∎

11institutetext: Giovanni Fusco22institutetext: Department of Mathematics
”Tullio Levi-Civita”, University of Padua
Via Trieste, 63, Padova 35121, Italy
Tel.: (+39) 049 827 1261
22email<EMAIL_ADDRESS>33institutetext: Monica Motta 44institutetext:
Department of Mathematics ”Tullio Levi-Civita”, University of Padua
Via Trieste, 63, Padova 35121, Italy
Tel.: (+39) 049 827 1368
Fax: (+39) 049 827 1499
44email<EMAIL_ADDRESS>
# No infimum gap and normality in optimal impulsive control under state
constraints ††thanks: This research is partially supported by the INdAM-GNAMPA
Project 2020 “Extended control problems: gap, higher order conditions and
Lyapunov functions” and by the Padua University grant SID 2018
“Controllability, stabilizability and infimum gaps for control systems”, prot.
BIRD 187147.

Giovanni Fusco  Monica Motta

(Received: date / Accepted: date)

###### Abstract

In this paper we consider an impulsive extension of an optimal control problem
with unbounded controls, subject to endpoint and state constraints. We show
that the existence of an extended-sense minimizer that is a normal extremal
for a constrained Maximum Principle ensures that there is no gap between the
infima of the original problem and of its extension. Furthermore, we translate
such relation into verifiable sufficient conditions for normality in the form
of constraint and endpoint qualifications. Links between existence of an
infimum gap and normality in impulsive control have previously been explored
for problems without state constraints. This paper establishes such links in
the presence of state constraints and of an additional ordinary control, for
locally Lipschitz continuous data.

###### Keywords:

Impulsive optimal control problems Maximum Principle State constraints Gap
phenomena Normality Degeneracy

###### MSC:

49N25 34K45 49K15

## Introduction

In Optimal Control Theory it is quite common practice to extend the domain of
a minimum problem to ensure the existence of the minimum or to identify
optimality conditions. In doing this, it is of course desirable to avoid the
so-called infimum gap phenomenon, i.e. that the minimum of the extended
problem is different from the minimum of the original problem. This is
relevant not only for theoretical reasons of ‘well-posedness’ of the
extension, but also for the actual usefulness of the extended problem in order
to identify, for instance, necessary optimality conditions or a non degenerate
Hamilton-Jacobi-Bellman equation for the original minimum problem. For the
classical extension of a minimum problem by convex relaxation –where the
original velocity set of the trajectories is replaced by its convexification–
it has emerged that a sufficient condition to avoid the infimum gap is the
normality of an extended sense minimizer, namely, that all sets of multipliers
verifying a Maximum Principle have cost multiplier, $\lambda$ in the
following, different from zero warga ; warga1 ; warga2 ; PV1 ; PV2 ; V19 . In
MRV  the ‘normality test’ has proved sufficient to guarantee the absence of an
infimum gap also for the impulsive extension of an optimal control problem
with unbounded controls. Very recently, in PR19 ; PR20  this link between
normality and no-infimum-gap has been established for the extension of an
abstract minimum control problem, which includes both relaxation and impulsive
extension. All the above results in the case of the impulsive extension
concern problems without state constraints, with $C^{1}$ data, and no ordinary
controls in the dynamics. However , state constraints, together with
nonsmoothness of the data and additional ordinary controls, arise very
frequently in the applications of impulsive optimal control (see e.g. MiRu ;
ST00 ; HW11 ; AKP15 ; KDPS15  and the references therein).

This paper provides ‘normality type’ sufficient conditions to avoid a gap
between the infima of the following optimization problem (P) and the extended
optimization problem ${\rm(P_{e})}$ below:

$\qquad\qquad\text{minimize}\,\,\,\Psi(t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2}))$
(P)

over $t_{1},\,t_{2}\in\mathbb{R}$, $t_{1}<t_{2}$, $(x,v,u)\in
W^{1,1}([t_{1},t_{2}];\mathbb{R}^{n+1+m})$, $a\in L^{1}([t_{1},t_{2}];A)$
satisfying

$\begin{cases}\dfrac{dx}{dt}(t)=f(t,x(t),a(t))+\sum_{j=1}^{m}g_{j}(t,x(t))\,\dfrac{du^{j}}{dt}(t)\qquad\text{
a.e. }t\in[t_{1},t_{2}],\\\\[6.45831pt]
\dfrac{dv}{dt}(t)=\left|\dfrac{du}{dt}(t)\right|\qquad\text{ a.e.
}t\in[t_{1},t_{2}],\\\\[6.45831pt] \dfrac{du}{dt}(t)\in\mathcal{C}\qquad\text{
a.e. }t\in[t_{1},t_{2}],\\\\[6.45831pt] h_{1}(t,x(t))\leq
0,\dots,h_{N}(t,x(t))\leq 0\qquad\text{ for all }t\in[t_{1},t_{2}],\\\
v(t_{1})=0,\quad v(t_{2})\leq
K,\quad\left(t_{1},x(t_{1}),t_{2},x(t_{2})\right)\in\mathcal{T}_{0},\end{cases}$

where $K>0$ is a fixed constant (possibly equal to $+\infty$),
$A\subset\mathbb{R}^{q}$ is a compact subset,
$\mathcal{C}\subseteq\mathbb{R}^{m}$ is a closed convex cone,
$\mathcal{T}_{0}\subseteq\mathbb{R}^{1+n+1+n}$ is a closed subset, and the
data are locally Lipschitz continuous in $t$, $x$ (the precise assumptions
will be given in Section 1). Problem (P) is a free end-time minimization
problem depending on an ordinary control $a$ and on a control $u$ whose
derivatives appear linearly in the dynamics. Furthermore, there are time-
dependent state constraints in the form of $N$ inequalities, endpoint
constraints, and we may have a bound $K$ on the total variation of $u$ –notice
that $v$ is nothing but the total variation function of $u$. Due to a lack of
coerciveness, minimizers for problem (P) do not exist in general. Hence,
adopting a by now standard extension, we embed the original problem into the
space-time problem ${\rm(P_{e})}$ below, where the extended state variable is
$(y^{0},y,\nu):=(t,x,v)$, and the extended trajectories are $(t,x,v)$-paths
which are (reparameterized) $C^{0}$-limits of graphs of the original
trajectories Ris:65 ; War:65 ; BR:88 ; Mi:94 ; MR:95 : 111As it is well-known,
a distributional approach, where $\frac{du}{dt}$ is interpreted as a Radon
measure, does not work unless $g_{i}=g_{i}(x)$ and the Lie brackets
$[g_{i},g_{j}](x)\equiv 0$ for every $i,j=1,\dots,m$ (see e.g. Haj85 ; BR:88
).

$\qquad\qquad\text{minimize }\,\,\,\Psi(y^{0}(0),y(0),y^{0}(S),y(S),\nu(S))$
(Pe)

over $S>0$, $({y^{0}},y,\nu)\in W^{1,1}([0,S];\mathbb{R}^{1+n+1})$,
$(\omega^{0},\omega,\alpha)\in L^{1}([0,S];{\mathbb{C}}\times A)$ satisfying

$\begin{cases}\dfrac{dy^{0}}{ds}(s)=\omega^{0}(s)\qquad\text{a.e.
}s\in[0,S],\\\
\dfrac{dy}{ds}(s)=f(y^{0},y,\alpha)(s)\omega^{0}(s)+\sum_{j=1}^{m}g_{j}(y^{0},y)(s)\omega^{j}(s)\quad\text{
a.e. }s\in[0,S],\\\ \dfrac{d\nu}{ds}(s)=|\omega(s)|\qquad\text{ a.e.
}s\in[0,S],\\\\[6.45831pt] h_{1}(y^{0}(s),y(s))\leq
0,\dots,h_{N}(y^{0}(s),y(s))\leq 0\qquad\text{ for all }s\in[0,S],\\\
\displaystyle\nu(0)=0,\quad\nu(S)\leq
K,\quad\left(y^{0}(0),y(0),y^{0}(S),y(S)\right)\in\mathcal{T}_{0},\end{cases}$

where ${\mathbb{C}}:=\left\\{(\omega^{0},\omega):\ \
(\omega^{0},\omega)\in[0,+\infty[\times\mathcal{C},\ \
\omega^{0}+|\omega|=1\right\\}.$

To any process $(t_{1},t_{2},u,a,x,v)$ of problem (P), by setting
$\sigma(t):=t-t_{1}+v(t)$, $t\in[t_{1},t_{2}]$, through the time-change
$y^{0}:=\sigma^{-1}$ we can associate a process
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ of the extended problem with
$\omega^{0}=dy^{0}(s)/ds>0$ a.e.. In particular, problem (P) can be identified
with the restriction of problem ${\rm(P_{e})}$ to the set of processes with
$\omega^{0}>0$ a.e. (see Section 1). Let us refer to such processes as
embedded strict sense processes in the following. The extension consists
therefore in considering extended sense processes with $\omega^{0}=0$ on non-
degenerate intervals, where the time $t=y^{0}(s)$ is constant but the extended
state $y(s)$ evolves according to the ‘fast’ dynamics
$dy(s)/ds=\sum_{j=1}^{m}g_{j}(y^{0}(s),y(s))\omega^{j}(s)$. This explains why
(Pe) is also called the impulsive problem, although it is a conventional
optimization problem with bounded controls. In fact, one could give an
equivalent $t$-based description of this extension using bounded variation
trajectories and controls MiRu ; Sa91 ; WZ07 ; KDPS15 ; AKP15 ; AR15 ; MS18 ;
MS20 .

The main result of the paper, obtained in Theorem 3.1 below, establishes that
the existence of an extended sense minimizer for problem ${\rm(P_{e})}$ which
is a normal extremal for a constrained version of the Maximum Principle, is a
sufficient condition for the infimum gap avoidance. The occurrence of a gap is
strictly related to the presence of endpoints and state constraints. In
particular, since the set of trajectories corresponding to embedded strict
sense processes is $C^{0}$-dense in the set of trajectories of the extended
system, the infimum gap phenomenon can show up only when some extended sense
process verifying the constraints is isolated, namely cannot be approximated
by trajectories of the original system that satisfy the constraints. From this
observation, Theorem 3.1 will be derived from a general result on the
properties of isolated processes (see Theorem 2.2). The proof makes use of
perturbation and penalization techniques and of the Ekeland’s variational
principle, in the same spirit of PV1 ; MRV . This approach is very different
from that of warga ; warga1 ; warga2 ; PR20 , which is based on the
construction of approximating cones to reachable sets and on set separation
arguments. We recall that normality is not necessary to exclude the gap
phenomenon: for example, it is known that without the drift $f$ in the
dynamics, gap never occurs (MRV, , Lemma 4.1) (see also Mar00 ).

The normality criterium for the absence of an infimum gap has some
disadvantages. First of all, it requires to know a priori a minimizing
extended sense process, information that is not always available. Then, it is
necessary to verify that all sets of multipliers associated to the minimizer
that meet the conditions of the Maximum Principle have $\lambda>0$. In
addition, in the presence of state constraints the normality condition may
never be met, making the criterium in fact useless. In particular, it is well
known that when the state constraint is active at the initial point of a
minimizing process, sets of degenerate multipliers with $\lambda=0$ may always
exist. Rather surprisingly, it seems that no attention has been paid to this
‘degeneracy question’ in previous articles on the relationship between gap and
normality in the presence of state constraints.

Based on the above considerations, in the second part of the paper we first
introduce a nondegenerate version of the Maximum Principle and provide simple
geometrical conditions on endpoint and state constraints, under which abnormal
–namely, not normal– extremals for the original Maximum Principle turn out to
be abnormal extremals also for the nondegenerate Maximum Principle. In this
case, Theorem 3.1 can be rephrased as follows: ‘normality among nondegenerate
multipliers implies no infimum gap’. This ‘nondegenerate normality test’ is
useful especially because in certain special cases it allows to deduce the
absence of the infimum gap from easily verifiable conditions, some examples of
which we will provide. In particular, these are constraint and endpoint
qualification conditions.

