# A Maximum Principle for Optimal Control Problems involving Sweeping
Processes with a Nonsmooth Set

M. d. R. de Pinho, M. Margarida A. Ferreira and Georgi Smirnov MdR de Pinho
and MMA Ferreira are at Faculdade de Engenharia da Universidade do Porto,
DEEC, SYSTEC. Portugal, mrpinho<EMAIL_ADDRESS>Smirnov is at Universidade do
Minho, Dep. Matemática, Physics Center of Minho and Porto Universities (CF-UM-
UP), Campus de Gualtar, Braga, Portugal<EMAIL_ADDRESS>
###### Abstract

We generalize a Maximum Principle for optimal control problems involving
sweeping systems previously derived in [14] to cover the case where the moving
set may be nonsmooth. Noteworthy, we consider problems with constrained end
point. A remarkable feature of our work is that we rely upon an ingenious
smooth approximating family of standard differential equations in the vein of
that used in [10].

Keywords: Sweeping Process Optimal Control, Maximum Principle, Approximations

##  1 Introduction

In recent years, there has been a surge of interest in optimal control
problems involving the controlled sweeping process of the form

$\dot{x}(t)\in f(t,x(t),u(t))-N_{C(t)}(x(t)),~{}u(t)\in U,~{}~{}x(0)\in
C_{0}.$ (1.1)

In this respect, we refer to, for example, [3], [4], [5], [8], [9], [16],
[23], [10] (see also accompanying correction [11]), [6], [15] and [14].
Sweeping processes first appeared in the seminal paper [18] by J.J. Moreau as
a mathematical framework for problems in plasticity and friction theory. They
have proved of interest to tackle problems in mechanics, engineering,
economics and crowd motion problems; to name but a few, see [1], [5], [16],
[17] and [21]. In the last decades, systems in the form (1.1) have caught the
attention and interest of the optimal control community. Such interest resides
not only in the range of applications but also in the remarkable challenge
they rise concerning the derivation of necessary conditions. This is due to
the presence of the normal cone $N_{C(t)}(x(t))$ in the dynamics. Indeed, the
presence of the normal cone renders the discontinuity of the right hand of the
differential inclusion in (1.1) destroying a regularity property central to
many known optimal control results.

Lately, there has been several successful attempts to derive necessary
conditions for optimal control problems involving (1.1). Assuming that the set
$C$ is time independent, necessary conditions for optimal control problems
with free end point have been derived under different assumptions and using
different techniques. In [10], the set $C$ has the form $C=\\{x:~{}\psi(x)\leq
0\\}$ and an approximating sequence of optimal control problems, where (1.1)
is approximated by the differential equation

$\dot{x}_{\gamma_{k}}(t)=f(t,x_{\gamma_{k}}(t),u(t))-\gamma_{k}e^{\gamma_{k}\psi(x_{\gamma_{k}}(t))}\nabla\psi(x_{\gamma_{k}}(t)),$
(1.2)

for some positive sequence $\gamma_{k}\to+\infty$, is used. Similar techniques
are also applied to somehow more general problems in [23]. A useful feature of
those approximations is explored in [12] to define numerial schemes to solve
such problems.

More recently, an adaptation of the family of approximating systems (1.2) is
used in [14] to generalize the results in [10] to cover problems with
additional end point constraints and with a moving set of the form
$C(t)=\\{x:~{}\psi(t,x)\leq 0\\}$.

In this paper we generalize the Maximum Principle proved in [14] to cover
problems with possibly nonsmooth sets. Our problem of interest is

$(P)\left\\{\begin{array}[]{l}\mbox{Minimize }\;\phi(x(T))\\\\[5.69054pt]
\mbox{over processes $(x,u)$ such that }\\\\[5.69054pt] \hskip
22.76219pt\dot{x}(t)\in f(t,x(t),u(t))-N_{C(t)}(x(t)),\hskip
5.69046pt\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptu(t)\in U,\
\ \;\,\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219pt(x(0),x(T))\in
C_{0}\times C_{T}{~{}\subset C(0)\times C(T)},\end{array}\right.$

where $T>0$ is fixed, $\phi:R^{n}\to R$, $f:[0,T]\times R^{n}\times R^{m}\to
R^{n}$, $U\subset R^{m}$ and

$C(t):=\left\\{x\in R^{n}:~{}\psi^{i}(t,x)\leq 0,\;i=1,\ldots,I\right\\}$
(1.3)

for some functions $\psi^{i}:[0,T]\times R^{n}\to R$, $i=1,\ldots,I$.

The case where $I=1$ in (1.3) and $\psi^{1}$ is $C^{2}$ is covered in [14].
Here, we assume $I>1$ and that the functions $\psi^{i}$ are also $C^{2}$.
Although going from $I=1$ in (1.3) to $I>1$ may be seen as a small
generalization, it demands a significant revision of the technical approach
and, plus, the introduction of a constraint qualification. This is because the
set (1.3) may be nonsmooth. We focus on sets (1.3), satisfying a certain
constraint qualification, introduced in assumption (A1) in section 2 below.
This is, indeed, a restriction on the nonsmoothness of (1.3). A similar
problem with nonsmooth moving set is considered in [15]. Our results cannot be
obtained from the results of [15] and do not generalize them.

This paper is organized in the following way. In section 2, we introduce the
main notation and we state and discuss the assumptions under which we work. In
this same section, we also introduce the family of approximating systems to
$\dot{x}(t)\in f(t,x(t),u(t))-N_{C(t)}(x(t))$ and establish a crucial
convergence result, Theorem 2.2. In section 3, we dwell on the approximating
family of optimal control problems to $(P)$ and we state the associated
necessary conditions. The Maximum Principle for $(P)$ is then deduced and
stated in Theorem 4.1, covering additionally, problems in the form of $(P)$
where the end point constraint $x(T)\in C_{T}$ is absent. Before finishing, we
present an illustrative example of our main result, Theorem 4.1.

##  2 Preliminaries

In this section, we introduce a summary of the notation and state the
assumptions on the data of $(P)$ enforced throughout. Furthermore, we extract
information from the assumptions establishing relations crucial for the
forthcoming analysis.

Notation

For a set $S\subset R^{n}$, $\partial S$, $\text{cl}\,S$ and $\text{int}\,S$
denote the boundary, closure and interior of $S$.

If $g:R^{p}\to R^{q}$, $\nabla g$ represents the derivative and $\nabla^{2}g$
the second derivative. If $g:R\times R^{p}\to R^{q}$, then $\nabla_{x}g$
represents the derivative w.r.t. $x\in R^{p}$ and $\nabla^{2}_{x}g$ the second
derivative, while $\partial_{t}g(t,x)$ represents the derivative w.r.t. $t\in
R$.

The Euclidean norm or the induced matrix norm on $R^{p\times q}$ is denoted by
$|\cdot|$. We denote by $B_{n}$ the closed unit ball in $R^{n}$ centered at
the origin. The inner product of $x$ and $y$ is denoted by $\langle
x,y\rangle$. For some $A\subset R^{n}$, $d(x,A)$ denotes the distance between
$x$ and $A$. We denote the support function of $A$ at $z$ by
$S(z,A)=\sup\\{\langle z,a\rangle\mid a\in A\\}$

The space $L^{\infty}([a,b];R^{p})$ (or simply $L^{\infty}$ when the domains
are clearly understood) is the Lebesgue space of essentially bounded functions
$h:[a,b]\to R^{p}$. We say that $h\in BV([a,b];R^{p})$ if $h$ is a function of
bounded variation. The space of continuous functions is denoted by
$C([a,b];R^{p})$.

Standard concepts from nonsmooth analysis will also be used. Those can be
found in [7], [19] or [22], to name but a few. The Mordukhovich normal cone to
a set $S$ at $s\in S$ is denoted by $N_{S}(s)$ and $\partial f(s)$ is the
Mordukhovich subdifferential of $f$ at $s$ (also known as limiting
subdifferential).

For any set $A\subset R^{n}$, $\text{cone}\,A$ is the cone generated by the
set $A$.

We now turn to problem $(P)$. We first state the definition of admissible
processes for $(P)$ and then we describe the assumptions under which we will
derive our main results.

###### Definition 2.1

A pair $(x,u)$ is called an admissible process for $(P)$ when $x$ is an
absolutely continuous function and $u$ is a measurable function satisfying the
constraints of $(P)$.

Assumptions on the data of $\mathbf{(P)}$

  * A1:

The function $\psi^{i}$, $i=1,\ldots,I$, are $C^{2}$. The graph of $C(\cdot)$
is compact and it is contained in the interior of a ball $rB_{n+1}$, for some
$r>0$. There exist constants $\beta>0$, $\eta>0$ and $\rho\in]0,1[$ such that

$\psi^{i}(t,x)\in[-\beta,\beta]\Longrightarrow|\nabla_{x}\psi^{i}(t,x)|>\eta\;\;{\rm
forall}\;(t,x)\in[0,T]\times R^{n},$ (2.1)

and, for $I(t,x)=\\{i=1,\ldots,I\mid\psi^{i}(t,x)\in]-2\beta,\beta]\\}$,

$\langle\nabla_{x}\psi^{i}(t,x),\nabla_{x}\psi^{j}(t,x)\rangle\geq
0,\;\;i,j\in I(t,x).$ (2.2)

Moreover, if $i\in I(t,x)$, then

$\sum_{j\in
I(t,x)\setminus\\{i\\}}\big{|}\langle\nabla_{x}\psi^{i}(t,x),\nabla_{x}\psi^{j}(t,x)\rangle\big{|}\leq\rho|\nabla_{x}\psi^{i}(t,x)|^{2}$
(2.3)

and

$\psi^{i}(t,x)\leq-2\beta~{}\Longrightarrow~{}\nabla\psi^{i}(t,x)=0~{}\text{
for }i=1,\ldots I.$ (2.4)

  * A2:

The function $f$ is continuous, $x\to f(t,x,u)$ is continuously differentiable
for all $(t,u)\in[0,T]\times R^{m}$. The constant $M>0$ is such that
$|f(t,x,u)|\leq M$ and $|\nabla_{x}f(t,x,u)|\leq M$ for all $(t,x,u)\in
rB_{n+1}\times U$.

