^{t_{f}}h(Tp(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}h(Tp_{k}(t),t)\text{d}t;$
(30)

and Lemma 5 implies

$\textstyle\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t=\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}\text{Tr}(\Pi^{k}(t))\text{d}t.$
(31)

To prove (28), based on (29)–(31), it suffices to show

$\textstyle\int_{0}^{t_{f}}g(p(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}g(p_{k}(t),t)\text{d}t.$
(32)

By contradiction, assume there is $\lambda>0$ such that

$\liminf_{k\rightarrow\infty}\textstyle\int_{o}^{t_{f}}g(p_{k}(t),t)\text{d}t<\lambda<\textstyle\int_{0}^{t_{f}}g(p(t),t)\text{d}t.$
(33)

There exists a subsequence $\\{p_{k_{j}}\\}\subset\\{p_{k}\\}$ such that
${\\{p_{k_{j}}\\}\subset O_{\lambda}=\\{q\in
L^{2}([0,t_{f}];\mathbb{R}^{m}):\int_{0}^{t_{f}}g(q(t),t)\text{d}t\leq\lambda\\}}$.
We wish to show that $O_{\lambda}$ is weakly sequentially closed. By [43,
Theorem 6.1.5], 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 $[0,t_{f}]$ [44, p. 53]. Now we have
$g(q_{k_{j}}(t),t)\geq 0$ for all $t\in[0,t_{f}]$ (assumption (A11)) and
$\lim_{j\rightarrow\infty}g(q_{k_{j}}(t),t)=g(q(t),t)$ almost everywhere on
$[0,t_{f}]$. By Fatou’s lemma [35],
$\int_{0}^{t_{f}}g(q(t),t)\text{d}t\leq\liminf_{k\rightarrow\infty}\textstyle\int_{0}^{t_{f}}g(q_{k_{j}}(t),t)\text{d}t\leq\lambda$,
where the last inequality holds due to the fact that $\\{q_{k_{j}}\\}\subset
O_{\lambda}$. 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 (33). Hence,
$J_{\eqref{prob: min covariance sensor guidance problem}}(p)\leq
J_{\eqref{prob: min covariance sensor guidance problem}}(p_{0})$ is proved,
and we conclude Condition-2.

Proof of Condition-3: We now show that the mapping $J_{\eqref{prob: min
covariance sensor guidance problem}}(\cdot):\mathcal{P}\rightarrow\mathbb{R}$
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 establish $J_{\eqref{prob: min covariance
sensor guidance problem}}(p)\leq\liminf_{k\rightarrow\infty}J_{\eqref{prob:
min covariance sensor guidance problem}}(p_{k})$, which can be shown using the
technique of proving Condition-2 (starting from (28)).

Thus we conclude the existence of a solution of problem (P). ∎

##  Appendix B Proof of Theorem 9

Proof Since $\text{Tr}(\Pi_{N}(t))\geq 0$ ($\Pi_{N}(t)$ is nonnegative and
self-adjoint for all $t\in[0,t_{f}]$) 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 8, where $\Pi(t)$ is
replaced by $\Pi_{N}(t)$. ∎

##  Appendix C Proof of Theorem 10

Recall that the notation $J^{*}_{\eqref{prob: approximated min covariance
sensor guidance problem}}(p_{N}^{*})$ means the optimal value of (AP)
evaluated at its optimal solution $p_{N}^{*}$, where the dimension of
approximation applied to (AP) is $N$ (as indicated by the subscript of
$p_{N}^{*}$). In this section, we attach a subscript to $\eqref{prob:
approximated min covariance sensor guidance problem}$, such as
$J_{\eqref{prob: approximated min covariance sensor guidance
problem}_{N}}(p)$, to indicate its dimension when it is not reflected by the
argument, e.g., $J_{\eqref{prob: approximated min covariance sensor guidance
problem}_{N}}(p)$ means that the cost of (AP) using an $N$-dimensional
approximation evaluated at a guidance function $p$. Lemma 14 (see proof in the
supplementary material) will be used in the proof of Theorem 10.

######  Lemma 14.

Consider problems (P) and its approximation (AP). Let assumptions (A4)–(A12)
hold. Then the following results hold:

  1. 1.