Although this article is mainly focused on the infimum gap phenomenon, it also
establishes some new sufficient conditions for normality which extend previous
conditions in MS20 . In the literature on conventional, non-impulsive problems
with state constraints, a variety of constraint qualifications to avoid
degeneracy as well as to ensure normality are known (see e.g. FeV94 ; FeFoV99
; FoFr15 ; FrTo13 ; LFodaP11 ; PV2 ; RV99 ; AA97 ; A00 ; AK16 ; AKP17  and the
references therein). In impulsive control, instead, some nondegenerate Maximum
Principles have been obtained in AKP05 ; AKP15 ; K06 ; MS20 , while a Maximum
Principle in normal form has only recently been introduced in MS20 .

The paper is organized as follows: in Section 1 we introduce precisely
problems (P), ${\rm(P_{e})}$ and a constrained version of the Maximum
Principle for the extended problem. Section 2 is devoted to we prove that an
isolated extended sense extremal cannot be normal and, as a corollary, we
deduce that presence of an infimum gap implies abnormality of any extended
sense minimizer. In Section 3 we provide sufficient conditions for normality,
which guarantee a priori, without any knowledge of the multipliers associated
with the given extended sense minimizer, the non occurrence of gap-phenomena.
In Section 4, we present some examples to illustrate the results.

###  0.1 Notations and preliminaries

Given an interval $I\subseteq\mathbb{R}$ and a set $X\subseteq\mathbb{R}^{k}$,
we write $W^{1,1}(I;X)$, $C^{0,1}(I;X)$, $C^{0,1}_{loc}(I;X)$ for the space of
absolutely continuous functions, Lipschitz continuous functions, locally
Lipschitz continuous functions defined on $I$ and with values in $X$,
respectively. For all the classes of functions introduced so far, we will not
specify domain and codomain when the meaning is clear. Furthermore, we denote
by $\ell(X)$, $co(X)$, Int$(X)$, $\partial X$ the Lebesgue measure, the convex
hull, the interior and the boundary of $X$, respectively. As customary,
$\chi_{{}_{X}}$ is the characteristic function of $X$, namely
$\chi_{{}_{X}}(x)=1$ if $x\in X$ and $\chi_{{}_{X}}(x)=0$ if
$x\in\mathbb{R}^{k}\setminus X$; $I\cdot X$ denotes the set $\\{r\,x\,|\,r\in
I,\,x\in X\\}$. Given two nonempty subsets $X_{1}$, $X_{2}$ of
$\mathbb{R}^{k}$, we denote by $X_{1}+X_{2}$ the set
$\\{x_{1}+x_{2}\,|\,x_{1}\in X_{1},\,x_{2}\in X_{2}\\}$. Let
$X\subseteq\mathbb{R}^{k_{1}+k_{2}}$ for some natural numbers $k_{1}$,
$k_{2}$, and write
$x=(x_{1},x_{2})\in\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}$ for any $x\in
X$. Then $\text{{\rm proj}}_{x_{i}}X$ will denote the projection of $X$ on
$\mathbb{R}^{k_{i}}$, for $i=1,2$. We denote the closed unit ball in
$\mathbb{R}^{k}$ by $\mathbb{B}_{k}$, omitting the dimension when it is clear
from the context. Given a closed set ${\mathcal{O}}\subseteq\mathbb{R}^{k}$
and a point $z\in\mathbb{R}^{k}$, we define the distance of $z$ from
${\mathcal{O}}$ as $d_{\mathcal{O}}(z):=\min_{y\in{\mathcal{O}}}|z-y|$. We set
$\mathbb{R}_{\geq 0}:=[0,+\infty[$. For any $a,b\in\mathbb{R}$, we write
$a\vee b:=\max\\{a,b\\}$.

For all $\tau_{1}$, $\tau_{2}$, $\bar{\tau}_{1}$,
$\bar{\tau}_{2}\in\mathbb{R}$, $\tau_{1}<\tau_{2}$,
$\bar{\tau}_{1}<\bar{\tau}_{2}$, and for any pair $(z_{1},z_{2})\in
C^{0}([\tau_{1},\tau_{2}],\mathbb{R}^{k})\times
C^{0}([\bar{\tau}_{1},\bar{\tau}_{2}],\mathbb{R}^{k})$, let us define the
distance

$d_{\infty}\big{(}(\tau_{1},\tau_{2},z_{1}),(\bar{\tau}_{1},\bar{\tau}_{2},z_{2})\big{)}:=|\tau_{1}-\bar{\tau}_{1}|+|\tau_{2}-\bar{\tau}_{2}|+\|\tilde{z}_{1}-\tilde{z}_{2}\|_{L^{\infty}(\mathbb{R})},$
(1)

where for any $z\in C^{0}([a,b],\mathbb{R}^{k})$, $\tilde{z}$ denotes its
continuous constant extension to $\mathbb{R}$ and $\|\cdot\|_{L^{\infty}(I)}$
is the ess-sup norm on $I\subseteq\mathbb{R}$ interval. When the domain is
clear, we will sometimes simply write $\|\cdot\|_{L^{\infty}}$.

We denote by $NBV^{+}([0,S];\mathbb{R})$ the space of increasing, real valued
functions $\mu$ on $[0,S]$ of bounded variation, vanishing at the point 0 and
right continuous on $]0,S[$. Each $\mu\in NBV^{+}([0,S];\mathbb{R})$ defines a
Borel measure on $[0,S]$, still denoted by $\mu$, its total variation function
is indicated by $\|\mu\|_{TV}$ or equivalently by $\mu([0,S])$, and its
support by spt$\\{\mu\\}$.

Some standard constructs from nonsmooth analysis are employed in this paper.
For background material we refer the reader for instance to OptV . A set
$K\subseteq\mathbb{R}^{k}$ is a cone if $\alpha k\in K$ for any $\alpha>0$,
whenever $k\in K$. Take a closed set $D\subseteq\mathbb{R}^{k}$ and a point
$\bar{x}\in D$, the proximal normal cone $N^{P}_{D}(\bar{x})$ of $D$ at
$\bar{x}$ is defined as

$N^{P}_{D}(\bar{x}):=\left\\{\eta\in\mathbb{R}^{k}\text{ : }\exists
M>0\,\,\text{ such that }\,\,\eta\cdot(x-\bar{x})\leq
M|x-\bar{x}|^{2}\,\,\forall x\in D\right\\}.$

The limiting normal cone $N_{D}(\bar{x})$ of $D$ at $\bar{x}$ is given by

$N_{D}(\bar{x}):=\left\\{\eta\in\mathbb{R}^{k}\text{ : }\exists
x_{i}\stackrel{{\scriptstyle D}}{{\to}}\bar{x},\,\eta_{i}\to\eta\,\,\text{
s.t. }\,\,\eta_{i}\in N^{P}_{D}(x_{i})\text{ for each
}i\in\mathbb{N}\right\\},$

in which the notation $x_{i}\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\bar{x}$ is used to indicate that all points in the
converging sequence $(x_{i})_{i\in\mathbb{N}}$ lay in $D$. In general,
$N^{P}_{D}(\bar{x})\subseteq N_{D}(\bar{x})$. Take a lower semicontinuous
function $G:\mathbb{R}^{k}\to\mathbb{R}$ and a point
$\bar{x}\in\mathbb{R}^{k}$, the limiting subdifferential of $G$ at $\bar{x}$
is

$\partial G(\bar{x}):=\left\\{\xi\in\mathbb{R}^{k}\text{ : }(\xi,-1)\in
N_{epi(G)}(\bar{x},G(\bar{x}))\right\\},$

where $epi(G)$ is the epigraph of $G$. If
$G:\mathbb{R}^{k}\times\mathbb{R}^{h}\to\mathbb{R}$ is a lower semicontinuous
function and $(\bar{x},\bar{y})\in\mathbb{R}^{k}\times\mathbb{R}^{h}$, we
write $\partial_{x}G(\bar{x},\bar{y})$, $\partial_{y}G(\bar{x},\bar{y})$ to
denote the partial limiting subdifferential of $G$ at $(\bar{x},\bar{y})$
w.r.t. $x$, $y$, respectively. Given $G\in
C^{0,1}_{loc}(\mathbb{R}^{k};\mathbb{R})$ and $\bar{x}\in\mathbb{R}^{k}$, the
reachable hybrid subdifferential of $G$ at $\bar{x}$ is

$\partial^{*>}G(\bar{x}):=\left\\{\xi\in\mathbb{R}^{k}\,:\,\exists\;x_{i}\rightarrow\bar{x}\mbox{
s. t. }\ G(x_{i})>0\ \text{$\forall i$ and}\ \ \nabla
G(x_{i})\to\xi\right\\},$

while the reachable gradient of $G$ at $\bar{x}$ is

$\partial^{*}G(\bar{x}):=\,\left\\{\xi\in\mathbb{R}^{k}\text{ :
}\exists(x_{i})_{i}\subset\text{diff}(G)\setminus\\{\bar{x}\\}\text{ s.t.
}x_{i}\to\bar{x}\text{ and }\nabla G(x_{i})\to\xi\right\\}$

where diff($G$) denotes the set of differentiability points of $G$ and
$\nabla$ is the usual gradient operator. We define the hybrid subdifferential
as $\partial^{>}G(\bar{x}):=$co$\,\partial^{*>}G(\bar{x})$. The set
$\partial^{*}G(\bar{x})$ is nonempty, closed, in general non convex, and its
convex hull coincides with the Clarke subdifferential
$\partial^{C}G(\bar{x})$, that is
$\partial^{C}G(\bar{x})=$co$\,\partial^{*}G(\bar{x})$. Finally, when $G$ is
locally Lipschitz continuous, $\partial^{C}G(\bar{x})=$co$\,\partial
G(\bar{x})$.

##  1 Optimal control problems and a Maximum Principle

In this section we introduce rigorously the constrained optimization problem
over $W^{1,1}$-controls $u$ and its embedding in an extended, or impulsive,
problem. Furthermore, we state a Maximum Principle for the extended problem.
For simplicity, we will establish all the results for a single state
constraint, explaining from time to time with remarks how to adapt these
results to the case with $N$ constraints. Throughout the paper we shall
consider the following hypotheses.

  * (H0) The control set $\mathcal{C}\subseteq\mathbb{R}^{m}$ is a convex cone, the set of ordinary controls $A\subset\mathbb{R}^{q}$ is compact, and the endpoint constraint set $\mathcal{T}_{0}\subseteq\mathbb{R}^{1+n+1+n}$ is closed.

  * (H1)The drift function $f\in C(\mathbb{R}^{1+n}\times A;\mathbb{R}^{n})$ and, for every $a\in A$, $f(\cdot,a)\in C_{loc}^{0,1}(\mathbb{R}^{1+n};\mathbb{R}^{n})$, uniformly w.r.t. $A$; $g_{j}\in C_{loc}^{0,1}(\mathbb{R}^{1+n};\mathbb{R}^{n})$ for any $j=1,\dots,m$; $h\in C_{loc}^{0,1}(\mathbb{R}^{1+n};\mathbb{R})$.

  * (H2) The cost function $\Psi\in C^{0,1}((\bar{t}_{1},\bar{x}_{1},\bar{t}_{2},\bar{x}_{2},\bar{v}_{2})+\delta\mathbb{B};\mathbb{R})$ for some $\delta>0$, where $(\bar{t}_{1},\bar{x}_{1},\bar{t}_{2},\bar{x}_{2},\bar{v}_{2})$ denotes the endpoints of the optimal trajectory that we will consider in all our results; moreover, for every $(t_{1},x_{1},t_{2},x_{2})$, the map $v_{2}\mapsto\Psi(t_{1},x_{1},t_{2},x_{2},v_{2})$ is monotone non-decreasing.