  * A3:

For each $(t,x)$, the set $f(t,x,U)$ is convex.

  * A4:

The set $U$ is compact.

  * A5:

The sets $C_{0}$ and $C_{T}$ are compact.

  * A6:

There exists a constant $L_{\phi}$ such that $|\phi(x)-\phi(x^{\prime})|\leq
L_{\phi}|x-x^{\prime}|$ for all $x,x^{\prime}\in R^{n}$.

Assumption (A1) concerns the functions $\psi^{i}$ defining the set $C$ and it
plays a crucial role in the analysis. All $\psi^{i}$ are assumed to be smooth
with gradients bounded away from the origin when $\psi^{i}$ takes values in a
neighorhood of zero. Moreover, the boundary of $C$ may be nonsmooth at the
intersection points of the level sets $\left\\{x:\psi^{i}(t,x)=0\right\\}$.
However, nonsmoothness at those corner points is restricted to (2.2) which
excludes the cases where the angle between the two gradients of the functions
defining the boundary of $C$ is obtuse; see figure 1.

Figure 1: Examples of two diferent sets $C$. On the left size, a set that does
not satisfies (2.2). On the right side, the set $C$ is nonsmooth and it
fulfils (2.2).

On the other hand, (2.3) guarantees that the Gramian matrix of the gradients
of the functions taking values near the boundary of $C(t)$ is diagonally
dominant and, hence, the gradients are linearly independent.

In many situations, as in the example we present in the last section, we can
guarantee the fulfillment of (A1), in particular (2.4), replacing the function
$\psi^{i}$ by

$\tilde{\psi}^{i}(t,x)=h\circ\psi^{i}(t,x),$ (2.5)

where

$h(z)=\left\\{\begin{array}[]{lcl}z&\text{ if}&z>-\beta,\\\ h_{s}(z)&\text{
if}&-2\beta\leq z\leq-\beta,\\\ -2\beta&\text{
if}&z<-2\beta,\end{array}\right.$

Here, $h$ is an $C^{2}$ function, with $h_{s}$ an increasing function defined
on $[-2\beta,-\beta]$. For example, $h$ may be a cubic polinomial with
positive derivative on the interval $]-2\beta,-\beta[$. For all $t\in[0,T]$,
set

$\tilde{C}(t):=\left\\{x\in R:~{}\tilde{\psi}^{i}(t,x)\leq
0,~{}i=1,\ldots,I\right\\}.$

It is then a simple matter to see that

$C(t)=\tilde{C}(t)\text{ for all }t\in[0,T].$

and that the functions $\tilde{\psi}^{i}(\cdot)$ satisfy the assumption (A1).

The assumption that the graph of $C(\cdot)$ is compact and contained in the
interior of a ball is introduced to avoid technicalities in our forthcoming
analysis. In applied problems, this may be easily side tracked by considering
the intersection of the graph of $C(\cdot)$ with a tube around the optimal
trajectory.

We now proceed introducing an approximation family of controlled systems to
(1.1). Let $x(\cdot)$ be a solution to the differential inclusion

$\dot{x}(t)\in f(t,x(t),U)-N_{C(t)}(x(t)).$

Under our assumptions, measurable selection theorems assert the existence of
measurable functions $u$ and $\xi^{i}$ such that $u(t)\in U$, $\xi^{i}(t)\geq
0$ a.e. $t\in[0,T]$, $\xi^{i}(t)=0$ if $\psi^{i}(t,x(t))<0$, and

$\dot{x}(t)=f(t,x(t),u(t))-\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\psi^{i}(t,x(t))\;{\rm
a.e.}\;t\in[0,T].$

Considering the trajectory $x$, some observations are called for. Let $\mu$ be
such that

$\max\left\\{(|\nabla_{x}\psi^{i}(t,x)||f(t,x,u)|+|\partial_{t}\psi^{i}(t,x)|)+1:\right.\\\\[2.84526pt]
\left.~{}t\in[0,T],\;u\in U,\;x\in C(t)+B_{n},\;i=1,\ldots,I\right\\}\leq\mu.$
(2.6)

The properties of the graph of $C(\cdot)$ in (A1) guarantee the existence of
such maximum.

Consider now some $t$ such that, for some $j\in\\{1,\ldots I\\}$,
$\psi^{j}(t,x(t))=0$ and $\dot{x}(t)$ exists. Since the trajectory $x$ is
always in $C$, we have (see (2.2))

$\begin{split}0&=\frac{d}{dt}\psi^{j}(t,x(t))=\langle\nabla_{x}\psi^{j}(t,x(t)),\dot{x}(t)\rangle+\partial_{t}\psi^{j}(t,x(t))\\\\[5.69054pt]
&=\langle\nabla_{x}\psi^{j}(t,x(t)),f(t,x(t),u(t))\rangle-\xi^{j}(t)|\nabla_{x}\psi^{j}(t,x(t))|^{2}\\\\[5.69054pt]
&-\sum_{i\in
I(t,x(t))\setminus\\{j\\}}\xi^{i}(t)\langle\nabla_{x}\psi^{i}(t,x(t)),\nabla_{x}\psi^{j}(t,x(t))\rangle+\partial_{t}\psi^{j}(t,x(t))\\\\[5.69054pt]
&\leq\langle\nabla_{x}\psi^{j}(t,x(t)),f(t,x(t),u(t))\rangle-\xi^{j}(t)|\nabla_{x}\psi^{j}(t,x(t))|^{2}+\partial_{t}\psi^{j}(t,x(t)),\end{split}$

and, hence (see (2.1)),

$\xi^{j}(t)\leq\frac{1}{|\nabla_{x}\psi^{j}(t,x(t))|^{2}}(\langle\nabla_{x}\psi^{j}(t,x(t)),f(t,x(t),u(t))\rangle+\partial_{t}\psi^{j}(t,x(t)))\leq\frac{\mu}{\eta^{2}}.$

Define the function

$\mu(\gamma)=\frac{1}{\gamma}\log\left(\frac{\mu}{\eta^{2}\gamma}\right),\quad\gamma>0,$

consider a sequence $\\{\sigma_{k}\\}$ such that $\sigma_{k}\downarrow 0$ and
choose another sequence $\\{\gamma_{k}\\}$ with $\gamma_{k}\uparrow+\infty$
and

$C(t)\subset{\rm int}\>C^{k}(t)={\rm
int}\>\left\\{x:\psi^{i}(t,x)-\sigma_{k}\leq\mu_{k},\;i=1,\ldots,I\right\\},$

where

$\mu_{k}=\mu(\gamma_{k}).$

Let $x_{k}$ be a solution to the differential equation

$\dot{x}_{k}(t)=f(t,x_{k}(t),u_{k}(t))-\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\nabla_{x}\psi^{i}(t,x_{k}(t))$
(2.7)

for some $u_{k}(t)\in U$ a.e. $t\in[0,T]$. Take any $t\in[0,T]$ such that
$\dot{x}_{k}(t)$ exists and $\psi^{j}(t,x_{k}(t))-\sigma_{k}=\mu_{k}$. Assume
$k$ is such that $j\in I(t,x_{k}(t))$. Then, whenever $\gamma_{k}$ is
sufficiently large, we have