For $p\in C([0,t_{f}];\mathbb{R}^{m})$,
$\lim_{N\rightarrow\infty}J_{\eqref{prob: approximated min covariance sensor
guidance problem}_{N}}(p)=J_{\eqref{prob: min covariance sensor guidance
problem}}(p)$;

  2. 2.

The mapping $J_{\eqref{prob: min covariance sensor guidance
problem}}:C([0,t_{f}];\mathbb{R}^{m})\rightarrow\mathbb{R}^{+}$ such that
$J_{\eqref{prob: min covariance sensor guidance
problem}}(p)=\int_{0}^{t_{f}}\text{Tr}(\Pi(t))\text{d}t+J_{\text{m}}(\zeta,p)$
is continuous, where $\zeta$ is the sensor state steered by $p$ under the
dynamics (2) and $\Pi(\cdot)$ is the covariance operator obtained through (13)
with sensor state $\zeta$.

Proof of Theorem 10 We start with proving (24), i.e., $|J^{*}_{\eqref{prob:
approximated min covariance sensor guidance
problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})|\rightarrow 0$ as $N\rightarrow\infty$. First,

$\displaystyle J^{*}_{\eqref{prob: approximated min covariance sensor guidance
problem}}(p_{N}^{*})=$ $\displaystyle\
\underset{p\in\mathcal{P}(p_{\max},a_{\max})}{\min}J_{\eqref{prob:
approximated min covariance sensor guidance problem}_{N}}(p)$
$\displaystyle\leq$ $\displaystyle\ J_{\eqref{prob: approximated min
covariance sensor guidance problem}_{N}}(p^{*})$ $\displaystyle\leq$
$\displaystyle\ |J_{\eqref{prob: approximated min covariance sensor guidance
problem}_{N}}(p^{*})-J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})|+J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*}).$

It follows that

$\displaystyle\limsup_{N\rightarrow\infty}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}}(p_{N}^{*})\leq$ $\displaystyle\
J^{*}_{\eqref{prob: min covariance sensor guidance problem}}(p^{*}),$ (34)

because $|J_{\eqref{prob: approximated min covariance sensor guidance
problem}_{N}}(p^{*})-J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})|\rightarrow 0$ as $N\rightarrow 0$ (see Lemma 14-1).

To proceed with proving (24), in addition to (34), we shall show
$\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob: approximated min covariance
sensor guidance problem}}(p_{N}^{*})\geq J^{*}_{\eqref{prob: min covariance
sensor guidance problem}}(p^{*})$. Choose a convergent subsequence
$\\{J^{*}_{\eqref{prob: approximated min covariance sensor guidance
problem}}(p_{N_{k}}^{*})\\}_{k=1}^{\infty}$ such that

$\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: approximated min covariance
sensor guidance
problem}}(p_{N_{k}}^{*})=\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}}(p_{N}^{*}).$ (35)

Since the subsequence
$\\{p_{N_{k}}^{*}\\}_{k=1}^{\infty}\subset\mathcal{P}(p_{\max},a_{\max})$
which is uniformly equicontinuous and uniformly bounded, by the Arzelà-Ascoli
Theorem [35], there is a (uniformly) convergent subsequence of
$\\{p_{N_{k}}^{*}\\}_{k=1}^{\infty}$. We denote this convergent subsequence
with the same indices $\\{N_{k}\\}_{k=1}^{\infty}$ to simplify notation.
Denote the limit of $\\{p_{N_{k}}^{*}\\}_{k=1}^{\infty}$ by $p_{\inf}^{*}$,
i.e.,

$\lim_{k\rightarrow\infty}\left\lVert
p_{N_{k}}^{*}-p_{\inf}^{*}\right\rVert_{C([0,t_{f}];\mathbb{R}^{m})}=0.$ (36)

Next, we show

$\lim_{k\rightarrow\infty}|J^{*}_{\eqref{prob: approximated min covariance
sensor guidance problem}}(p_{N_{k}}^{*})-J_{\eqref{prob: min covariance sensor
guidance problem}}(p_{\inf}^{*})|=0.$ (37)