###  1.1 The original optimal control problem

We set $\mathcal{T}:=\mathcal{T}_{0}\times]-\infty,K]$ and define the set
${\mathcal{U}}$ of strict sense controls as

${\mathcal{U}}:=\left\\{\begin{array}[]{l}(t_{1},t_{2},u,a):t_{1},t_{2}\in\mathbb{R},\
t_{1}<t_{2},\ (du/dt,a)\in L^{1}([t_{1},t_{2}];\mathbb{R}^{m}\times
A),\\\\[6.45831pt] \quad\qquad\qquad\qquad\qquad\ du(t)/dt\in\mathcal{C}\
\text{ a.e. }\ t\in[t_{1},t_{2}]\end{array}\right\\}.$

######  Definition 1 (Strict sense processes)

Let $(t_{1},t_{2},u,a)\in{\mathcal{U}}$ be a strict sense control, we call
$(t_{1},t_{2},u,a,x,v)$ a strict sense process if the pair $(x,v)\in
W^{1,1}([t_{1},t_{2}];\mathbb{R}^{n+1})$ verifies

$\begin{cases}\displaystyle\dfrac{dx}{dt}(t)=f(t,x(t),a(t))+\sum_{j=1}^{m}g_{j}(t,x(t))\dfrac{du^{j}}{dt}(t)\\\\[6.45831pt]
\dfrac{dv}{dt}(t)=\left|\dfrac{du}{dt}(t)\right|\end{cases}\qquad\text{a.e.
}t\in[t_{1},t_{2}].$

Furthermore, we say that $(t_{1},t_{2},u,a,x,v)$ is feasible if $v(t_{1})=0$,
$h(t,x(t))\leq 0$ for each $t\in[t_{1},t_{2}]$ and
$\big{(}t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2})\big{)}\in\mathcal{T}$.

The original optimal control problem is defined as

$\left\\{\begin{array}[]{l}\displaystyle\text{minimize}\,\,\,\Psi(t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2}))\\\\[6.45831pt]
\text{over the set of feasible strict sense processes
$(t_{1},t_{2},u,a,x,v)$.}\end{array}\right.$ (P)

We consider the following concept of local minimizer.

###### Definition 2

We call a feasible strict sense process
$(\bar{t}_{1},\bar{t}_{2},\bar{u},\bar{a},\bar{x},\bar{v})$ a local strict
sense minimizer of $\eqref{P}$ if there exists $\delta>0$ such that

$\Psi(\bar{t}_{1},\bar{x}(\bar{t}_{1}),\bar{t}_{2},\bar{x}(\bar{t}_{2}),\bar{v}(\bar{t}_{2}))\,\leq\,\Psi(t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2}))$
(2)

for any feasible $(t_{1},t_{2},u,a,x,v)$ verifying
$d_{\infty}\Big{(}(\bar{t}_{1},\bar{t}_{2},\bar{x},\bar{v}),(t_{1},t_{2},x,v)\Big{)}<\delta,$
where $d_{\infty}$ is the distance defined in (1). If relation (2) is
satisfied for all feasible strict sense processes, we say that
$(\bar{t}_{1},\bar{t}_{2},\bar{u},\bar{a},\bar{x},\bar{v})$ is a (global)
strict sense minimizer.

###### Remark 1

Arguing similarly to MS20 , we could consider a more general cost of the form

$\Psi(t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2}))+\int_{t_{1}}^{t_{2}}\ell_{0}(t,x(t),a(t))+\ell_{1}(t,x(t),|\dot{u}(t)|)\,dt,$

where $\ell_{0}$, $\ell_{1}$ are nonnegative and the extended Lagrangian $L$,
defined by

$L(t,x,\omega^{0},r,a):=\ell_{0}(t,x,a)\omega^{0}+\lim_{\rho\to\omega^{0}}\ell_{1}\left(t,x,\rho^{-1}r\right)\rho,$

verifies $L\in C(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}_{\geq
0}\times\mathbb{R}_{\geq 0}\times A;\mathbb{R}_{\geq 0})$ and, for every $a\in
A$, $L(\cdot,a)\in
C_{loc}^{0,1}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}_{\geq
0}\times\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})$, uniformly w.r.t. $A$. The
results of this article can also be applied to the case where dynamics, cost,
and constraints depend on the variable $u$. In fact, it is sufficient to add
to the control system in $(P)$ the equations
$dx_{n+1}(t)/dt=du_{1}(t)/dt,\dots,dx_{n+m}(t)/dt=du_{m}(t)/dt$.

###### Remark 2

As is not difficult to see, given a closed, Hausdorff-Lipschitz continuous
multifunction $X:[t_{1},t_{2}]\rightsquigarrow\mathbb{R}^{n}$, the function
$h(t,x):=d_{X(t)}(x)$ belongs to $C^{0,1}(\mathbb{R}^{1+n};\mathbb{R})$.
Therefore, we could allow implicit time-dependent state constraints of the
form $x(t)\in X(t)$ for all $t\in[t_{1},t_{2}]$, since one clearly has that
$x(t)\in X(t)$ if and only if $h(t,x(t))\leq 0$ on $[t_{1},t_{2}]$.

###  1.2 The extended optimal control problem

We set

${\mathbb{C}}:=\left\\{(\omega^{0},\omega)\in\mathbb{R}_{\geq
0}\times\mathcal{C}\text{ : }\omega^{0}+|\omega|=1\right\\}$ (3)

and introduce the set of extended sense controls, defined as follows:

$\mathcal{W}:=\bigcup_{S>0}\\{S\\}\times L^{1}([0,S];{\mathbb{C}}\times A).$

######  Definition 3 (Extended sense processes)

For any extended sense control $(S,\omega^{0},\omega,\alpha)\in\mathcal{W},$
we refer to $(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ as an extended sense
process if $(y^{0},y,\nu)\in W^{1,1}([0,S];\mathbb{R}^{1+n+1}$ verifies

$\left\\{\begin{split}\frac{d{y^{0}}}{ds}(s)&=\omega^{0}(s)\\\
\frac{dy}{ds}(s)&=f(y^{0},y,\alpha)(s)\omega^{0}(s)+\sum_{i=1}^{m}g_{i}(y^{0},y)(s)\omega^{i}(s)\\\
\frac{d\nu}{ds}(s)&=|\omega(s)|\end{split}\right.\quad{\rm a.e.}\,s\in[0,S].$
(4)

We say that $(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ is feasible if
$\nu(0)=0$, $h(y^{0}(s),y(s))\leq 0$ for each $s\in[0,S]$ and
$\left(y^{0}(0),y(0),y^{0}(S),y(S),\nu(S)\right)\in\mathcal{T}$.

The set of strict sense processes, say $\Sigma$, can be embedded into the set
of extended sense processes, $\Sigma_{e}$, through the following map
${\mathcal{I}}:\Sigma\to\Sigma_{e}$, defined as

${\mathcal{I}}(t_{1},t_{2},u,a,x,v):=(S,\omega^{0},\omega,\alpha,y^{0},y,\nu),$
(5)

where, setting $\sigma(t):=t-t_{1}+v(t)$, $S:=\sigma(t_{2})$, and
$y^{0}:=\sigma^{-1}:[0,S]\to[t_{1},t_{2}]$, we associate to any strict sense
process $(t_{1},t_{2},u,a,x,v)$ the extended sense process

$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu):=\left(S,\frac{dy^{0}}{ds},\left(\frac{du}{dt}\circ
y_{0}\right)\cdot\frac{dy^{0}}{ds},a\circ y^{0},y^{0},x\circ y^{0},v\circ
y^{0}\right),\,$ (6)

where clearly $\omega^{0}>0$ a.e.. Conversely, if
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ is an extended sense process with
$\omega^{0}>0$ a.e., the absolutely continuous, increasing and surjective
inverse $\sigma:[t_{1},t_{2}]\to[0,S]$ of $y^{0}$, allows us to define the
strict sense process

$(t_{1},t_{2},u,a,x,v):=\left(t_{1},t_{2},\int_{\sigma(t_{1})}^{\sigma(t)}\omega(s)\,ds,\alpha\circ\sigma,y\circ\sigma,\nu\circ\sigma\right).$
(7)

Therefore, ${\mathcal{I}}$ is injective, 333Of course, up to translations of
$u$.
${\mathcal{I}}(\Sigma)=\\{(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)\in\Sigma_{e}:\
\omega^{0}>0\ \text{a.e.}\\}$, and the extension consists in considering also
extended sense processes with $w^{0}$ possibly zero on some non-degenerate
intervals. As anticipated in the Introduction, we will sometimes refer to the
processes in ${\mathcal{I}}(\Sigma)$ as embedded strict sense processes.

We define the extended problem as

$\left\\{\begin{array}[]{l}\text{minimize
}\,\,\,\Psi(y^{0}(0),y(0),y^{0}(S),y(S),\nu(S))\\\\[6.45831pt] \text{over
feasible estended-sense processes
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$.}\end{array}\right.$ (Pe)

###### Definition 4

A feasible extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is said to be a local minimizer for the extended problem (Pe) if there exists
$\delta>0$ such that

$\Psi(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(S),\bar{y}(S),\bar{\nu}(S)))\leq\Psi(y^{0}(0),y(0),y^{0}(S),y(S),\nu(S))$
(8)

for all feasible extended sense processes
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ that satisfy
$d_{\infty}\Big{(}(y^{0}(0),y^{0}(S),y,\nu),(\bar{y}^{0}(0),\bar{y}^{0}(\bar{S}),\bar{y},\bar{\nu})\Big{)}$
$<\delta,$ where $d_{\infty}$ is as in (1). If (8) is satisfied for all
feasible extended sense processes, we call
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
a (global) extended sense minimizer.

###### Remark 3

The notion of extended sense local minimizer is the natural extension of the
definition of strict sense local minimizer. Indeed, in view of (AMR19, , Prop.
2.7), $(\bar{T},\bar{u},\bar{a},\bar{x},\bar{v})$ is a strict sense local
minimizer for (P) if and only if
${\mathcal{I}}(\bar{T},\bar{u},\bar{a},\bar{x},\bar{v})$ is an extended sense
local minimizer for (Pe) among the feasible embedded strict sense processes.

###  1.3 A Maximum Principle for the extended problem

Consider the unmaximized Hamiltonian $H$, defined by

$\begin{array}[]{l}H(t,x,p_{0},p,\pi,\omega^{0},\omega,a):=p_{0}\omega^{0}+p\cdot\Big{(}f(t,x,a)\omega^{0}+\sum_{i=1}^{m}g_{i}(t,x)\omega^{i}\Big{)}+\pi|\omega|,\end{array}$

for all
$(t,x,p_{0},p,\pi,\omega^{0},\omega,a)\in\mathbb{R}^{1+n+1+n+1}\times{\mathbb{C}}\times
A$.