$\begin{split}\frac{d}{dt}\psi^{j}(t,x_{k}(t))&=\langle\nabla_{x}\psi^{j}(t,x_{k}(t)),f(t,x_{k}(t),u_{k}(t))\rangle\\\
&\hskip
28.45274pt-\gamma_{k}e^{\gamma_{k}(\psi^{j}(t,x_{k}(t))-\sigma_{k})}|\nabla_{x}\psi^{j}(t,x_{k}(t))|^{2}\\\
&\hskip 8.5359pt-\sum_{i\in
I(t,x_{k}(t))\setminus\\{j\\}}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\langle\nabla_{x}\psi^{i}(t,x_{k}(t)),\nabla_{x}\psi^{j}(t,x_{k}(t))\rangle\\\
&\hskip 8.5359pt-\sum_{i\not\in
I(t,x_{k}(t))}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\langle\nabla_{x}\psi^{i}(t,x_{k}(t)),\nabla_{x}\psi^{j}(t,x_{k}(t))\rangle\\\
&\hskip 8.5359pt+\partial_{t}\psi^{j}(t,x_{k}(t))\\\
&\leq\langle\nabla_{x}\psi^{j}(t,x_{k}(t)),f(t,x_{k}(t),u_{k}(t))\rangle\\\
&\hskip
8.5359pt-\gamma_{k}e^{\gamma_{k}(\psi^{j}(t,x_{k}(t))-\sigma_{k})}|\nabla_{x}\psi^{j}(t,x_{k}(t))|^{2}\\\
&\hskip 8.5359pt-\sum_{i\not\in
I(t,x_{k}(t))}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\langle\nabla_{x}\psi^{i}(t,x_{k}(t)),\nabla_{x}\psi^{j}(t,x_{k}(t))\rangle\\\
&\hskip 8.5359pt+\partial_{t}\psi^{j}(t,x_{k}(t))\\\
&\leq\langle\nabla_{x}\psi^{j}(t,x_{k}(t)),f(t,x_{k}(t),u_{k}(t))\rangle\\\
&\hskip
8.5359pt-\gamma_{k}e^{\gamma_{k}(\psi^{j}(t,x_{k}(t))-\sigma_{k})}|\nabla_{x}\psi^{j}(t,x_{k}(t))|^{2}\\\
&\hskip 8.5359pt+\sum_{i\not\in
I(t,x_{k}(t))}\gamma_{k}e^{\gamma_{k}(-{\color[rgb]{1,0,0}2}\beta-\sigma_{k})}|\langle\nabla_{x}\psi^{i}(t,x_{k}(t)),\nabla_{x}\psi^{j}(t,x_{k}(t))\rangle|\\\
&\hskip 8.5359pt+\partial_{t}\psi^{j}(t,x_{k}(t))\\\
&\leq\mu-\frac{1}{2}-\eta^{2}\gamma_{k}e^{\gamma_{k}\mu_{k}}\\\
&=-\frac{1}{2}.\end{split}$

Above, we have used the definition of $\mu$ and the inequality

$\sum_{i\not\in
I(t,x_{k}(t))}\gamma_{k}e^{\gamma_{k}(-{\color[rgb]{1,0,0}2}\beta-\sigma_{k})}|\langle\nabla_{x}\psi^{i}(t,x_{k}(t)),\nabla_{x}\psi^{j}(t,x_{k}(t))\rangle|\leq\frac{1}{2},$

which holds for $\gamma_{k}$ sufficiently large.

Now, if $x_{k}(0)\in C^{k}(0)$, we assure that $x_{k}(t)\in C^{k}(t)$, for all
$t\in[0,T]$, and

$\gamma_{k}e^{\gamma_{k}(\psi^{j}(t,x_{k}(t))-\sigma_{k})}\leq\gamma_{k}e^{\gamma_{k}\mu_{k}}=\frac{\mu}{\eta^{2}}.$
(2.8)

It follows that, for $k$ sufficienttly large, we have

$|\dot{x}_{k}(t)|\leq({\rm const}).$

We are now a in position to state and prove our first result, Theorem 2.2
below. This is in the vein of Theorem 4.1 in [23] (see also Lemma 1 in [10]
when $\psi$ is independent of $t$ and convex) deviating from it in so far as
the approximating sequence of control systems (2.7) differs from the one
introduced in [10]111See also Theorem 2.2 in [14]. The proof of Theorem 2.2
relies on (2.8).

###### Theorem 2.2

Let $\\{(x_{k},u_{k})\\}$, with $u_{k}(t)\in U$ a.e., be a sequence of
solutions of Cauchy problems

$\left\\{\begin{array}[]{rcl}\dot{x}_{k}(t)&=&f(t,x_{k}(t),u_{k}(t))-\displaystyle\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\nabla_{x}\psi^{i}(t,x_{k}(t)),\\\\[5.69054pt]
x_{k}(0)&=&b_{k}\in C^{k}(0).\end{array}\right.$ (2.9)

If $b_{k}\rightarrow x_{0}$, then there exists a subsequence $\\{x_{k}\\}$ (we
do not relabel) converging uniformly to $x$, a unique solution to the Cauchy
problem

$\dot{x}(t)\in f(t,x(t),u(t))-N_{C(t)}(x(t)),\;\;\;x(0)=x_{0},$ (2.10)

where $u$ is a measurable function such that $u(t)\in U$ a.e. $t\in[0,T]$.

If, moreover, all the controls $u_{k}$ are equal, i.e., $u_{k}=u$, then the
subsequence converges to a unique solution of (2.10), i.e., any solution of

$\dot{x}(t)\in f(t,x(t),U)-N_{C(t)}(x(t)),\;\;\;x(0)=x_{0}\in C(0)$ (2.11)

can be approximated by solutions of (2.9).

Proof Consider the sequence $\\{x_{k}\\}$, where $(x_{k},u_{k})$ solves (2.9).
Recall that $x_{k}(t)\in C^{k}(t)$ for all $t\in[0,T]$, and

$|\dot{x}_{k}(t)|\leq({\rm const})\;\;\;{\rm
and}\;\;\;\xi_{k}^{i}(t)=\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}\leq({\rm
const}).$ (2.12)

Then there exist subsequences (we do not relabel) weakly-$*$ converging in
$L^{\infty}$ to some $v$ and $\xi^{i}$. Hence

$x_{k}(t)=x_{0}+\int_{0}^{t}\dot{x}_{k}(s)ds\longrightarrow
x(t)=x_{0}+\int_{0}^{t}v(s)ds,~{}\forall~{}t\in[0,T],$

for an absolutely continuous function $x$. Obviously, $x(t)\in C(t)$ for all
$t\in[0,T]$. Considering the sequence $\\{x_{k}\\}$, recall that

$\dot{x}_{k}(t)\in
f(t,x_{k}(t),U)-\sum_{i=1}^{I}\xi_{k}^{i}(t)\nabla_{x}\psi^{i}(t,x_{k}(t)).$
(2.13)

Inclusion (2.13) is equivalent to

$\langle z,\dot{x}_{k}(t)\rangle\leq
S(z,f(t,x_{k}(t),U))-\sum_{i=1}^{I}\xi_{k}^{i}(t)\langle
z,\nabla_{x}\psi^{i}(t,x_{k}(t))\rangle,\;\;\;\forall\>z\in R^{n}.$

Integrating this inequality, we get

$\left\langle z,\frac{x_{k}(t+\tau)-x_{k}(t)}{\tau}\right\rangle\\\
\leq\frac{1}{\tau}\int_{t}^{t+\tau}\left(S(z,f(s,x_{k}(s),U))-\sum_{i=1}^{I}\xi_{k}^{i}(s)\langle
z,\nabla_{x}\psi^{i}(s,x_{k}(s))\rangle\right)ds\\\
=\frac{1}{\tau}\int_{t}^{t+\tau}\left(S(z,f(s,x_{k}(s),U))-\sum_{i=1}^{I}\xi_{k}^{i}(s)\langle
z,\nabla_{x}\psi^{i}(s,x(s))\rangle\right.\\\
+\left.\sum_{i=1}^{I}\xi_{k}^{i}(s)\langle
z,\nabla_{x}\psi^{i}(s,x(s))-\nabla_{x}\psi^{i}(s,x_{k}(s))\rangle\right)ds.$
(2.14)

Passing to the limit as $k\rightarrow\infty$, we obtain

$\left\langle z,\frac{x(t+\tau)-x(t)}{\tau}\right\rangle\\\
\leq\frac{1}{\tau}\int_{t}^{t+\tau}\left(S(z,f(s,x(s),U))-\sum_{i=1}^{I}\xi^{i}(s)\langle
z,\nabla_{x}\psi^{i}(s,x(s))\rangle\right)ds.$ (2.15)

Let $t\in[0,T]$ be a Lebesgue point of $x$ and $\xi$. Passing in the last
inequality to the limit as $\tau\downarrow 0$, it leads to

$\langle z,\dot{x}(t)\rangle\leq
S(z,f(t,x(t),U))-\sum_{i=1}^{I}\xi^{i}(t)\langle
z,\nabla_{x}\psi^{i}(t,x(t))\rangle.$

Since $z\in R^{n}$ is an arbitrary vector and the set $f(t,x(t),U)$ is convex,
we conclude that

$\dot{x}(t)\in
f(t,x(t),U)-\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\psi^{i}(t,x(t)).$

By the Filippov lemma there exists a measurable control $u(t)\in U$ such that

$\dot{x}(t)=f(t,x(t),u(t))-\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\psi^{i}(t,x(t)).$

Furthermore, observe that $\xi^{i}$ is zero if $\psi^{i}(t,x(t))<0$. If for
some $u$ such that $u(t)\in U$ a.e., $u_{k}=u$ for all $k$, then the sequence
$x_{k}$ converges to the solution of

$\dot{x}(t)=f(t,x(t),u(t))-\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\psi^{i}(t,x(t)).$

Indeed, to see this, it suffices to pass to the limit as $k\rightarrow\infty$
and then as $\tau\downarrow 0$, in the equality

$\frac{x_{k}(t+\tau)-x_{k}(t)}{\tau}=\frac{1}{\tau}\int_{t}^{t+\tau}\left(f(s,x_{k}(s),u(s))-\sum_{i=1}^{I}\xi_{k}^{i}(s)\nabla_{x}\psi^{i}(s,x_{k}(s))\right)ds.$

We now prove the uniqueness of the solution. We follow the proof of Theorem
4.1 in [23]. Notice, however, that we now consider a special case and not the
general case treated in [23]. Suppose that there exist two different solutions
of (2.10): $x_{1}$ and $x_{2}$. We have