First notice that for all $p\in\mathcal{P}(p_{\max},a_{\max})$,
$J_{\eqref{prob: approximated min covariance sensor guidance problem}_{N}}(p)$
converges to $J_{\eqref{prob: min covariance sensor guidance problem}}(p)$
pointwise as the dimension of approximation $N$ goes to infinity (see Lemma
14-1). Furthermore, since the sequence of approximated uncertainty cost
$\\{\int_{0}^{t_{f}}\text{Tr}(\Pi_{N}(t))\text{d}t\\}_{N=1}^{\infty}$ is a
monotonically increasing sequence, the sequence $\\{J_{\eqref{prob:
approximated min covariance sensor guidance
problem}_{N}}(p)\\}_{N=1}^{\infty}$ is a monotonically increasing sequence for
each $p$ on the compact set $\mathcal{P}(p_{\max},a_{\max})$. By Dini’s
Theorem [36, Theorem 7.13], $|J_{\eqref{prob: approximated min covariance
sensor guidance problem}_{N}}(p)-J_{\eqref{prob: min covariance sensor
guidance problem}}(p)|\rightarrow 0$ uniformly on
$\mathcal{P}(p_{\max},a_{\max})$ as $N\rightarrow\infty$. By Moore-Osgood
Theorem [36, Theorem 7.11], this uniform convergence and the convergence
$p_{N_{k}}^{*}\rightarrow p_{\inf}^{*}$ as $k\rightarrow\infty$ (see (36))
imply that

$\displaystyle\lim_{k\rightarrow\infty}J_{\eqref{prob: min covariance sensor
guidance
problem}}(p_{N_{k}}^{*})=\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}_{j}}(p_{N_{k}}^{*}).$
(38)

And the iterated limit in (38) equals the double limit [38, p. 140], i.e.,

$\displaystyle\lim_{j\rightarrow\infty}\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}_{j}}(p_{N_{k}}^{*})=$
$\displaystyle\lim_{\begin{subarray}{c}j\rightarrow\infty\\\
k\rightarrow\infty\end{subarray}}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}_{j}}(p_{N_{k}}^{*})$ $\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}}(p_{N_{k}}^{*}).$ (39)

By (38), (C), and the fact that $J_{\eqref{prob: min covariance sensor
guidance problem}}(p_{\inf}^{*})=\lim_{k\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{N_{k}}^{*})$ holds (due to the
continuity of $J_{\eqref{prob: min covariance sensor guidance
problem}}(\cdot)$, see Lemma 14-2), we conclude that (37) holds and

$\displaystyle\liminf_{N\rightarrow\infty}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}}(p_{N}^{*})=$ $\displaystyle\
\lim_{k\rightarrow\infty}J^{*}_{\eqref{prob: approximated min covariance
sensor guidance problem}}(p_{N_{k}}^{*})$ $\displaystyle=$ $\displaystyle\
J_{\eqref{prob: min covariance sensor guidance problem}}(p_{\inf}^{*})$
$\displaystyle\geq$ $\displaystyle\ J^{*}_{\eqref{prob: min covariance sensor
guidance problem}}(p^{*}).$ (40)

Therefore, we conclude $\lim_{N\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance
problem}}(p_{N}^{*})=J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})$ from (34) and (40).

Next, we prove (25), i.e., $|J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{N}^{*})-J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})|\rightarrow 0$ as $N\rightarrow\infty$. We start with
$J^{*}_{\eqref{prob: min covariance sensor guidance problem}}(p^{*})\leq
J_{\eqref{prob: min covariance sensor guidance problem}}(p_{N}^{*})$ for all
$N$, which implies that

$J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})\leq\liminf_{N\rightarrow\infty}J_{\eqref{prob: min covariance
sensor guidance problem}}(p_{N}^{*}).$ (41)

To prove (25), what remains to be shown is $J^{*}_{\eqref{prob: min covariance
sensor guidance
problem}}(p^{*})\geq\limsup_{N\rightarrow\infty}J_{\eqref{prob: min covariance
sensor guidance problem}}(p_{N}^{*})$. Choose a convergent subsequence
$\\{J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{N_{j}}^{*})\\}_{j=1}^{\infty}$ such that
$\lim_{j\rightarrow\infty}J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{N_{j}}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{N}^{*})$. Since
$\\{p_{N_{j}}^{*}\\}_{j=1}^{\infty}\subset\mathcal{P}(p_{\max},a_{\max})$ is
uniformly equicontinuous and uniformly bounded, by Arzelà-Ascoli Theorem [35],
$\\{p_{N_{j}}^{*}\\}_{j=1}^{\infty}$ has a (uniformly) convergent subsequence
which we denote with the same indices $\\{N_{j}\\}_{j=1}^{\infty}$ to simplify
notation. Denote the limit of $\\{p_{N_{j}}^{*}\\}_{j=1}^{\infty}$ by
$p_{\sup}^{*}$ such that