######  Theorem 1.1 (PMP)

Assume (H0)-(H2). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense local minimizer for (Pe). Then there exist a path
$(p_{0},p)\in W^{1,1}([0,\bar{S}];\mathbb{R}\times\mathbb{R}^{n})$,
$\lambda\geq 0$, $\pi\leq 0$, $\mu\in NBV^{+}([0,\bar{S}];\mathbb{R})$,
$(m_{0},m):[0,\bar{S}]\to\mathbb{R}^{1+n}$ Borel measurable and
$\mu$-integrable functions, verifying the following conditions:

  * (i)

(non-triviality)

$\|p_{0}\|_{L^{\infty}}+\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\lambda\neq 0;$ (9)

  * (ii)

(adjoint equation) for a.e. $s\in[0,\bar{S}]$,

$\left(-\frac{dp_{0}}{ds}(s),-\frac{dp}{ds}(s)\right)\in co\ \partial_{t,x}\
H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)},$

  * (iii)

(tranversality)

$\begin{split}&\left(p_{0}(0),p(0),-q_{0}(\bar{S}),-q(\bar{S}),-\pi\right)\in\lambda\partial\Psi\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}d}\qquad\qquad\qquad\qquad\qquad\qquad\qquad+N_{\mathcal{T}}\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right);\end{split}$
(10)

  * (iv)

(maximization and vanishing of the Hamiltonian) for a.e. $s\in[0,\bar{S}]$,

$\begin{array}[]{l}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}\\\
\displaystyle\qquad\qquad\qquad=\max_{(\omega^{0},\omega,a)\in{\mathbb{C}}\times
A}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\omega^{0},\omega,a\Big{)}=0;\end{array}$
(11)

  * (v)

$(m_{0},m)(s)\in\partial_{t,x}^{>}\,h\left(\bar{y}^{0}(s),\bar{y}(s)\right)$
$\mu$-a.e. $s\in[0,\bar{S}]$;

  * (vi)

$spt(\mu)\subseteq\\{s\in[0,\bar{S}]\text{ :
}h\left(\bar{y}^{0}(s),\bar{y}(s)\right)=0\\}$,

where

$(q_{0},q)(s):=\begin{cases}\displaystyle(p_{0},p)(s)+\int_{[0,s[}(m_{0},m)(\tau)\mu(d\tau)\,\qquad\qquad\qquad\,\,s\in[0,\bar{S}[,\\\
\displaystyle(p_{0},p)(\bar{S})+\int_{[0,\bar{S}]}(m_{0},m)(\tau)\mu(d\tau)\,\qquad\qquad\qquad
s=\bar{S}.\end{cases}$

Furthermore:

  * (vii)

if
$\lambda\partial_{v}\Psi\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)=0$
and $\bar{\nu}(\bar{S})<K$, then $\pi=0$;

  * (viii)

if $\bar{y}^{0}(0)<\bar{y}^{0}(\bar{S})$, (i) can be strengthened to
$\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\lambda\neq 0$.

Proof:  The extended problem (Pe) is a conventional optimization problem in
the state-space $(y^{0},y,\nu)\in\mathbb{R}^{1+n+1}$, with endpoint constraint
$\mathcal{T}$ and state constraint $h(y^{0},y)\leq 0$, to which standard ‘free
end-time’ versions of the constrained Maximum Principle are applicable. In
particular, the current result can be deduced from (OptV, , Theorem 9.3.1) by
means of usual reparameterization techniques (see e.g. (OptV, , Theorem
8.7.1)). Actually, by these arguments it follows the existence of a further
multiplier $r\in W^{1,1}([0,\bar{S}];\mathbb{R})$ such that

$\displaystyle\frac{dr}{ds}(s)=-H_{s}\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}\quad$
$\displaystyle\text{a.e. }s\in[0,\bar{S}],$ $\displaystyle
r(\bar{S})\in-\lambda\Psi_{s}\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right),$
$\displaystyle\max_{(\omega^{0},\omega,a)\in{\mathbb{C}}\times
A}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\omega^{0},\omega,a\Big{)}=r(s)$
$\displaystyle\text{a.e. }s\in[0,\bar{S}],$

where $H_{s}$, $\Psi_{s}$ denote the partial derivatives of $H$ and $\Psi$
w.r.t. $s$, respectively. However, since the vector fields $f$ and
$(g_{j})_{j=1,\dots,m}$, the cost function $\Psi$, and the constraints do not
depend explicitly on the pseudo-time $s$, this yields the constancy of the
Hamiltonian with constant equal to 0 in (iv). Finally, the strengthened non-
triviality condition (9), which does not involve the multiplier $\pi$
associated to $\nu$, and the refinements (vii), (viii), can be proved as in
(MRV, , Theorem 3.1). ∎

######  Remark 4 (Multiple state constraints)

As observed in (OptV, , Section 9), when in problem (Pe) the single state
constraint is replaced with a collection of constraints $h_{i}(t,x)\leq 0$,
$h_{i}\in C^{0,1}_{loc}(\mathbb{R}^{1+n};\mathbb{R})$ for $i=1,\dots,N$, from
Theorem 1.1 one can deduce the following corollary: given an extended sense
local minimizer
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$,
there exist $(p_{0},p)\in W^{1,1}$, $\lambda\geq 0$, $\pi\leq 0$, $\mu_{i}\in
NBV^{+}([0,\bar{S}];\mathbb{R})$ for $i=1,\dots,N$ and Borel measurable and
$\mu_{i}$-integrable functions $(m_{0_{i}},m_{i})$, such that
$(m_{0_{i}},m_{i})(s)\in\partial^{>}_{t,x}h_{i}\left(\bar{y}^{0}(s),\bar{y}(s)\right)\
\ \mu_{i}\text{-a.e. }s\in[0,\bar{S}]$,
$spt(\mu_{i})\subseteq\\{s\in[0,\bar{S}]\text{ :
}h_{i}\left(\bar{y}^{0}(s),\bar{y}(s)\right)=0\\},$ and conditions (i)–(iv),
(vii) and (viii) of Theorem 1.1 are met with $(q_{0},q)$ and $\mu$ verifying

$\begin{split}&(q_{0},q)(s):=\begin{cases}(p_{0},p)(s)+\int_{[0,s[}\sum_{i=1}^{N}(m_{0_{i}},m)(\tau)\mu_{i}(d\tau)\quad\,\,\,\,\text{if
}s<\bar{S}\\\
(p_{0},p)(\bar{S})+\int_{[0,\bar{S}]}\sum_{i=1}^{N}(m_{0_{i}},m)(\tau)\mu_{i}(d\tau)\,\,\quad\text{if
}s=\bar{S},\end{cases}\\\ &\mu(A)=\mu_{1}(A)+\dots+\mu_{N}(A)\qquad\text{for
all Borel subsets }A\subseteq[0,\bar{S}].\end{split}$ (12)

######  Definition 5 (Normal and abnormal extremal)

We say that a feasible extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an (extended sense) extremal if there exists a set of multipliers
$(p_{0},p,\pi,\lambda,\mu)$ and functions $m_{0}$ and $m$ which meet the
conditions of Theorem 1.1. We will call an extremal normal if all possible
choices of multipliers as above have $\lambda>0$, and abnormal when there
exists at least one set of such multipliers with $\lambda=0$.

##  2 Infimum gap and abnormality

Write $J(t_{1},t_{2},u,a,x,v):=\Psi(t_{1},x(t_{1}),t_{2},x(t_{2}),v(t_{2}))$
for the cost of a strict sense process $(t_{1},t_{2},u,a,x,v)$ in problem (P),
and
$J_{e}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu):=\Psi(y^{0}(0),y(0),y^{0}(S),y(S),\nu(S))$
for the cost of an extended sense process
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ in problem (Pe). We also write
$\Sigma^{f}\subseteq\Sigma$ and $\Sigma_{e}^{f}\subseteq\Sigma_{e}$ for the
subset of feasible strict sense processes and for the subset of feasible
extended sense processes, respectively.

######  Definition 6 (Infimum gap)

We shall say that there is infimum gap if

$\displaystyle\inf_{\Sigma_{e}^{f}}\,J_{e}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)<\inf_{\Sigma^{f}}\,J(t_{1},t_{2},u,a,x,v).$

Furthermore, if
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an extended sense local minimizer, we shall say that there is local infimum
gap at
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
if, for some $\delta>0$,

$\displaystyle
J_{e}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})<\,\inf_{B^{\delta}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})\cap\Sigma^{f}}\,J(t_{1},t_{2},u,a,x,v),$

where we have set

$\begin{array}[]{l}B^{\delta}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu}):=\Big{\\{}(t_{1},t_{2},u,a,x,v)\in\Sigma:\
(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)\\\ \quad:={\cal
I}(t_{1},t_{2},u,a,x,v)\hbox{ and
}d_{\infty}\Big{(}(y^{0}(0),y^{0}(S),y,\nu),(\bar{y}^{0}(0),\bar{y}^{0}(\bar{S}),\bar{y},\bar{\nu})\Big{)}<\delta\Big{\\}}.\end{array}$

To prove that, in the presence of a gap, extended sense local minimizers for
problem (Pe) are abnormal extremals, it is convenient to rephrase Definition 6
only in terms of extended sense processes. Precisely, using the above
notation, by the properties of the map ${\mathcal{I}}$ (see (6)) it follows
that

$\displaystyle\inf_{\Sigma^{f}}\,J(t_{1},t_{2},u,a,x,v)=\inf_{{\mathcal{I}}(\Sigma^{f})}\,J_{e}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu),$
$\begin{split}&\inf_{B^{\delta}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})\cap\Sigma^{f}}\,J(t_{1},t_{2},u,a,x,v)\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}kjhgfldhlkjhglkdjfh}=\inf_{B^{\delta}_{e}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})\cap{\mathcal{I}}(\Sigma^{f})}\,J_{e}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu),\end{split}$

where

$\begin{array}[]{l}B^{\delta}_{e}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu}):=\Big{\\{}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)\in\Sigma_{e}:\\\
\quad\qquad\qquad\qquad\qquad\qquad
d_{\infty}\Big{(}(y^{0}(0),y^{0}(S),y,\nu),(\bar{y}^{0}(0),\bar{y}^{0}(\bar{S}),\bar{y},\bar{\nu})\Big{)}<\delta\Big{\\}}.\end{array}$

Even if the set of embedded strict sense processes is dense into the set of
extended sense processes with respect to the distance $d_{\infty}$, the
infimum gap can actually occur, since all embedded strict sense processes
close to a given feasible extended sense process might violate either the
endpoint constraints or the state constraint. This leads us to the following
definition:

######  Definition 7 (Isolated feasible extended sense process)

A feasible extended sense process $(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$
is called isolated if, for some $\delta>0$, one has

$B^{\delta}_{e}(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)\cap{\mathcal{I}}(\Sigma^{f})=\emptyset.$

The following result relates isolated feasible extended sense processes and
infimum gap.

###### Proposition 1

Assume (H0)-(H1). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense minimizer [resp., local minimizer] for the extended
problem (Pe) and assume $\Psi\in
C^{0}((\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S}))+\delta\mathbb{B};\mathbb{R})$
for some $\delta>0$. If there is infimum gap [resp., local infimum gap at
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$],
then
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an isolated feasible extended sense process.

Proof:  Suppose by contradiction that
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is not isolated. Then we can take a sequence $\delta_{j}\downarrow 0$ and, for
each $j\in\mathbb{N}$, there exists
$\left(S_{j},\omega^{0}_{j},\omega_{j},\alpha_{j},y^{0}_{j},y_{j},\nu_{j}\right)\in
B^{\delta_{j}}_{e}(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})\cap{\mathcal{I}}(\Sigma^{f})$.
By the definition of $d_{\infty}$ and the continuity of the cost function
$\Psi$, this implies that no infimum gap may occur. ∎

In the following theorem we establish the main result of this section:

###### Theorem 2.1

Assume (H0)-(H1). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense minimizer [resp., local minimizer] for the extended
problem (Pe) and assume $\Psi\in
C^{0}((\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S}))+\delta\mathbb{B};\mathbb{R})$
for some $\delta>0$. If there is infimum gap [resp., local infimum gap at
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$],
then
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an abnormal extremal.

Thanks to Proposition 1, Theorem 2.1 is a straightforward consequence of the
following result, which extends (MRV, , Th. 4.4) to the case with state
constraints, an additional ordinary control in the drift, and nonsmooth data.

###### Theorem 2.2

Assume (H0)-(H1). If
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an isolated feasible extended sense process, then it is an abnormal
extremal.