$\frac{1}{2}\frac{d}{dt}|x_{1}(t)-x_{2}(t)|^{2}=\langle
x_{1}(t)-x_{2}(t),\dot{x}_{1}(t)-\dot{x}_{2}(t)\rangle\\\ =\langle
x_{1}(t)-x_{2}(t),f(t,x_{1}(t),u(t))-f(t,x_{2}(t),u(t))\rangle\\\
-\left\langle
x_{1}(t)-x_{2}(t),\sum_{i=1}^{I}\xi_{1}^{i}(t)\nabla\psi^{i}(t,x_{1}(t))-\sum_{i=1}^{I}\xi_{2}^{i}(t)\nabla\psi^{i}(t,x_{2}(t))\right\rangle.$
(2.16)

If, for all $i$, $\psi^{i}(t,x_{1}(t))<0$ and $\psi^{i}(t,x_{2}(t))<0$, then
$\xi_{1}^{i}(t)=\xi_{2}^{i}(t)=0$ and we obtain

$\frac{1}{2}\frac{d}{dt}|x_{1}(t)-x_{2}(t)|^{2}\leq
L_{f}|x_{1}(t)-x_{2}(t)|^{2}.$

Suppose that $\psi^{j}(t,x_{1}(t))=0$. Then by the Taylor formula we get

$\psi^{j}(t,x_{2}(t))=\psi^{j}(t,x_{1}(t))+\langle\nabla_{x}\psi^{j}(t,x_{1}(t)),x_{2}(t)-x_{1}(t)\rangle\\\
+\frac{1}{2}\langle x_{2}(t)-x_{1}(t),\nabla_{x}^{2}\psi^{j}(t,\theta
x_{2}(t)+(1-\theta)x_{1}(t))(x_{2}(t)-x_{1}(t))\rangle,$ (2.17)

where $\theta\in[0,1]$. Since $\psi^{j}(t,x_{2}(t))\leq 0$, we have

$\langle\nabla_{x}\psi^{j}(t,x_{1}(t)),x_{2}(t)-x_{1}(t)\rangle\\\
\leq-\frac{1}{2}\langle x_{2}(t)-x_{1}(t),\nabla_{x}^{2}\psi^{j}(t,\theta
x_{2}(t)+(1-\theta)x_{1}(t))(x_{2}(t)-x_{1}(t))\rangle\\\ \leq({\rm
const})|x_{1}(t)-x_{2}(t)|^{2}.$ (2.18)

Now, if $\psi^{j}(t,x_{2}(t))=0$, we deduce in the same way that

$\langle\nabla_{x}\psi^{j}(t,x_{2}(t)),x_{1}(t)-x_{2}(t)\rangle\leq({\rm
const})|x_{1}(t)-x_{2}(t)|^{2}.$

Thus we have

$\frac{1}{2}\frac{d}{dt}|x_{1}(t)-x_{2}(t)|^{2}\leq({\rm
const})|x_{1}(t)-x_{2}(t)|^{2}.$

Hence $|x_{1}(t)-x_{2}(t)|=0$. $\Box$

##  3 Approximating Family of Optimal Control Problems

In this section we define an approximating family of optimal control problems
to $(P)$ and we state the corresponding necessary conditions.

Let $(\hat{x},\hat{u})$ be a global solution to $(P)$ and consider sequences
$\\{\gamma_{k}\\}$ and $\\{\sigma_{k}\\}$ as defined above. Let
$\hat{x}_{k}(\cdot)$ be the solution to

$\left\\{\begin{array}[]{rcl}&&\dot{x}(t)=f(t,x(t),\hat{u}(t))-\displaystyle\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x(t))-\sigma_{k})}\nabla_{x}\psi^{i}(t,x(t)),\\\
&&x(0)=\hat{x}(0).\end{array}\right.$ (3.1)

Set $\epsilon_{k}=|\hat{x}_{k}(T)-\hat{x}(T)|$. It follows from Theorem 2.2
that $\epsilon_{k}\downarrow 0$. Take $\alpha>0$ and define the problem

$(P_{k}^{\alpha})\left\\{\begin{array}[]{l}\mbox{Minimize
}\;\phi(x(T))+|x(0)-\hat{x}(0)|^{2}+\alpha\displaystyle\int_{0}^{T}|u(t)-\hat{u}(t)|dt\\\\[5.69054pt]
\mbox{over processes $(x,u)$ such that }\\\\[5.69054pt] \hskip
22.76219pt\dot{x}(t)=f(t,x(t),u(t))-\displaystyle\sum_{i=1}^{I}\nabla_{x}e^{\gamma_{k}(\psi^{i}(t,x(t))-\sigma_{k})}\hskip
5.69046pt\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptu(t)\in
U~{}~{}~{}~{}\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptx(0)\in
C_{0},~{}~{}x(T)\in C_{T}+\epsilon_{k}B_{n},\end{array}\right.$

Clearly, the problem $(P_{k}^{\alpha})$ has admissible solutions. Consider the
space

$W=\\{(c,u)\mid c\in C_{0},\;u\in L^{\infty}\;{\rm with}\;u(t)\in U\\}$

and the distance

$d_{W}((c_{1},u_{1}),(c_{2},u_{2}))=|c_{1}-c_{2}|+\int_{0}^{T}|u_{1}(t)-u_{2}(t)|dt.$

Endowed with $d_{W}$, $W$ is a complete metric space. Take any $(c,u)\in W$
and a solution $y$ to the Cauchy problem

$\left\\{\begin{array}[]{rcl}\dot{y}(t)&=&f(t,y(t),u(t))-\displaystyle\sum_{i=1}^{I}\nabla_{x}e^{\gamma_{k}(\psi^{i}(t,y(t))-\sigma_{k})}\hskip
5.69046pt\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] y(0)&=&c.\end{array}\right.$

Under our assumptions, the function

$(c,u)~{}\rightarrow~{}\phi(y(T))+|c-\hat{x}(0)|^{2}+\alpha\int_{0}^{T}|u-\hat{u}|~{}dt$

is continuous on $(W,d_{W})$ and bounded below. Appealing to Ekeland’s Theorem
we deduce the existence of a pair $(x_{k},u_{k})$ solving the following
problem

$(AP_{k})\left\\{\begin{array}[]{l}\mbox{Minimize
}\;\Phi(x,{u})=\phi(x(T))+|x(0)-\hat{x}(0)|^{2}+\alpha\displaystyle\int_{0}^{T}|u(t)-\hat{u}(t)|dt\\\\[8.53581pt]
\qquad\qquad+\epsilon_{k}\left(|x(0)-x_{k}(0)|+\displaystyle\int_{0}^{T}|u(t)-u_{k}(t)|dt\right),\\\\[5.69054pt]
\mbox{over processes $(x,u)$ such that }\\\\[5.69054pt] \hskip
22.76219pt\dot{x}(t)=f(t,x(t),u(t))-\displaystyle\sum_{i=1}^{I}\nabla_{x}e^{\gamma_{k}(\psi^{i}(t,x(t))-\sigma_{k})}\hskip
5.69046pt\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptu(t)\in
U~{}~{}~{}~{}\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptx(0)\in
C_{0},~{}~{}x(T)\in C_{T}+\epsilon_{k}B_{n},\end{array}\right.$

###### Lemma 3.1

Take $\gamma_{k}\to\infty$, $\sigma_{k}\to 0$ and $\epsilon_{k}\to 0$ as
defined above. For each $k$, let $(x_{k},u_{k})$ be the solution to
$(AP_{k})$. Then there exists a subsequence (we do not relabel) such that

$u_{k}(t)\rightarrow\hat{u}(t)~{}{a.e.},\quad
x_{k}\rightarrow\hat{x}\;\rm{uniformly}\;in\;[0,T].$

Proof We deduce from Theorem 2.2 that $\\{x_{k}\\}$ uniformly converges to an
admissible solution $\tilde{x}$ to $(P)$. Since $U$ and $C_{0}$ are compact,
we have $U\subset KB_{m}$ and $C_{0}\subset KB_{n}$. Without loss of
generality, $u_{k}$ weakly-$*$ converges to a function $\tilde{u}\in
L_{\infty}([0,T],U)$. Hence it weakly converges to $\tilde{u}$ in $L_{1}$.
From optimality of the processes $(x_{k},u_{k})$ we have

$\phi(x_{k}(T))+|x_{k}(0)-\hat{x}(0)|^{2}+\alpha\int_{0}^{T}|u_{k}(t)-\hat{u}(t)|dt$
$\leq\phi(\hat{x}_{k}(T))+\epsilon_{k}\left(|\hat{x}_{k}(0)-x_{k}(0)|+\int_{0}^{T}|u_{k}(t)-\hat{u}(t)|dt\right)$
$\leq\phi(\hat{x}_{k}(T))+2K(1+T)\epsilon_{k}.$

Since $(\hat{x},\hat{u})$ is a global solution of the problem, passing to the
limit, we get

$\phi(\tilde{x}(T))+|\tilde{x}(0)-\hat{x}(0)|^{2}+\alpha\int_{0}^{T}|\tilde{u}(t)-\hat{u}(t)|dt$
$\leq\lim_{k\rightarrow\infty}(\phi(x_{k}(T))+|x_{k}(0)-\hat{x}(0)|^{2})+\alpha\liminf_{k\rightarrow\infty}\int_{0}^{T}|u_{k}(t)-\hat{u}(t)|dt$
$\leq\lim_{k\rightarrow\infty}\phi(\hat{x}_{k}(T))=\phi(\hat{x}(T))\leq\phi(\tilde{x}(T)).$

Hence $\tilde{x}(0)=\hat{x}(0)$, $\tilde{u}=\hat{u}$ a.e., and $u_{k}$
converges to $\hat{u}$ in $L_{1}$, and some subsequence converges to $\hat{u}$
almost everywhere (we do not relabel). $\Box$

We now finish this section with the statement of the optimality necessary
conditions for the family of problems $(AP_{k})$. These can be seen as a
direct consequence of Theorem 6.2.1 in [22].