$\lim_{j\rightarrow\infty}\left\lVert
p_{N_{j}}^{*}-p_{\sup}^{*}\right\rVert_{C([0,t_{f}];\mathbb{R}^{m})}=0.$ (42)

Due to the continuity of $J_{\eqref{prob: min covariance sensor guidance
problem}}(\cdot)$ (see Lemma 14-1), we have $J_{\eqref{prob: min covariance
sensor guidance
problem}}(p_{\sup}^{*})=\lim_{j\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance
problem}}(p_{N_{j}}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{N}^{*})$. Now we have

$\displaystyle J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{\sup}^{*})$ $\displaystyle\leq$ $\displaystyle|J_{\eqref{prob:
min covariance sensor guidance problem}}(p_{\sup}^{*})-J^{*}_{\eqref{prob: min
covariance sensor guidance problem}}(p^{*})|+J^{*}_{\eqref{prob: min
covariance sensor guidance problem}}(p^{*})$ $\displaystyle=$
$\displaystyle|J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{\sup}^{*})-\lim_{N\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance
problem}}(p_{N}^{*})|+J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*})$ $\displaystyle=$ $\displaystyle|J_{\eqref{prob: min
covariance sensor guidance
problem}}(p_{\sup}^{*})-\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance
problem}}(p_{N_{j}}^{*})|+J^{*}_{\eqref{prob: min covariance sensor guidance
problem}}(p^{*}).$ (43)

Since the sequence of approximated uncertainty cost
$\\{\int_{0}^{t_{f}}\Pi_{N}(t)\text{d}t\\}_{N=1}^{\infty}$ is a monotonically
increasing sequence, the sequence $\\{J_{\eqref{prob: approximated min
covariance sensor guidance problem}_{N}}(p)\\}_{N=1}^{\infty}$ is a
monotonically increasing sequence for each $p$ on the compact set
$\mathcal{P}(p_{\max},a_{\max})$. Since
$\lim_{N\rightarrow\infty}J_{\eqref{prob: approximated min covariance sensor
guidance problem}_{N}}(p)=J_{\eqref{prob: min covariance sensor guidance
problem}}(p)$ for all $p\in\mathcal{P}(p_{\max},a_{\max})$ (see Lemma 14-1),
by Dini’s Theorem [36, Theorem 7.13], the limit holds uniformly on
$\mathcal{P}(p_{\max},a_{\max})$ as $N\rightarrow\infty$. By Moore-Osgood
Theorem [36, Theorem 7.11], this uniform convergence and the convergence
$p_{N_{j}}^{*}\rightarrow p_{\sup}^{*}$ as $j\rightarrow\infty$ (see (42))
imply that $J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{\sup}^{*})=\lim_{k\rightarrow\infty}\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}_{k}}(p_{N_{j}}^{*})$.
Furthermore, the iterated limit equals the double limit [38, p. 140], i.e.,

$\displaystyle\lim_{k\rightarrow\infty}\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}_{k}}(p_{N_{j}}^{*})=$
$\displaystyle\lim_{\begin{subarray}{c}j\rightarrow\infty\\\
k\rightarrow\infty\end{subarray}}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}_{k}}(p_{N_{j}}^{*})$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob: approximated min
covariance sensor guidance problem}}(p_{N_{j}}^{*}).$ (44)

Hence, $J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{\sup}^{*})=\lim_{j\rightarrow\infty}J^{*}_{\eqref{prob:
approximated min covariance sensor guidance problem}}(p_{N_{j}}^{*})$, which,
combined with (43), implies

$J^{*}_{\eqref{prob: min covariance sensor guidance problem}}(p^{*})\geq
J_{\eqref{prob: min covariance sensor guidance
problem}}(p_{\sup}^{*})=\limsup_{N\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{N}^{*}).$ (45)

The desired convergence $\lim_{N\rightarrow\infty}J_{\eqref{prob: min
covariance sensor guidance problem}}(p_{N}^{*})=J^{*}_{\eqref{prob: min
covariance sensor guidance problem}}(p^{*})$ follows from (41) and (45). ∎

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