Proof:  Since the proof involves only space-time trajectories $(y^{0},y)$
which are close to the reference space-time trajectory $(\bar{y}^{0},\bar{y})$
and the controls assume values in a compact set, using standard truncation and
mollification arguments we can assume that there exists some $L>0$ such that
the functions $f$, $g_{1},\dots,g_{m}$, and $h$ are $L$-Lipschitz continuous
and bounded by $L$. The proof is divided into several steps in which
successive sequences of optimization problems are introduced that have as
eligible controls only embedded strict controls, and costs that measure how
much a process violates the constraints. Using the Ekeland Principle,
minimizers are then built for these problems, which converge to the initial
isolated process. Furthermore, applying the PMP to these approximate problems
with reference to the above mentioned minimizers, we obtain in the limit a set
of multipliers with $\lambda=0$ for problem (Pe), with reference to the
isolated process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$.

Step 1. Define the function $\Phi:\mathbb{R}^{1+n+1+n+1}\to\mathbb{R}$, given
by

$\Phi\left(t_{1},x_{1},t_{2},x_{2},v_{2}\right):=d_{\mathcal{T}_{0}}(t_{1},x_{1},t_{2},x_{2})\vee[(v_{2}-K)\vee
0]$

and for any $(y^{0},y,\nu)\in W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n+1})$,
introduce the payoff

${\mathcal{J}}(y^{0},y,\nu):=\Phi\left(y^{0}(0),y(0),y^{0}(\bar{S}),y(\bar{S}),\nu(\bar{S})\right)\vee\max_{s\in[0,\bar{S}]}h(y^{0}(s),y(s)).$
(13)

Fixed a sequence $(\varepsilon_{i})_{i}$ such that $\varepsilon_{i}\downarrow
0$, for each $i\in\mathbb{N}$ we consider the fixed end-time optimal control
problem:

$\left(\hat{P}_{i}\right)\begin{cases}\qquad\qquad\qquad\qquad\qquad\text{minimize
}\,\,\,{\mathcal{J}}(y^{0},y,\nu)\\\ \text{over }({y^{0}},y,\nu)\in
W^{1,1}([0,\bar{S}]),\,\,\,(\zeta,\omega,\alpha)\in L^{1}([0,\bar{S}])\text{
satisfying}\\\
\displaystyle\frac{dy^{0}}{ds}(s)=(1+\zeta(s))(1-|\omega(s)|)\qquad\text{a.e.
}s\in[0,\bar{S}]\\\
\displaystyle\frac{dy}{ds}(s)=(1+\zeta)\Big{[}f(y^{0},y,\alpha)(1-|\omega|)+\sum_{j=1}^{m}g_{j}(y^{0},y)\omega^{j}\Big{]}(s)\,\,\text{
a.e. }s\in[0,\bar{S}]\\\
\displaystyle\frac{d\nu}{ds}(s)=(1+\zeta(s))|\omega(s)|\qquad\text{ a.e.
}s\in[0,\bar{S}]\\\ \nu(0)=0\\\
\omega(s)\in(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B}),\,\,\,\zeta(s)\in[-1/2,1/2],\,\,\,\alpha(s)\in
A\quad\text{ a.e. }s\in[0,\bar{S}].\end{cases}$

We will call an element $(\zeta,\omega,\alpha,{y^{0}},y,\nu)$ satisfying the
constraints in $(\hat{P}_{i})$ a feasible process for problem $(\hat{P}_{i})$.
For every $i\in\mathbb{N}$, let
$(\bar{S},\hat{\omega}_{i}^{0},\hat{\omega_{i}},\hat{\alpha}_{i},\hat{y}_{i}^{0},\hat{y_{i}},\hat{\nu_{i}})$
be the extended sense process in which
$(\hat{y}^{0}_{i},\hat{y}_{i},\hat{\nu}_{i})(0)=(\bar{y}^{0},\bar{y},\bar{\nu})(0)$
and

$\left(\hat{\omega}_{i}^{0},\hat{\omega_{i}},\hat{\alpha}_{i}\right)(s)\,:=\left\\{\begin{array}[]{l}\left(\varepsilon_{i},(1-\varepsilon_{i})\frac{\bar{\omega}(s)}{|\bar{\omega}(s)|},\bar{\alpha}(s)\right)\qquad\text{if
$\bar{\omega}^{0}(s)<\varepsilon_{i}$}\\\ \\\
\left(\bar{\omega}^{0}(s),{\bar{\omega}}(s),\bar{\alpha}(s)\right)\qquad\qquad\text{
if $\ {\bar{\omega}}^{0}(s)\geq\varepsilon_{i}$. }\end{array}\right.$

Notice that
$(\bar{S},\hat{\omega}_{i}^{0},\hat{\omega_{i}},\hat{\alpha}_{i},\hat{y}_{i}^{0},\hat{y_{i}},\hat{\nu_{i}})$
is an embedded strict sense process for the extended problem (Pe), since
${\hat{\omega}}_{i}^{0}(s)\geq\varepsilon_{i}>0$ for a.e. $s\in[0,\bar{S}]$.
Moreover,
$(0,{\hat{\omega}}_{i},\hat{\alpha}_{i},\hat{y}_{i}^{0},\hat{y}_{i},\hat{\nu}_{i})$
is a feasible process for $(\hat{P}_{i})$, since
${\hat{\omega}}_{i}(s)\in(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B})$ a.e.
$s\in[0,\bar{S}]$. Furthermore,

$\hat{\alpha}_{i}\equiv\bar{\alpha},\qquad\left\|\left({\hat{\omega}}_{i}^{0},{\hat{\omega}}_{i}\right)-\left({\bar{\omega}}^{0},{\bar{\omega}}^{0}\right)\right\|_{L^{\infty}([0,\bar{S}])}\to
0\qquad\text{as $i\to\infty$},$ (14)

therefore, by the continuity of the input-output map
$(\zeta,\omega,\alpha)\mapsto(y^{0},y,\nu)$, we have:

$\left\|\left(\hat{y}_{i}^{0},\hat{y}_{i},\hat{\nu}_{i}\right)-\left(\bar{y}^{0},\bar{y},\bar{\nu}\right)\right\|_{L^{\infty}([0,\bar{S}])}\to
0\qquad\text{as $i\to\infty$}.$ (15)

Since ${\mathcal{J}}$ is nonnegative and vanishes at
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$,
by the $L$-Lipschitz continuity of $h$ and the $1$-Lipschitz continuity of
$d_{\mathcal{T}_{0}}(\cdot)$, (15) implies that there exist a sequence
$\rho_{i}\downarrow 0$ such that, for every $i\in\mathbb{N}$,
$(0,{\hat{\omega}}_{i},\hat{\alpha}_{i},\hat{y}_{i}^{0},\hat{y}_{i},\hat{\nu}_{i})$
has cost not greater than $\rho_{i}^{2}$, namely is a $\rho_{i}^{2}$-minimizer
for the problem $(\hat{P}_{i})$.

Step 2. If we endow the set of feasible processes for problem $(\hat{P}_{i})$,
say $\hat{\Gamma}_{i}$, with the distance

$\begin{split}&\hat{d}\left((\zeta,\omega,\alpha,{y^{0}},y,\nu)\,,\,(\tilde{\zeta},\tilde{\omega},\tilde{\alpha},\tilde{y}^{0},\tilde{y},\tilde{\nu})\right):=|(y^{0},y)(0)-(\tilde{y}^{0},\tilde{y})(0)|\\\
&\qquad\qquad+\|\omega-\tilde{\omega}\|_{L^{1}([0,\bar{S}])}+\ell\\{s\in[0,\bar{S}]\,:\,(\zeta(s),\alpha(s))\neq(\tilde{\zeta}(s),\tilde{\alpha}(s))\\}\end{split}$

for every pair $(\zeta,\omega,\alpha,{y^{0}},y,\nu)$,
$(\tilde{\zeta},\tilde{\omega},\tilde{\alpha},\tilde{y}^{0},\tilde{y},\tilde{\nu})\in\hat{\Gamma}_{i}$,
$(\hat{P}_{i})$ can be seen as an optimization problem with continuous cost
over the complete metric space $(\hat{\Gamma}_{i},\hat{d})$. Hence, by
Ekeland’s Principle, if we introduce the function

$\ell_{i}(s,\zeta,a):=\chi_{\\{(\zeta,a)\neq(\zeta_{i}(s),\alpha_{i}(s))\\}}\qquad\forall(s,\zeta,a)\in[0,\bar{S}]\times\left[-\frac{1}{2},\frac{1}{2}\right]\times
A,$

for any $i\in\mathbb{N}$ there is a feasible process
$(\zeta_{i},\omega_{i},\alpha_{i},y_{i}^{0},y_{i},\nu_{i})$ for
$(\hat{P}_{i})$ which is a minimizer of

$\left(P_{i}\right)\begin{cases}\text{minimize}\,\,{\mathcal{J}}(y^{0},y,\nu)+\rho_{i}\Bigl{(}\left|(y^{0},y)(0)-(y_{i}^{0},y_{i})(0)\right|\\\
\qquad\qquad\qquad\qquad\qquad+\int_{0}^{\bar{S}}\left[|\omega-\omega_{i}|(s)+\ell_{i}(s,\zeta(s),\alpha(s))\right]\,ds\Bigr{)}\\\
\text{over }({y^{0}},y,\nu)\in
W^{1,1}([0,\bar{S}]),\,\,\,(\zeta,\omega,\alpha)\in L^{1}([0,\bar{S}])\text{
satisfying}\\\
\displaystyle\frac{dy^{0}}{ds}(s)=(1+\zeta(s))(1-|\omega(s)|)\qquad\text{a.e.
}s\in[0,\bar{S}]\\\
\displaystyle\frac{dy}{ds}(s)=(1+\zeta)\Big{[}f(y^{0},y,\alpha)(1-|\omega|)+\sum_{j=1}^{m}g_{j}(y^{0},y)\omega^{j}\Big{]}(s)\,\,\text{
a.e. }s\in[0,\bar{S}]\\\
\displaystyle\frac{d\nu}{ds}(s)=(1+\zeta(s))|\omega(s)|\qquad\text{ a.e.
}s\in[0,\bar{S}]\\\ \nu(0)=0\\\
\zeta(s)\in[-1/2,1/2],\,\,\,\omega(s)\in(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B}),\,\,\,\alpha(s)\in
A\,\,\,\text{ a.e. }s\in[0,\bar{S}]\end{cases}$

and verifies

$\displaystyle\left|\left(\hat{y}_{i}^{0},\hat{y}_{i}\right)(0)-\left(y_{i}^{0},y_{i}\right)(0)\right|+\int_{0}^{\bar{S}}\left[|\hat{\omega}_{i}-\omega_{i}|(s)+\ell_{i}(s,0,\bar{\alpha}(s))\right]\,ds\leq\rho_{i}\to
0.$ (16)

Thus, by (14) and (16) it follows that, as $i\to+\infty$,

$\left\|\left(y_{i}^{0},y_{i},\nu_{i}\right)-\left(\bar{y}^{0},\bar{y},\bar{\nu}\right)\right\|_{L^{\infty}([0,\bar{S}])}\to
0,\qquad\left\|\omega_{i}-\bar{\omega}\right\|_{L^{1}([0,\bar{S}])}\to 0$ (17)
$\ell\\{s\in[0,\bar{S}]\,:\,(\zeta_{i}(s),\alpha_{i}(s))\neq(0,\bar{\alpha}(s))\\}\to
0$ (18)

so that, eventually passing to a subsequence, $(\omega_{i})_{i}$ converges to
$\bar{\omega}$ almost everywhere.

Let us now show that, through suitable reparameterization techniques, the
sequence of minimizing processes
$(\zeta_{i},\omega_{i},\alpha_{i},y_{i}^{0},y_{i},\nu_{i})$ can be associated
to a sequence of embedded strict processes converging to the original isolated
process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$.