###### Proposition 3.2

For each k, let $(x_{k},u_{k})$ be a solution to $(AP_{k})$. Then there exist
absolutely continous functions $p_{k}$ and scalars $\lambda_{k}\geq 0$ such
that

  * (a)

(nontriviality condition)

$\lambda_{k}+|p_{k}(T)|=1,$ (3.2)

  * (b)

(adjoint equation)

$\begin{array}[]{c}\dot{p}_{k}={\color[rgb]{1,0,0}-}(\nabla_{x}f_{k})^{*}p_{k}+\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\nabla^{2}_{x}\psi_{k}^{i}p_{k}\\\\[5.69054pt]
+\sum_{i=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\nabla_{x}\psi_{k}^{i}\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle,\end{array}$
(3.3)

where the superscript $*$ stands for transpose,

  * (c)

(maximization condition)

$\max_{u\in U}\left\\{\langle
f(t,x_{k},u),p_{k}\rangle-\alpha\lambda_{k}|u-\hat{u}|-\epsilon_{k}\lambda_{k}|u-u_{k}|\right\\}$
(3.4)

is attained at $u_{k}(t)$, for almost every $t\in[0,T]$,

  * (d)

(transversality condition)

$\displaystyle(p_{k}(0),-p_{k}(T))\in\lambda_{k}\left(2(x_{k}(0)-\hat{x}(0))+\epsilon_{k}B_{n},\partial\phi(x_{k}(T))\right)$
$\displaystyle+N_{C_{0}}(x_{k}(0))\times
N_{C_{T}+\epsilon_{k}B_{n}}(x_{k}(T)).\hskip 28.45274pt$ (3.5)

To simplify the notation above, we drop the $t$ dependance in $p_{k}$,
$\dot{p}_{k}$, $x_{k}$, $u_{k}$, $\hat{x}$ and $\hat{u}$. Moreover, in (b), we
write $\psi_{k}$ instead of $\psi(t,x_{k}(t))$, $f_{k}$ instead of
$f(t,x_{k}(t),u_{k}(t))$. The same holds for the derivatives of $\psi$ and
$f$.

##  4 Maximum Principle for $(P)$

In this section, we establish our main result, a Maximum Principle for $(P)$.
This is done by taking limits of the conclusions of Proposition 3.2, following
closely the analysis done in the proof of [10, Theorem 2].

Observe that

$\begin{split}\frac{1}{2}\frac{d}{dt}|p_{k}(t)|^{2}&=-\langle\nabla_{x}f_{k}p_{k},p_{k}\rangle+\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}^{2}\psi_{k}^{i}p_{k},p_{k}\rangle\\\
&\hskip
28.45274pt+\sum_{i=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle^{2}\\\
&\geq-\langle\nabla_{x}f_{k}p_{k},p_{k}\rangle+\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}^{2}\psi_{k}^{i}p_{k},p_{k}\rangle\\\
&\geq\
-M|p_{k}|^{2}+\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}^{2}\psi_{k}^{i}p_{k},p_{k}\rangle,\end{split}$

where $M$ is the constant of (A2). Taking into account hypothesis (A1) and
(2.8) we deduce the existence of a constant $K_{0}>0$ such that

$\frac{1}{2}\frac{d}{dt}|p_{k}(t)|^{2}\geq-
K_{0}|p_{k}{\color[rgb]{0,0,1}(t)}|^{2}.$

This last inequality leads to

$|p_{k}(t)|^{2}\ \leq\ e^{2K_{0}(T-t)}|p_{k}(T)|^{2}\leq\
e^{2K_{0}T}|p_{k}(T)|^{2}.$

Since, by (a) of Proposition 3.2, $|p_{k}(T)|\leq 1$, we deduce from the above
that there exists $M_{0}>0$ such that

$|p_{k}(t)|\ \leq M_{0}.$ (4.1)

Now, we claim that the sequence $\\{\dot{p}_{k}\\}$ is uniformly bounded in
$L^{1}$. To prove our claim, we need to establish bounds for the three terms
in (3.3). Following [10] and [14], we start by deducing some inequalities that
will be of help.

Denote $I_{k}=I(t,x_{k}(t))$ and $S_{k}^{j}={\rm
sign}\left(\langle\nabla_{x}\psi_{k}^{j},p_{k}\rangle\right)$. We have

$\begin{split}\sum_{j=1}^{I}\frac{d}{dt}&\left|\langle\nabla_{x}\psi_{k}^{j},p_{k}\rangle\right|\\\
&=\sum_{j=1}^{I}\left(\langle\nabla^{2}_{x}\psi_{k}^{j}\dot{x}_{k},p_{k}\rangle+\langle\partial_{t}\nabla_{x}\psi_{k}^{j},p_{k}\rangle+\langle\nabla_{x}\psi_{k}^{j},\dot{p}_{k}\rangle\right)\,S_{k}^{j}\\\
&=\sum_{j=1}^{I}\left(\langle
p_{k},\nabla^{2}_{x}\psi_{k}^{j}f_{k}\rangle-\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle
p_{k},\nabla^{2}\psi_{k}^{j}\nabla_{x}\psi_{k}^{i}\rangle\right)S_{k}^{j}\\\
&\hskip
28.45274pt+\sum_{j=1}^{I}\left(\langle\partial_{t}\nabla_{x}\psi_{k}^{j},p_{k}\rangle-\langle\nabla_{x}\psi_{k}^{j},(\nabla_{x}f_{k})^{*}p_{k}\rangle\right)S_{k}^{j}\\\
&\hskip
28.45274pt+\sum_{j=1}^{I}\left(\sum_{i=1}^{I}\gamma_{k}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}\psi_{k}^{j},\nabla^{2}_{x}\psi_{k}^{i}p_{k}\rangle\right)S_{k}^{j}\\\
&\hskip
28.45274pt+\sum_{i=1}^{I}\sum_{j=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle
S_{k}^{j}\end{split}$

Observe that (see (2.3) and (2.4))

$\sum_{i=1}^{I}\sum_{j=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle
S_{k}^{j}\\\ =\sum_{i=1}^{I}\sum_{j\in
I_{k}}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle
S_{k}^{j}\\\ =\sum_{i\not\in
I_{k}}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\sum_{j\in
I_{k}}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle
S_{k}^{j}\\\ +\sum_{i\in
I_{k}}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\left(|\nabla_{x}\psi_{k}^{i}|^{2}+\sum_{j\in
I_{k}\setminus\\{i\\}}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle
S_{k}^{j}~{}S_{k}^{i}\right)|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|\\\
=\sum_{i\in
I_{k}}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\left(|\nabla_{x}\psi_{k}^{i}|^{2}+\sum_{j\in
I_{k}\setminus\\{i\\}}\langle\nabla_{x}\psi_{k}^{j},\nabla_{x}\psi_{k}^{i}\rangle
S_{k}^{j}~{}S_{k}^{i}\right)|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|\\\
\geq\displaystyle(1-\rho)\sum_{i\in
I_{k}}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|^{2}|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|\\\
=(1-\rho)\sum_{i=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|^{2}|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|.\hskip
56.9055pt$ (4.2)

Using this and integrating the previous equality, we deduce the existence of
$M_{1}>0$ such that:

$\int_{0}^{T}\sum_{i=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|^{2}|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|dt\leq
M_{1}.$ (4.3)

We are now in a position to show that

$\displaystyle\int_{0}^{T}\sum_{i=1}^{I}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|\left|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle\right|\
dt$

is bounded. For simplicity, set
$L_{k}^{i}(t)=\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|\left|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle\right|$.
Notice that

$\displaystyle\sum_{i=1}^{I}\int_{0}^{T}L_{k}^{i}(t)dt=\displaystyle\sum_{i=1}^{I}\left\\{\int_{\\{t:|\nabla_{x}\psi_{k}^{i}|<\eta\\}}L_{k}^{i}(t)~{}dt+\displaystyle\int_{\\{t:|\nabla_{x}\psi_{k}^{i}|\geq\eta\\}}L_{k}^{i}(t)dt\right\\}.$

Using (A1) and (4.3), we deduce that

$\begin{split}\displaystyle\sum_{i=1}^{I}\int_{0}^{T}L_{k}^{i}(t)~{}dt&\leq\displaystyle\sum_{i=1}^{I}\left(\gamma_{k}^{2}e^{-\gamma_{k}(\beta+\sigma_{k})}\eta^{2}\max_{t}|p_{k}(t)|\right)\\\
&\hskip
14.22636pt+\displaystyle\sum_{i=1}^{I}\left(\gamma_{k}^{2}\int_{\\{t:|\nabla_{x}\psi_{k}^{i}|\geq\eta\\}}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\frac{|\nabla_{x}\psi_{k}^{i}|^{2}}{|\nabla_{x}\psi_{k}^{i}|}\left|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle\right|\
dt\right)\\\\[2.84526pt]
&\leq\gamma_{k}^{2}I~{}e^{-\gamma_{k}(\beta+\sigma_{k})}\eta^{2}M_{0}\\\
&\hskip
14.22636pt+\frac{1}{\eta}\displaystyle\sum_{i=1}^{I}\left(\int_{0}^{T}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla_{x}\psi_{k}^{i}|^{2}\left|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle\right|\
dt\right)\\\\[2.84526pt] &\leq\
\eta^{2}M_{0}{\color[rgb]{0,0,1}I}+\frac{M_{1}}{\eta},\end{split}$

for $k$ large enough. Summarizing, there exists a $M_{2}>0$ such that

$\displaystyle\sum_{i=1}^{I}\gamma_{k}^{2}\int_{0}^{T}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\nabla\psi_{k}^{i}|\left|\langle\nabla\psi_{k}^{i},p_{k}\rangle\right|\
dt\ \ \leq M_{2}.$ (4.4)

Mimicking the analysis conducted in Step 1, b) and c) of the proof of Theorem
2 in [10] and taking into account (b) of Proposition 3.2 we conclude that
there exist constants $N_{1}>0$ such that

$\int_{0}^{T}\left|\dot{p}_{\gamma_{k}}(t)\right|dt\leq N_{1},$ (4.5)

for $k$ sufficiently large, proving our claim.

Before proceeding, observe that it is a simple matter to assert the existence
of a constant $N_{2}$ such that

$\displaystyle\sum_{i=1}^{I}\int_{0}^{T}\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}|\langle\nabla\psi_{k}^{i},p_{\gamma_{k}}\rangle|dt\leq
N_{2}.$ (4.6)

This inequality will be of help in what follows.

Let us now recall that

$\xi_{k}^{i}(t)=\gamma_{k}e^{\gamma_{k}(\psi^{i}(t,x_{k}(t))-\sigma_{k})}$

and that the second inequality in (2.12) holds. We turn to the analysis of
Step 2 in the proof of Theorem 2 in [10] (see also [14]). Adapting those
arguments, we can conclude the existence of some function $p\in
BV([0,T],R^{n})$ and, for $i=1,\ldots,I$, functions $\xi^{i}\in
L^{\infty}([0,T],R)$ with $\xi^{i}(t)\geq 0\ \mbox{ a. e. }t$, $\xi^{i}(t)=0,\
t\in I_{b}^{i}$, where

$I_{b}^{i}=\left\\{t\in[0,T]:~{}\psi^{i}(t,\hat{x}(t))<0\right\\},$

and finite signed Radon measures $\eta^{i}$, null in $I_{b}^{i}$, such that,
for any $z\in C([0,T],R^{n})$

$\int_{0}^{T}\langle z,dp\rangle=-\int_{0}^{T}\langle
z,(\nabla\hat{f})^{*}p\rangle
dt+\displaystyle\sum_{i=1}^{I}\left(\int_{0}^{T}\xi^{i}\langle
z,\nabla^{2}\hat{\psi}^{i}p\rangle dt+\int_{0}^{T}\langle
z,\nabla\hat{\psi}^{i}(t)\rangle d\eta^{i}\right),$

where $\nabla\hat{\psi}^{i}(t)=\nabla\psi^{i}(t,\hat{x}(t))$. The finite
signed Radon measures $\eta^{i}$ are weak-$*$ limits of

$\gamma_{k}^{2}e^{\gamma_{k}(\psi_{k}^{i}-\sigma_{k})}\langle\nabla\psi_{k}^{i}(x_{k}(t),p_{k}(t)\rangle
dt.$

Observe that the measures

$\langle\nabla\psi^{i}(\hat{x}(t),p(t)\rangle d\eta^{i}(t)$ (4.7)

are nonnegative.

For each $i=1,\ldots,I$, the sequence $\xi_{k}^{i}$ is weakly-$*$ convergent
in $L^{\infty}$ to $\xi^{i}\geq 0$. Following [14], we deduce from (4.6) that,
for each $i=1,\ldots,I$,

$\int_{0}^{T}|\xi^{i}\langle\nabla_{x}\hat{\psi}^{i},p\rangle|dt=\lim_{k\rightarrow\infty}\int_{0}^{T}|\xi_{k}^{i}\langle\nabla_{x}\hat{\psi}^{i},p\rangle|dt$
$\leq\lim_{k\rightarrow\infty}\left(\int_{0}^{T}\xi_{k}^{i}|\langle\nabla_{x}\hat{\psi}^{i},p\rangle-\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|dt+\int_{0}^{T}\xi_{k}^{i}|\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle|dt\right)$
$\leq\lim_{k\rightarrow\infty}\left(\Big{|}\xi_{k}^{i}\Big{|}_{L^{\infty}}\Big{|}\langle\nabla_{x}\hat{\psi}^{i},p\rangle-\langle\nabla_{x}\psi_{k}^{i},p_{k}\rangle\Big{|}_{L^{1}}+\frac{N_{2}}{\gamma_{k}}\right)=0.$

It turns out that

$\xi^{i}\langle\nabla_{x}\hat{\psi}^{i},p\rangle=0\;{\rm a.e.}.$ (4.8)

Consider now the sequence of scalars $\\{\lambda_{k}\\}$. It is an easy matter
to show that there exists a subsequence of $\\{\lambda_{k}\\}$ converging to
some $\lambda\geq 0$. This, together with the convergence of $p_{k}$ to $p$,
allows us to take limits in (a) and (c) of Proposition 3.2 to deduce that

$\lambda+|p(T)|=1$

and

$\langle p(t),f(t,\hat{x}(t),u)\rangle-\alpha\lambda|u-\hat{u}(t)|\leq\langle
p(t),f(t,\hat{x}(t),\hat{u}(t))\rangle~{}\forall u\in U,\text{ a.e.
}t\in[0,T].$

It remains to take limits of the transversality conditions (d) in Proposition
3.2. First, observe that

$C_{T}+\epsilon_{k}B_{n}=\left\\{x:~{}d(x,C_{T})\leq\epsilon_{k}\right\\}.$

From the basic properties of the Mordukhovich normal cone and subdifferential
(see [19], section 1.3.3) we have

$N_{C_{T}+\epsilon_{k}B_{n}}(x_{k}(T))\subset\text{ cl cone}\>\partial
d(x_{k}(T),C_{T})$

and

$N_{C_{T}}(\hat{x}(T))=\text{ cl cone}\>\partial d(\hat{x}(T),C_{T}).$

Passing to the limit as $k\to\infty$ we get

$(p(0),-p(T))\in N_{C_{0}}(\hat{x}(0))\times
N_{C_{T}}(\hat{x}(T))+\\{0\\}\times\lambda~{}\partial\phi(\hat{x}(T)).$

Finally, and mimicking Step 3 in the proof of Theorem 2 in [10], we remove the
dependence of the conditions on the parameter $\alpha$. This is done by taking
further limits, this time considering a sequence of $\alpha_{j}\downarrow 0$.

We then summarize our conclusions in the following Theorem.

###### Theorem 4.1

Let $(\hat{x},\hat{u})$ be the optimal solution to $(P)$. Suppose that
assumption A1–A6 are satisfied. For $i=1,\cdots,I$, set

$I^{i}_{b}=\\{t\in[0,T]:~{}\psi^{i}(t,\hat{x}(t))<0\\}.$

There exist $\lambda\geq 0$, $p\in BV([0,T],R^{n})$, finite signed Randon
measures $\eta^{i}$, null in $I^{i}_{b}$, for $i=1,\cdots,I$, $\xi^{i}\in
L^{\infty}([0,T],R)$, with $i=1,\cdots,I$, where $\displaystyle\xi^{i}(t)\geq
0\ \text{ a. e. }t$ and $\xi^{i}(t)=0,\ t\in I^{i}_{b},$ such that

  * a)

$\lambda+|p(T)|\neq 0$,

  * b)

$\dot{\hat{x}}(t)=f(t,\hat{x}(t),\hat{u}(t))-\displaystyle\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\hat{\psi}^{i}(t),$

  * c)

for any $z\in C([0,T];R^{n})$

$\begin{array}[]{c}\displaystyle\int_{0}^{T}\langle
z(t),dp(t)\rangle=-\displaystyle\int_{0}^{T}\langle
z(t),(\nabla_{x}\hat{f}(t))^{*}p(t)\rangle dt\\\\[8.53581pt]
\displaystyle+\sum_{i=1}^{I}\displaystyle\left(\int_{0}^{T}\xi^{i}(t)\langle
z(t),\nabla^{2}_{x}\hat{\psi}^{i}(t)p(t)\rangle dt\right.\
+\displaystyle\left.\int_{0}^{T}\langle
z(t),\nabla_{x}\hat{\psi}^{i}(t)\rangle d\eta_{i}\right),\end{array}$

where
$\nabla\hat{f}(t)=\nabla_{x}f(t,\hat{x}(t),\hat{u}(t)),~{}~{}\nabla\hat{\psi}^{i}(t)=\nabla\psi^{i}(t,\hat{x}(t))$
and $\nabla^{2}\hat{\psi}^{i}(t)=\nabla^{2}\psi^{i}(t,x(t)),$

  * d)

$\xi_{i}(t)\langle\nabla_{x}\psi^{i}(t,\hat{x}(t)),p(t)\rangle=0$, $a.e.\,t$
for all $i=1,\ldots,I$,

  * e)

for all $i=1,\ldots,I$, the meaures
$\langle\nabla\psi^{i}(\hat{x}(t),p(t)\rangle d\eta^{i}(t)$ are nonnegative,

  * f)

$\displaystyle\langle p(t),f(t,\hat{x}(t),u)\rangle\leq\langle
p(t),f(t,\hat{x}(t),\hat{u}(t))\rangle$ for all $u\in U,$ $~{}a.e.\,t$,

  * g)

$\displaystyle\begin{array}[]{c}(p(0),-p(T))\in N_{C_{0}}(\hat{x}(0))\times
N_{C_{T}}(\hat{x}(T))+\\{0\\}\times\lambda\partial\phi(\hat{x}(T)).\end{array}$

Noteworthy, condition e) is not considered in any of our previous works.