Precisely, for each $i\in\mathbb{N}$, let us consider the surjective, bi-
Lipschitz continuous, and strictly increasing function
$\sigma_{i}:[0,\bar{S}]\to[0,\tilde{S}_{i}]$, given by

$\displaystyle\sigma_{i}(s):=\int_{0}^{s}(1+\zeta_{i}(r))\text{
d}r,\qquad\tilde{S}_{i}:=\sigma_{i}(\bar{S}).$

Using as reparameterization the inverse function
$\sigma^{-1}_{i}:[0,\tilde{S}_{i}]\to[0,\bar{S}]$, we derive that the
corresponding process
$(\tilde{S}_{i},\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i},\tilde{y}^{0}_{i},\tilde{y}_{i},\tilde{\nu}_{i})$,
where

$(\tilde{y}^{0}_{i},\tilde{y}_{i},\tilde{\nu}_{i}):=(y^{0}_{i},y_{i},\nu_{i},)\circ\sigma_{i}^{-1},\quad(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i}):=(1-|\omega_{i}|,\omega_{i},\alpha_{i})\circ\sigma_{i}^{-1},$
(19)

is an embedded strict sense process for problem (Pe). In particular, we have

$\begin{split}&{\tilde{\omega}}_{i}^{0}(s)=1-\left|{\tilde{\omega}}_{i}(s)\right|,\qquad{\tilde{\omega}}_{i}(s)\in(1-\epsilon_{i})\,(\mathcal{C}\cap\mathbb{B}_{m})\
\mbox{ a.e. }s\in[0,\tilde{S}_{i}],\\\
&(\tilde{y}^{0}_{i}(0),\tilde{y}_{i}(0),\tilde{y}^{0}_{i}(\tilde{S}_{i}),\tilde{y}_{i}(\tilde{S}_{i}),\tilde{\nu}_{i}(\tilde{S}_{i}))\;=\;(y^{0}_{i}(0),y_{i}(0),y^{0}_{i}(\bar{S}),y_{i}(\bar{S}),\nu_{i}(\bar{S}))\,.\end{split}$
(20)

Hence, we deduce from (17) that, for $i$ sufficiently large,

$d_{\infty}\Big{(}(\tilde{y}^{0}_{i}(0),\tilde{y}^{0}_{i}(\tilde{S}_{i}),\tilde{y}_{i},\tilde{\nu}_{i}),(\bar{y}^{0}(0),\bar{y}^{0}(\bar{S}),\bar{y},\bar{\nu})\Big{)}<\delta,$
(21)

where $\delta>0$ is the constant appearing in Definition 7, with reference to
the isolated feasible extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$.
As a consequence, for all $i$ large enough,
$(\tilde{S}_{i},\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i},\tilde{y}^{0}_{i},\tilde{y}_{i},\tilde{\nu}_{i})$
cannot be a feasible embedded strict sense process, namely, it must violate
either the endpoint constraints or the state constraint. By (19), (20) this
implies that ${\mathcal{J}}(y_{i}^{0},y_{i},\nu_{i})>0$, namely, at least one
of the following three inequalities holds true:

$d_{\mathcal{T}_{0}}(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}))>0,\quad\nu_{i}(\bar{S})>K,\quad\max_{s\in[0,\bar{S}]}\,h(y_{i}^{0}(s),y_{i}(s))>0.$
(22)

In the following, as is clearly not restrictive, we will always assume that
the properties valid from a certain index onwards, apply to each index
$i\in\mathbb{N}$.

Step 3. For each $i\in\mathbb{N}$, define
$c_{i}:=\max_{s\in[0,\bar{S}]}\,h(y_{i}^{0}(s),y_{i}(s))$ and set

$\tilde{h}(t,x,c):=h(t,x)-c\qquad\forall(t,x,c)\in\mathbb{R}^{1+n+1}.$

The process $(\zeta_{i},\omega_{i},\alpha_{i},y_{i}^{0},y_{i},\nu_{i},c_{i})$
turns out to be a minimizer for

$\left(Q_{i}\right)\begin{cases}\text{minimize}\,\,\,\Phi(y^{0}(0),y(0),y^{0}(\bar{S}),y(\bar{S}),\nu(\bar{S}))\vee
c(\bar{S})\\\
\,\,\,\,\,+\rho_{i}\left(\left|(y^{0},y)(0)-(y_{i}^{0},y_{i})(0)\right|+\int_{0}^{\bar{S}}\left[|\omega-\omega_{i}|(s)+\ell_{i}(s,\zeta(s),\alpha(s))\right]\,ds\right)\\\
\text{over }({y^{0}},y,\nu,c)\in
W^{1,1}([0,\bar{S}]),\,\,\,(\zeta,\omega,\alpha)\in L^{1}([0,\bar{S}])\text{
satisfying}\\\
\displaystyle\frac{dy^{0}}{ds}(s)=(1+\zeta(s))(1-|\omega(s)|)\qquad\text{a.e.
}s\in[0,\bar{S}]\\\
\displaystyle\frac{dy}{ds}(s)=(1+\zeta)\Big{[}f(y^{0},y,\alpha)(1-|\omega|)+\sum_{j=1}^{m}g_{j}(y^{0},y)\omega^{j}\Big{]}(s)\,\,\text{
a.e. }s\in[0,\bar{S}]\\\
\displaystyle\frac{d\nu}{ds}(s)=(1+\zeta(s))|\omega(s)|\qquad\text{ a.e.
}s\in[0,\bar{S}]\\\\[6.45831pt] \displaystyle\frac{dc}{ds}(s)=0\qquad\text{
a.e. }s\in[0,\bar{S}]\\\ \nu(0)=0\\\ \tilde{h}(y^{0}(s),y(s),c(s))\leq
0\qquad\text{for all }s\in[0,\bar{S}]\\\
(\zeta,\omega,\alpha,c)(s)\in[-1/2,1/2]\times(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B})\times
A\times\mathbb{R}\quad\text{ a.e. }s\in[0,\bar{S}].\end{cases}$

Our aim is now to apply the Pontryagin Maximum Principle to problem $(Q_{i})$
with reference to the minimizer
$(\zeta_{i},\omega_{i},\alpha_{i},y_{i}^{0},y_{i},\nu_{i},c_{i})$.
Preliminarily, let us observe that, passing eventually to a subsequence, we
may assume that either $c_{i}>0$ for each $i\in\mathbb{N}$ or $c_{i}\leq 0$
for each $i\in\mathbb{N}$.

Assume first that $c_{i}>0$ for each $i\in\mathbb{N}$. Fix $i\in\mathbb{N}$
and set $\omega_{0_{i}}:=1-|\omega_{i}|$. In the Maximum Principle, several
generalized subdifferentials are involved which it is convenient to make as
explicit as possible. First of all, the condition
‘$h(y_{i}^{0}(s),y_{i}(s))-c_{i}>0$’ implies
‘$h(y_{i}^{0}(s),y_{i}(s))>c_{i}>0$’, so that
$\partial_{t,x,c}^{{}^{>}}\tilde{h}(y_{i}^{0}(s),y_{i}(s),c_{i})=\partial_{t,x}^{{}^{>}}h(y_{i}^{0}(s),y_{i}(s))\times\\{-1\\}.$
Furthermore, by the ‘max rule’ of subdifferential calculus (see (OptV, , Th.
5.5.2)), the properties of the subdifferential of the distance function (see
(OptV, , Lemma 4.8.3)), and (22), we have that
$(\gamma_{0_{i}}^{1},\gamma_{i}^{1},\gamma_{0_{i}}^{2},\gamma_{i}^{2},\gamma_{\nu_{i}}^{2},\gamma_{c_{i}}^{2})\in\partial\left(\Phi(y^{0}(0),y(0),y^{0}(\bar{S}),y(\bar{S}),\nu(\bar{S}))\vee
c(\bar{S})\right)$ implies that there are some $\sigma^{1}_{i}$,
$\sigma_{i}^{2}$, $\sigma_{i}^{3}\geq 0$ with
$\sum_{k=1}^{3}\sigma_{i}^{k}=1$, such that

$\begin{array}[]{l}\displaystyle(\gamma_{0_{i}}^{1},\gamma_{i}^{1},\gamma_{0_{i}}^{2},\gamma_{i}^{2})\in\sigma_{i}^{1}\,\left(\partial
d_{\mathcal{T}_{0}}(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}))\cap\partial\mathbb{B}_{1+n+1+n}\right),\\\\[6.45831pt]
\quad\displaystyle\gamma_{\nu_{i}}^{2}=\sigma_{i}^{2}\quad\text{(since
$\partial\left((\nu_{i}(\bar{S})-K)\vee 0\right)=1$ when
$\nu_{i}(\bar{S})>K$),}\qquad\gamma_{c_{i}}^{2}=\sigma_{i}^{3},\end{array}$

and $\sigma_{i}^{k}=0$ when the maximum in
$\Phi\left(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}),\nu_{i}(\bar{S})\right)\vee
c_{i}(\bar{S})$ is strictly greater than the $k$-th term in the maximization.
Thus, the Maximum Principle in (OptV, , Th. 9.3.1) yields the existence of
some multipliers $(p_{0_{i}},p_{i},\pi_{i},r_{i})\in
W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n+1+1})$ associated with
$(y_{i}^{0},y_{i},\nu_{i},c_{i})$, $\mu_{i}\in
NBV^{+}([0,\bar{S}];\mathbb{R})$, $\lambda_{i}\geq 0$, $\sigma^{1}_{i}$,
$\sigma_{i}^{2}$, $\sigma_{i}^{3}\geq 0$ with
$\sum_{k=1}^{3}\sigma_{i}^{k}=1$, and Borel-measurable, $\mu_{i}$-integrable
functions $(m_{0_{i}},m_{i}):[0,\bar{S}]\to\mathbb{R}^{1+n}$, such that:

  * (i)′

$\|p_{0_{i}}\|_{L^{\infty}}+\|p_{i}\|_{L^{\infty}}+\|\mu_{i}\|_{TV}+\lambda_{i}+\|r_{i}\|_{L^{\infty}}+\|\pi_{i}\|_{L^{\infty}}=1$;

  * (ii)′

$\displaystyle\left(-\frac{dp_{0_{i}}}{ds}(s),-\frac{dp_{i}}{ds}(s)\right)\in
co\text{
}\partial_{t,x}\Big{\\{}\Big{[}q_{i}(s)\cdot\big{(}f((y_{i}^{0},y_{i},\alpha_{i})(s))\omega_{i}^{0}(s)$
r
$+\sum_{j=1}^{m}g_{j}((y_{i}^{0},y_{i})(s))\omega_{i}^{j}(s)\big{)}\Big{]}(1+\zeta_{i}(s))\Big{\\}}$
and $d\pi_{i}(s)/ds=dr_{i}(s)/ds=0$ for a.e. $s\in[0,\bar{S}]$;

  * (iii)′

$\left(p_{0_{i}}(0),p_{i}(0),-q_{0_{i}}(\bar{S}),-q_{i}(\bar{S})\right)\in\lambda_{i}\rho_{i}\mathbb{B}_{1+n}\times\\{0_{1+n}\\}$
sdfkssdfsddjf$+\lambda_{i}\sigma_{i}^{1}\left(\partial
d_{\mathcal{T}_{0}}(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}))\cap\partial\mathbb{B}_{1+n+1+n}\right)$,
$-\pi_{i}=\lambda_{i}\,\sigma_{i}^{2}$, $r_{i}(0)=0$,
$-r_{i}(\bar{S})+\int_{[0,\bar{S}]}\mu_{i}(ds)=\lambda_{i}\sigma_{i}^{3}$;