We now turn to the free end point case, i. e., to the problem

$(P_{f})\left\\{\begin{array}[]{l}\mbox{Minimize }\;\phi(x(T))\\\\[5.69054pt]
\mbox{over processes $(x,u)$ such that }\\\\[5.69054pt] \hskip
22.76219pt\dot{x}(t)\in f(t,x(t),u(t))-N_{C(t)}(x(t)),\hskip
5.69046pt\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptu(t)\in U,\
\ \;\,\mbox{a.e.}\ \ t\in[0,T],\\\\[5.69054pt] \hskip 22.76219ptx(0)\in
C_{0}\subset C(0).\end{array}\right.$

Problem $(P_{f})$ differs from $(P)$ because $x(T)$ is not constrained to take
values in $C_{T}$. We apply Theorem 4.1 to $(P_{f})$. Since $x(T)$ is free, we
deduce from (f) in the above Theorem that
$-p(T)=\lambda\partial\phi(\hat{x}(T))$. Suppose that $\lambda=0$. Then
$p(T)=0$ contradicting the nontriviality condition (a) of Theorem 4.1. Without
loss of generality, we then conclude that the conditions of Theorem 4.1 hold
with $\lambda=1$. We summarize our findings in the following Corollary.

###### Corollary 4.2

Let $(\hat{x},\hat{u})$ be the optimal solution to $(P_{f})$. Suppose that
assumption A1–A6 are satisfied. For $i=1,\cdots,I$, set

$I^{i}_{b}=\\{t\in[0,T]:~{}\psi^{i}(t,\hat{x}(t))<0\\}.$

There exist $p\in BV([0,T],R^{n})$, finite signed Randon measures $\eta_{i}$,
null in $I^{i}_{b}$, for $i=1,\cdots,I$, $\xi^{i}\in L^{\infty}([0,T],R)$,
with $i=1,\cdots,I$, where $\displaystyle\xi^{i}(t)\geq 0\ \text{ a.e. }t$ and
$\xi^{i}(t)=0$ for $t\in I^{i}_{b},$ such that

  * a)

$\dot{\hat{x}}(t)=f(t,\hat{x}(t),\hat{u}(t))-\displaystyle\sum_{i=1}^{I}\xi^{i}(t)\nabla_{x}\hat{\psi}^{i}(t),$

  * b)

for any $z\in C([0,T];R^{n})$

$\begin{array}[]{c}\displaystyle\int_{0}^{T}\langle
z(t),dp(t)\rangle=-\displaystyle\int_{0}^{T}\langle
z(t),(\nabla_{x}\hat{f}(t))^{*}p(t)\rangle dt\\\\[8.53581pt]
\displaystyle+\sum_{i=1}^{I}\displaystyle\left(\int_{0}^{T}\xi^{i}(t)\langle
z(t),\nabla^{2}_{x}\hat{\psi}^{i}(t)p(t)\rangle dt\right.\
+\displaystyle\left.\int_{0}^{T}\langle
z(t),\nabla_{x}\hat{\psi}^{i}(t)\rangle d\eta_{i}\right),\end{array}$

where
$\nabla\hat{f}(t)=\nabla_{x}f(t,\hat{x}(t),\hat{u}(t)),~{}~{}\nabla\hat{\psi}^{i}(t)=\nabla\psi^{i}(t,\hat{x}(t))$
and $\nabla^{2}\hat{\psi}^{i}(t)=\nabla^{2}\psi^{i}(t,x(t)),$

  * c)

$\xi^{i}(t)\langle\nabla_{x}\psi^{i}(t,\hat{x}(t)),p(t)\rangle=0$ for
$a.e.~{}t$ and for all $i=1,\ldots,I$,

  * d)

for all $i=1,\ldots,I$, the meaures
$\langle\nabla\psi^{i}(\hat{x}(t),p(t)\rangle d\eta^{i}(t)$ are nonnegative,

  * e)

$\displaystyle\langle p(t),f(t,\hat{x}(t),u)\rangle\leq\langle
p(t),f(t,\hat{x}(t),\hat{u}(t))\rangle$ for all $u\in U$, $a.e.\,t$,

  * f)

$\displaystyle\begin{array}[]{c}(p(0),-p(T))\in
N_{C_{0}}(\hat{x}(0))\times\\{0\\}+\\{0\\}\times\partial\phi(\hat{x}(T)).\end{array}$

##  5 Example

Let us consider the following problem

$\left\\{\begin{array}[]{l}\mbox{Minimize }\;-x(T)\\\\[5.69054pt] \mbox{over
processes $((x,y,z),u)$ such that }\\\\[5.69054pt] \hskip
22.76219pt\begin{bmatrix}\dot{x}(t)\\\ \dot{y}(t)\\\
\dot{z}(t)\end{bmatrix}\in\begin{bmatrix}0&\sigma&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}+\begin{bmatrix}0\\\
u\\\ 0\end{bmatrix}-N_{C}(x,y,z),\\\\[5.69054pt] \hskip
22.76219ptu\in[-1,1],\\\\[2.84526pt] \hskip
22.76219pt(x,y,z)(0)=(x_{0},y_{0},z_{0}),\\\\[2.84526pt] \hskip
22.76219pt(x,y,z)(T)\in C_{T},\end{array}\right.$

where

  * $\bullet$

$0<\sigma\ll 1$,

  * $\bullet$

$C=\\{(x,y,z)\mid x^{2}+y^{2}+(z+h)^{2}\leq 1,\;x^{2}+y^{2}+(z-h)^{2}\leq
1\\},\;\;2h^{2}<1$,

  * $\bullet$

$(x_{0},y_{0},z_{0})\in{\rm int}C$, with $x_{0}<-\delta$, $y_{0}=0$ and
$z_{0}>0$,

  * $\bullet$

$C_{T}=\\{(x,y,z)\mid x\leq 0,\;y\geq 0,\;\delta y-y_{2}x\leq\delta
y_{2}\\}\cap C$, where

$\delta<\frac{y_{2}|x_{0}|}{y_{1}},~{}{\rm
with}~{}y_{1}=\sqrt{1-x_{0}^{2}-(z_{0}+h)^{2}}\text{ and
}y_{2}=\sqrt{1-h^{2}}.$

We choose $T>0$ small and, nonetheless, sufficiently large to guarantee that,
when $\sigma=0$, the system can reach the interior of $C_{T}$ but not the
segment $\\{(x,0,0)\mid x\in[-\delta,0]\\}$. Since $\sigma$ and $T$ are small,
it follows that the optimal trajectory should reach $C_{T}$ at the face
$\delta y-y_{2}x=\delta y_{2}$ of $C_{T}$.

To significantly increase the value of the $x(T)$, the optimal trajectory
needs to live on the boundary of $C$ for some interval of time. Then, before
reaching and after leaving the boundary of $C$, the optimal trajectory lives
in the interior of $C$. Since $\delta$ is small, the trajectory cannot reach
$C_{T}$ from any point of the sphere $x^{2}+y^{2}+(z+h)^{2}=1$ with $z>0$.
This means that, while on the boundary of $C$ the trajectory should move on
the sphere $x^{2}+y^{2}+(z+h)^{2}=1$ untill reaching the plane $z=0$ and then
it moves on the intersection of the two spheres.

While in the interior of $C$, the control can change sign from $-1$ to $1$ or
from $1$ to $-1$. Certainly, the control should be $1$ right before reaching
the boundary and $-1$ right before arriving at $C_{T}$. Changes of the control
from $1$ to $-1$ or $-1$ to $1$ before reaching the boundary translate into
time waste and leads to smaller values of $x(T)$. It then follows that the
optimal control should be of the form

$u(t)=\left\\{\begin{array}[]{cl}1,&t\in[0,\tilde{t}],\\\\[2.84526pt]
-1,&t\in\ ]\tilde{t},T],\end{array}\right.$ (5.1)

for some value $\tilde{t}\in]0,T[$.

After the modification (2.5), the data of the problem satisfy the conditions
under which Theorem 4.1 holds. We now show that the conclusions of Theorem 4.1
completly identify the structure (5.1) of the optimal control.