  * (iv)′

$\int_{0}^{\bar{S}}\\{[q_{i}\cdot(f(y_{i}^{0},y_{i},\alpha_{i})\omega_{i}^{0}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega_{i}^{j})+q_{0_{i}}\omega_{i}^{0}+\pi_{i}|\omega_{i}|](1+\zeta_{i})\\}\text{d}s$
e$\geq\int_{0}^{\bar{S}}\\{[q_{i}\cdot(f(y_{i}^{0},y_{i},\alpha)\omega^{0}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega^{j})+q_{0_{i}}\omega^{0}+\pi_{i}|\omega|](1+\zeta)$
e
$-\lambda_{i}\rho_{i}\left[|\omega-\omega_{i}|+\ell_{i}(s,\zeta(s),\alpha(s))\right]\\}\text{
d}s$
for all measurable selectors $(\zeta,\omega,\alpha)$ of
$[-\frac{1}{2},\frac{1}{2}]\times(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B}_{m})\times
A$;

  * (v)′

$(m_{0_{i}},m_{i})(s)\in\partial_{t,x}^{{}^{>}}h\left(y_{i}^{0}(s),y_{i}(s)\right)$
$\mu_{i}$-a.e. $s\in[0,\bar{S}]$,

  * (vi)′

$spt(\mu_{i})\subseteq\\{s\text{ :
}h\left(y_{i}^{0}(s),y_{i}(s)\right)-c_{i}=0\\}$,

where

$(q_{0_{i}},q_{i})(s):=\begin{cases}\displaystyle(p_{0_{i}},p_{i})(s)+\int_{[0,s[}(m_{0_{i}},m_{i})(\tau)\mu_{i}(d\tau)\qquad\,\,s\in[0,\bar{S}[,\\\
\displaystyle(p_{0_{i}},p_{i})(\bar{S})+\int_{[0,\bar{S}]}(m_{0_{i}},m_{i})(\tau)\mu_{i}(d\tau)\qquad
s=\bar{S}.\end{cases}$

Observe that, for each $i\in\mathbb{N}$, by (ii)′ and (iii)′ we have

$r_{i}\equiv
0,\qquad\|\mu_{i}\|_{TV}=\int_{[0,\bar{S}]}\mu_{i}(ds)=\lambda_{i}\sigma_{i}^{3},\qquad|\pi_{i}|=\lambda_{i}\,\sigma_{i}^{2};$
(23)

furthermore, $\|(m_{0_{i}},m_{i})\|_{L^{\infty}}\leq L$ by (v)′ and the
$L$-Lipschitz continuity of $h$. Then by (iii)′ and (23), we get

$\begin{split}\lambda_{i}(1-\sigma_{i}^{3})-\lambda_{i}\rho_{i}&\leq\left|\left(p_{0_{i}}(0),p_{i}(0),-q_{0_{i}}(\bar{S}),-q_{i}(\bar{S})\right)\right|+|\pi_{i}|\\\
&\leq
2\|p_{0_{i}}\|_{L^{\infty}}+2\|p_{i}\|_{L^{\infty}}+2L\|\mu_{i}\|_{TV}+|\pi_{i}|.\end{split}$

By this estimate, (23), the non-triviality condition (i)′, and using the facts
that $\rho_{i}\leq\frac{1}{2}$ for $i$ sufficiently large and
$\lambda_{i}\in[0,1]$, for such $i$ we get

$\begin{array}[]{l}3\|p_{0_{i}}\|_{L^{\infty}}+3\|p_{i}\|_{L^{\infty}}+(2L+2)\|\mu_{i}\|_{TV}+3|\pi_{i}|\\\\[6.45831pt]
\qquad\qquad\geq\lambda_{i}(1-\sigma_{i}^{3})-\lambda_{i}\rho_{i}+1-\lambda_{i}+\lambda_{i}\sigma_{i}^{2}+\lambda_{i}\sigma_{i}^{3}=1+\lambda_{i}\sigma_{i}^{2}-\lambda_{i}\rho_{i}\geq\frac{1}{2}.\end{array}$

Hence, scaling the multipliers, we obtain

$\|p_{0_{i}}\|_{L^{\infty}}+\|p_{i}\|_{L^{\infty}}+\|\mu_{i}\|_{TV}+|\pi_{i}|=1,\qquad\lambda_{i}\leq\tilde{L}:=6\vee
4(1+L).$ (24)

Suppose now $c_{i}\leq 0$ for each $i\in\mathbb{N}$. In this case, by (22),
either $\nu_{i}(\bar{S})>K$ or
$d_{\mathcal{T}_{0}}(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}))>0$.
Thus, for $\varepsilon>0$ suitably small, the process
$(\zeta_{i},\omega_{i},\alpha_{i},y_{i}^{0},y_{i},\nu_{i},\hat{c}_{i})$ with
$\hat{c}_{i}:=c_{i}+\varepsilon$ is still a minimizer for problem $(Q_{i})$
and, in addition, it verifies $h(y_{i}^{0}(s),y_{i}(s))-\hat{c}_{i}<0$ for all
$s\in[0,\bar{S}]$ (namely, the state constraint is inactive on $[0,\bar{S}]$).
Hence, by applying the Maximum Principle for problem $(Q_{i})$ with reference
to this minimizer we deduce the existence of multipliers
$(p_{0_{i}},p_{i},\pi_{i},r_{i})\in
W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n+1+1})$, which satisfy conditions
(i)′–(vi)′ with $\mu_{i}=0$, $\sigma_{i}^{3}=0$, $\pi_{i}\leq 0$, and
$\lambda_{i}>0$. In this case, by considering again $i$ sufficiently large to
have $\rho_{i}\leq\frac{1}{2}$, from (iii)′ we get

$\lambda_{i}(1-\rho_{i})\leq\left|\left(p_{0_{i}}(0),p_{i}(0),-q_{0_{i}}(\bar{S}),-q_{i}(\bar{S})\right)\right|+|\pi_{i}|\leq
2\|p_{0_{i}}\|_{L^{\infty}}+2\|p_{i}\|_{L^{\infty}}+|\pi_{i}|,$

and, scaling the multipliers appropriately after summing (i)′, we finally
obtain

$\|p_{0_{i}}\|_{L^{\infty}}+\|p_{i}\|_{L^{\infty}}+|\pi_{i}|=1,\qquad\lambda_{i}\leq
6\quad(\leq\tilde{L}).$ (25)

Step 4. From the previous step, we arrive at the following properties (for
either the case where $c_{i}>0$ for each $i\in\mathbb{N}$ or the case where
$c_{i}\leq 0$ for each $i\in\mathbb{N}$): for any $i\in\mathbb{N}$, there
exist $(p_{0_{i}},p_{i})\in W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n})$,
$\pi_{i}\leq 0$, $\mu_{i}\in NBV^{+}([0,\bar{S}];\mathbb{R})$ and Borel-
measurable, $\mu_{i}$-integrable functions
$(m_{0_{i}},m_{i}):[0,\bar{S}]\to\mathbb{R}^{1+n}$, such that:

  * (i)

$\|p_{0_{i}}\|_{L^{\infty}}+\|p_{i}\|_{L^{\infty}}+\|\mu_{i}\|_{TV}+|\pi_{i}|=1$,

  * (ii)

$\displaystyle\left(-\frac{dp_{0_{i}}}{ds}(s),-\frac{dp_{i}}{ds}(s)\right)\in
co\text{
}\partial_{t,x}\Big{\\{}\Big{[}q_{i}(s)\cdot\big{(}f(y_{i}^{0}(s),y_{i}(s),\alpha_{i}(s))\omega_{i}^{0}(s)$
r
$+\sum_{j=1}^{m}g_{j}(y_{i}^{0}(s),y_{i}(s))\omega_{i}^{j}(s)\big{)}\Big{]}(1+\zeta_{i}(s))\Big{\\}}$
for a.e. $s\in[0,\bar{S}]$;

  * (iii)

$\left(p_{0_{i}}(0),p_{i}(0),-q_{0_{i}}(\bar{S}),-q_{i}(\bar{S}),-\pi_{i}\right)$
r
$\in[0,\tilde{L}]\cdot\partial\Phi(y_{i}^{0}(0),y_{i}(0),y_{i}^{0}(\bar{S}),y_{i}(\bar{S}),\nu_{i}(\bar{S}))+\tilde{L}\rho_{i}\mathbb{B}_{1+n}\times\\{0_{1+n}\\}\times\\{0\\}$;

  * (iv)

$\int_{0}^{\bar{S}}\\{[q_{i}\cdot(f(y_{i}^{0},y_{i},\alpha_{i})\omega_{i}^{0}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega_{i}^{j})+q_{0_{i}}\omega_{i}^{0}+\pi_{i}|\omega_{i}|](1+\zeta_{i})\\}\text{d}s$
r
$\geq\int_{0}^{\bar{S}}\\{[q_{i}\cdot(f(y_{i}^{0},y_{i},\alpha)\omega^{0}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega^{j})$
r
$+q_{0_{i}}\omega^{0}+\pi_{i}|\omega|](1+\zeta)-3\tilde{L}\rho_{i}\\}\text{d}s$,
for all measurable selectors $(\zeta,\omega,\alpha)$ of
$[-\frac{1}{2},\frac{1}{2}]\times(1-\varepsilon_{i})(\mathcal{C}\cap\mathbb{B}_{m})\times
A$;

  * (v)

$(m_{0_{i}},m_{i})(s)\in\partial_{t,x}^{>}h\left(y_{i}^{0}(s),y_{i}(s)\right)$
$\mu_{i}$-a.e. $s\in[0,\bar{S}]$,

  * (vi)

$spt(\mu_{i})\subseteq\\{s\text{ :
}h\left(y_{i}^{0}(s),y_{i}(s)\right)-c_{i}=0\\}$,

where

$(q_{0_{i}},q_{i})(s):=\begin{cases}\displaystyle(p_{0_{i}},p_{i})(s)+\int_{[0,s[}(m_{0_{i}},m_{i})(\tau)\mu_{i}(d\tau)\qquad\,\,s\in[0,\bar{S}[,\\\
\displaystyle(p_{0_{i}},p_{i})(\bar{S})+\int_{[0,\bar{S}]}(m_{0_{i}},m)(\tau)\mu_{i}(d\tau)\qquad\,\,s=\bar{S}.\end{cases}$

Here $\tilde{L}$ is the same constant as in (24). By Banach-Alaoglu’s Theorem,
there exist a subsequence of $(\mu_{i})_{i}$, $\mu\in
NBV^{+}([0,\bar{S}];\mathbb{R})$,
$(m_{0},m):[0,\bar{S}]\to\mathbb{R}\times\mathbb{R}^{n}$ Borel measurable and
$\mu$-integrable, such that $\mu_{i}\overset{*}{\rightharpoonup}\mu$ weakly*
in $C^{*}([0,\bar{S}])$ and
$m_{i}\mu_{i}(ds)\overset{*}{\rightharpoonup}m\mu(ds)$,
$m_{0_{i}}\mu_{i}(ds)\overset{*}{\rightharpoonup}m_{0}\mu(ds)$ (see (OptV, ,
Proposition 9.2.1)). Furthermore,
$\int_{[0,s[}(m_{0_{i}},m_{i})(\tau)\mu_{i}(d\tau)\to\int_{[0,s[}(m_{0},m)(\tau)\mu(d\tau)$
for a.e. $s\in[0,\bar{S}]$; the real sequence $(\pi_{i})$ is bounded; the
functions $(p_{0_{i}},p_{i})$ are uniformly bounded and have uniformly
integrable, bounded derivatives. Hence, there exist $\pi\leq 0$ and
$(p_{0},p)\in W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n})$ (see e.g. (OptV, , Th
2.5.3 and Ch. 9)) such that, eventually for a further subsequence,
$\pi_{i}\to\pi$, $(p_{0_{i}},p_{i})\to(p_{0},p)$ in $L^{\infty}$, and
$\left(\frac{dp_{0_{i}}}{ds},\frac{dp_{i}}{ds}\right)\rightharpoonup\left(\frac{dp_{0}}{ds},\frac{dp}{ds}\right)$
weakly in $L^{1}$, as $i\to+\infty$. By this analysis it also follows that the
functions $(q_{0_{i}},q_{i})$ are uniformly integrably bounded and verify for
a.e. $s\in[0,\bar{S}]$,