From Theorem 4.1 we deduce the existence of $\lambda\geq 0$, $p,~{}q,~{}r\in
BV([0,T],R)$, finite signed Randon measures $\eta_{1}$ and $\eta_{2}$, null
respectively in

$I^{1}_{b}=\left\\{(x,y,z)\mid x^{2}+y^{2}+(z+h)^{2}-1<0\right\\}$

and

$I^{2}_{b}=\left\\{(x,y,z)\mid x^{2}+y^{2}+(z-h)^{2}-1<0\right\\},$

$\xi_{i}\in L^{\infty}([0,T],R)$, with $i=1,2$, where
$\displaystyle\xi_{i}(t)\geq 0\ \text{ a. e. }t$ and $\xi_{i}(t)=0,\ t\in
I^{i}_{b},$ such that

$\begin{array}[]{rl}\text{(i)}&\begin{bmatrix}\dot{x}(t)\\\ \dot{y}(t)\\\
\dot{z}(t)\end{bmatrix}=\begin{bmatrix}0&\sigma&0\\\ 0&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}+\begin{bmatrix}0\\\
u\\\ 0\end{bmatrix}-2\xi_{1}\begin{bmatrix}x\\\ y\\\
z+h\end{bmatrix}-2\xi_{2}\begin{bmatrix}x\\\ y\\\
z-h\end{bmatrix}\\\\[17.07164pt] \text{(ii)}&d\begin{bmatrix}p\\\ q\\\
r\end{bmatrix}=\begin{bmatrix}0&0&0\\\ -\sigma&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}p\\\ q\\\ r\end{bmatrix}dt\\\ &\hskip
56.9055pt+2(\xi_{1}+\xi_{2})\begin{bmatrix}p\\\ q\\\
r\end{bmatrix}dt+2\begin{bmatrix}x\\\ y\\\
z+h\end{bmatrix}d\eta_{1}+2\begin{bmatrix}x\\\ y\\\
z-h\end{bmatrix}d\eta_{2},\\\\[17.07164pt] \text{(iii)}&\begin{bmatrix}p\\\
q\\\ r\end{bmatrix}(T)=\begin{bmatrix}\lambda\\\ 0\\\
0\end{bmatrix}+\mu\begin{bmatrix}y_{2}\\\ -\delta\\\ 0\end{bmatrix},\text{
where }\mu\geq 0,\\\\[14.22636pt]
\text{(iv)}&\xi_{1}(xp+yq+(z+h)r)=0,\;\xi_{2}(xp+yq+(z-h)r)=0,\\\\[5.69054pt]
\text{(v)}&\text{the meaures }(xp+yq+(z+h)r)d\eta_{1}\text{ and
}(xp+yq+(z-h)r)d\eta_{2}\\\ &\hskip 14.22636pt\text{ are
nonnegative,}\\\\[5.69054pt]
\text{(vi)}&\max_{u\in[-1,1]}uq=\hat{u}q.\end{array}$

where $\hat{u}$ is the optimal control.

Let $t_{1}$ be the instant of time when the trajectory reaches the shere
$x^{2}+y^{2}+(z+h)^{2}=1$, $t_{2}$ the instant of time when the trajectory
reaches the intersection of the two spheres and $t_{3}$ be the instant of time
the trajectory leaves the boundary of $C$. We have $0<t_{1}<t_{2}<t_{3}<T$.

Next we show that the multiplier $q$ changes sign only once and so identifing
the structure (5.1) of the optimal control in a unique way. We start by
looking at the case when $t=T$. We have

$\left[\begin{array}[]{c}p\\\
q\end{array}\right](T)=\left[\begin{array}[]{c}\lambda\\\
0\end{array}\right]+\mu\left[\begin{array}[]{c}y_{2}\\\
-\delta\end{array}\right].$

Starting from $t=T$, let us go backwards in time until the instant $t_{3}$
when the trajectory leaves the boundary of $C$. If $q(T)=0$, then
$p(T)=\lambda>0$ and we would have $q(t)>0$ for $t\in]t_{3},T[$ (see (ii)
above), which is impossible. We then have $p(T)>0$ and $q(T)<0$ and, in
$]t_{3},T[$, since $\sigma$ is small, the vector $(p(t),q(t))$ does not change
much. At $t=t_{3}$, the vector $(p,q)$ has a jump and such jump can only occur
along the vector $(x(t_{3}),y(t_{3}))$. Therefore, we have $p(t_{3}-0)>0$ and
$q(t_{3}-0)<0$.

Let us now consider $t\in]t_{2},t_{3}[$. We have the following

  1. 1.

when $t\in[t_{2},t_{3}]$, we have $z=0$;

  2. 2.

condition (i) above implies that $\xi_{1}=\xi_{2}=\xi$, $\xi>0$ since,
otherwise the motion along $x^{2}+y^{2}=1-h^{2}$ would not be possible;

  3. 3.

from $0=\frac{d}{dt}(x^{2}+y^{2})=\sigma 2xy-8\xi x^{2}+2uy-8\xi y^{2}$ we get
$\xi=\frac{\sigma xy+uy}{4(1-h^{2})}$;

  4. 4.

condition (iv) implies that $r=0$ leading to $xp+yq=0$. Since $x<0$, $y>0$,
then $q=0$ implies $p=0$;

  5. 5.

condition (ii) implies $d\eta_{1}=d\eta_{2}=d\eta$;

  6. 6.

$0=d(xp+yq)=uqdt+4(1-h^{2})d\eta$ $\Rightarrow$
$\frac{d\eta}{dt}=-\frac{uq}{4(1-h^{2})}$;

  7. 7.

from the above analysis we deduce that

$\displaystyle\dot{p}=\frac{\sigma
xy+uy}{(1-h^{2})}~{}p-\frac{xuq}{(1-h^{2})},$ $\displaystyle\dot{q}=-\sigma
p+\frac{\sigma xy}{(1-h^{2})}~{}q.$

Thus, $(p,q)$ is a solution to a linear system and it can never be equal to
zero. It follows that $q$ cannot be zero because $q=0$ implies $p=0$. Since
$q\neq 0$, we have $q>0$.

Let us consider the case when $t=t_{2}$. We claim that

$(p(t_{2}-0),q(t_{2}-0))\neq(0,0).$

Seeking a contradiction, assume that it is $(p(t_{2}-0),q(t_{2}-0))=(0,0)$.
Then we have

$(p(t_{2}+0),q(t_{2}+0))=(0,0)+(2x_{2}(t_{2}),2y_{2}(t_{2}))(d\eta_{1}+d\eta_{2})$

and such jump has to be normal to $(x(t_{2}),y(t_{2}))$ since $r(t_{2}+0)=0$
(see (iv)). It follows that
$(x^{2}(t_{2})+y^{2}(t_{2}))(d\eta_{1}+d\eta_{2})=0$ and, since
$x^{2}(t_{2})+y^{2}(t_{2})>0$, we get $d\eta_{1}+d\eta_{2}=0$, proving our
claim.

We now consider $t\in]t_{1},t_{2}[$. It is easy to see that $\xi_{2}=0$ and
$d\eta_{2}=0$. We also deduce that

  1. 1.

$0=\frac{d}{dt}(x^{2}+y^{2}+(z+h)^{2})=2\sigma
xy+2uy-4\xi_{1}y^{2}-4\xi_{1}x^{2}-4\xi_{1}(z+h)^{2}$ which implies that
$\xi_{1}=\frac{\sigma xy+uy}{2}$;

  2. 2.

also $0=d(xp+yq+(z+h)r)=uqdt+2d\eta_{1}$ implies that
$\frac{d\eta_{1}}{dt}=-\frac{uq}{2}$;

  3. 3.

from the above we deduce that

$\displaystyle\dot{p}=(\sigma xy+uy)p-xuq,$ $\displaystyle\dot{q}=-\sigma
p+\sigma xyq.$

Thus $(p,q)$ is a solution to a linear system and never is equal to zero.
Second equation implies that if $q=0$ then $\dot{q}\neq 0$. Hence $q>0$.

Now we need to consider $t=t_{1}$. We claim that

$(p(t_{1}-0),q(t_{1}-0),r(t_{1}-0))\neq(0,0,0).$

Let us then assume that it is $(p(t_{1}-0),q(t_{1}-0),r(t_{1}-0))=(0,0,0)$. It
then follows that
$(p(t_{1}+0),q(t_{1}+0),r(t_{1}+0))=(0,0,0)+(2x(t_{1})d\eta_{1},2y(t_{1})d\eta_{1},2(z(t_{1})+h)d\eta_{1})$.
We now show that there is no such jump. Set $r(t_{1}-0)=r_{0}$. Then it
follows from (iv) that $(x(t_{1})\cdot 0+y(t_{1})\cdot 0+(z(t_{1})+h))r_{0}=0$
which implies that $r_{0}=0$. We also have
$(x^{2}(t_{1})+y^{2}(t_{1})+(z(t_{1})+h)^{2})d\eta_{1}=0$ from (v). But this
implies that $d\eta_{1}=0$. Consequently, the multipliers do not exhibit a
jump at $t_{1}$.

From the previous analysis we deduce that $q$ should be positive almost
everywhere on the boundary. It then follows that to find the optimal solution
we have to analyze admissible trajectories with the controls with the
structure (5.1) and choose the optimal value of $\tilde{t}$.

Acknowledgements

The authors gratefully thank the support of Portuguese Foundation for Science
and Technology (FCT) in the framework of the Strategic Funding
UIDB/04650/2020.

Also we thank the support by the ERDF - European Regional Development Fund
through the Operational Programme for Competitiveness and Internationalisation
- COMPETE 2020, INCO.2030, under the Portugal 2020 Partnership Agreement and
by National Funds, Norte 2020, through CCDRN and FCT, within projects To Chair
(POCI-01-0145-FEDER-028247), Upwind (PTDC/EEI-AUT/31447/2017 -
POCI-01-0145-FEDER-031447) and Systec R&D unit (UIDB/00147/2020).

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