$\displaystyle\lim_{i\to+\infty}(q_{0_{i}},q_{i})(s)=(q_{0},q)(s):=(p_{0},p)(s)+\int_{[0,s[}(m_{0},m)(\tau)\mu(d\tau).$

Hence, by the dominated convergence theorem, one has

$(q_{0_{i}},q_{i})\to(q_{0},q)\qquad\text{in $L^{1}([0,\bar{S}])$.}$ (26)

Passing to the limit as $i\to+\infty$ and using (17), by (i),(v), and (vi) we
get

$\|p_{0}\|_{L^{\infty}}+\|p\|_{L^{\infty}}+\|\mu\|_{TV}+|\pi|=1,$ (27)
$(m_{0},m)(s)\in\partial_{t,x}^{>}h\left(\bar{y}^{0}(s),\bar{y}(s)\right)\qquad\text{$\mu$-a.e.
}s\in[0,\bar{S}],$ (28) $spt(\mu)\subseteq\\{s\text{ :
}h\left(\bar{y}^{0}(s),\bar{y}(s)\right)=0\\}.$ (29)

Furthermore, using that
$\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)\in\mathcal{T}$,
the properties of distance function, and the ‘max-rule’ for subdifferentials,
by (iii) we have

$\begin{array}[]{l}(p_{0}(0),p(0),-q_{0}(\bar{S}),-q(\bar{S}),-\pi)\in[0,\tilde{L}]\cdot\partial\Phi\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)\\\
\qquad\qquad\subseteq[0,\tilde{L}]\cdot\left[\left(N_{\mathcal{T}_{0}}\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)\cap\mathbb{B}_{1+n+1+n}\right)\times\\{0\\}\right.\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+\\{0_{1+n+1+n}\\}\times\partial\left((\nu_{i}(\bar{S})-K)\vee
0\right)\right]\\\ \qquad\qquad\qquad\subseteq
N_{\mathcal{T}_{0}}\left(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})\right)\times
N_{]-\infty,K]}(\bar{\nu}(\bar{S})),\end{array}$ (30)

Incidentally, from this relation we immediately deduce that $\pi=0$ if
$\bar{\nu}(\bar{S})<K$. Passing to the limit in (iv), with the help of a
measurable selection theorem, using (17), (18) and the dominated convergence
Theorem, we deduce that, for a.e. $s\in[0,\bar{S}]$,

$\begin{array}[]{l}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}\\\
\displaystyle\qquad\qquad=\max_{(\zeta,\omega^{0},\omega,a)\in[-\frac{1}{2},\frac{1}{2}]\times{\mathbb{C}}\times
A}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\omega^{0},\omega,a\Big{)}(1+\zeta).\end{array}$

Since 0 is in the interior of $[-\frac{1}{2},\frac{1}{2}]$, this implies that,
for a.e. $s\in[0,\bar{S}]$,

$\begin{array}[]{l}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}\\\
\displaystyle\quad\qquad\qquad\qquad\qquad=\max_{(\omega^{0},\omega,a)\in{\mathbb{C}}\times
A}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\omega^{0},\omega,a\Big{)}=0.\end{array}$
(31)

To prove that $(p_{0},p)$ verifies the adjoint equation in Theorem 1.1, for
each $i\in\mathbb{N}$ we set

$\begin{array}[]{l}\mathcal{A}_{i}:=\Big{\\{}s\in[0,\bar{S}]\text{ :
}\left(\zeta_{i}(s),\alpha_{i}(s)\right)=\left(0,\bar{\alpha}(s)\right)\Big{\\}}\subseteq[0,\bar{S}].\end{array}$

By (18) it follows that $\ell(\mathcal{A}_{i})\to\bar{S}$ as $i\to+\infty.$
Hence, by the $L$-boundedness of $f$ and $g_{1},\dots,g_{m}$, we deduce that
the functions $(y_{i}^{0},y_{i})$ are uniformly integrable, so that by (OptV,
, Th. 2.5.3) and (17),
$\left(\frac{dy^{0}_{i}}{ds},\frac{dy_{i}}{ds}\right)\rightharpoonup\left(\frac{dy^{0}}{ds},\frac{dy}{ds}\right)$
weakly in $L^{1}$, as $i\to+\infty$. Moreover, for a.e. $s\in\mathcal{A}_{i}$,
we have

$\begin{array}[]{l}\displaystyle\left(\frac{dy^{0}_{i}}{ds},\frac{dy_{i}}{ds}\right)(s)=\Big{(}\omega_{i}^{0},f(y_{i}^{0},y_{i},\bar{\alpha})\omega^{0}_{i}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega^{j}_{i}\Big{)}(s),\\\
\displaystyle\left(-\frac{dp_{0_{i}}}{ds},-\frac{dp_{i}}{ds}\right)(s)\in
co\,\partial_{t,x}\Bigl{\\{}q_{i}(s)\cdot\Big{(}f(y_{i}^{0},y_{i},\bar{\alpha})\omega^{0}_{i}+\sum_{j=1}^{m}g_{j}(y_{i}^{0},y_{i})\omega^{j}_{i}\Big{)}(s)\Bigr{\\}},\end{array}$

By (17), (26), and (OptV, , Theorem 2.5.3) we can conclude that, for a.e.
$s\in[0,\bar{S}]$,444Notice that, since the convex hull of a compact subset of
some space $\mathbb{R}^{k}$ is compact, our hypotheses on the vector fields
$f,g_{1},\dots,g_{m}$ guarantee that the multifunction in the differential
inclusion is closed.

$\left(-\frac{dp_{0}}{ds},-\frac{dp}{ds}\right)(s)\in
co\,\partial_{t,x}\Bigl{\\{}q(s)\cdot\Big{(}f(\bar{y}^{0},\bar{y},\bar{\alpha})\bar{\omega}^{0}+\sum_{j=1}^{m}g_{j}(\bar{y}^{0},\bar{y})\bar{\omega}^{j}\Big{)}(s)\Bigr{\\}}.$
(32)

In view of relations (28), (29), (30), (31), and (32), to conclude the proof
that the isolated feasible process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an abnormal extremal, it remains only to show that

$\|p_{0}\|_{L^{\infty}}+\|p\|_{L^{\infty}}+\|\mu\|_{TV}\neq 0.$ (33)

Suppose by contradiction that (33) is not true. Then $q_{0}\equiv 0$, $q\equiv
0$ a.e., and by (27) we deduce that $\pi\neq 0$, which in turn implies
$\bar{\nu}(\bar{S})=K>0$. Thanks to these information and integrating (31) in
$[0,\bar{S}]$ we find that
$0=\int_{0}^{\bar{S}}\pi\,\left|\bar{\omega}\right|\text{d}s=\pi\,\bar{\nu}(\bar{S})=\pi\,K$,
which is not possible. ∎

######  Remark 5 (Multiple state constraints)

In order to allow multiple state constraints $h_{i}$ for $i=1,\dots,N$, it is
sufficient replacing the payoff $\mathcal{J}$ in (13) by the function
$\Phi\left(y^{0}(0),y(0),y^{0}(\bar{S}),y(\bar{S}),\nu(\bar{S})\right)\vee\max_{s\in[0,\bar{S}]}h_{1}(y^{0}(s),y(s))\vee\dots\vee\max_{s\in[0,\bar{S}]}h_{N}(y^{0}(s),y(s)),$
and making obvious changes to the preceding proof.

##  3 Nondegeneracy, normality and no infimum gap

As a consequence of Theorem 2.1, ‘normality implies no infimum gap’.
Precisely, as a corollary of the results in Section 2, we have:

###### Theorem 3.1

Assume hypotheses (H0)-(H2) are satisfied.

  * (i)

Suppose that there exists an extended sense minimizer for (Pe) which is a
normal extremal. Then there is no infimum gap.

  * (ii)

Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense local minimizer for the extended problem (Pe) which is a
normal extremal. Then there is no local infimum gap at
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$.

As observed in the Introduction, the above ‘normality test’ is of more
theoretical than practical interest (specially in the presence of state
constraints). In this section we identify some verifiable conditions
guaranteeing that every set of multipliers is normal. To begin with, let us
introduce the notion of nondegenerate estremal.

######  Definition 8 (Nondegenerate Maximum Principle)

Given an extended sense local minimizer for problem (Pe),
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$,
we say that the Maximum Principle is nondegenerate when there is a choice of
the multipliers $(p_{0},p,\pi,\lambda,\mu)$ and of the functions $m_{0}$, $m$
that meets the conditions (i)–(vi) of Theorem 1.1, and such that

$\displaystyle\mu(]0,\bar{S}])+\|q_{0}\|_{L^{\infty}}+\|q\|_{L^{\infty}}+\lambda\neq
0\qquad$ $\displaystyle\text{if }\qquad\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0),$
(34) $\displaystyle\mu(]0,\bar{S}])+\|q\|_{L^{\infty}}+\lambda\neq 0\qquad$
$\displaystyle\text{if }\qquad\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0),$

where $q_{0}$, $q$ are defined as in Theorem 1.1.

######  Definition 9 (Nondegenerate normal and abnormal extremals)

We say that a feasible extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is an extremal of the nondegenerate Maximum Principle, in short, a
nondegenerate extremal, if there exists a set of multipliers
$(p_{0},p,\pi,\lambda,\mu)$ and functions $m_{0}$, $m$ which meet the
conditions of Theorem 1.1 and also satisfy (34). We call
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
a nondegenerate normal extremal if all possible choices of multipliers as
above have $\lambda>0$, and a nondegenerate abnormal extremal when there
exists at least one set of such multipliers with $\lambda=0$.

As it is easy to see, a nondegenerate abnormal extremal is always an abnormal
extremal, and, on the contrary, any normal extremal is also nondegenerate
normal. To obtain the converse implications, we introduce condition (CNa)
below. In the following, we will often use the notation

$\Omega:=\\{(t,x)\,:\,h(t,x)\leq 0\\}.$

Condition for nondegenerate abnormality (CNa). A feasible extended sense
process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is said to verify condition (CNa) if

$\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))\,\cap\,\left(-\text{{\rm
proj}}_{(t_{1},x_{1})}(N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})))\right)=\emptyset.$
(35)

###### Remark 6

To clarify the geometrical meaning of condition (CNa), let us notice that, if
$(\bar{y}^{0}(0),\bar{y}(0))\in\text{Int}(\Omega)$, condition (35) is
trivially satisfied, since the hybrid subdifferential
$\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))=\emptyset$. Incidentally, observe
that $\\{(t,x)\,:\,h(t,x)<0\\}\subseteq\text{Int}(\Omega)$ but the inclusion
is in general strict. When instead
$(\bar{y}^{0}(0),\bar{y}(0))\in\partial\Omega$, (35) implies that
$0\notin\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))$. If $h\in C^{2}$ in a
neighborhood of $(\bar{y}^{0}(0),\bar{y}(0))\in\partial\Omega$, (35) simply
reads ($\nabla h(\bar{y}^{0}(0),\bar{y}(0))\neq 0$ and)

$\nabla h(\bar{y}^{0}(0),\bar{y}(0))\notin-\text{{\rm
proj}}_{(t_{1},x_{1})}(N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))).$
(36)

Condition (36) is satisfied at $(\bar{y}^{0}(0),\bar{y}(0))$ with
$h(\bar{y}^{0}(0),\bar{y}(0))=0$ and $\nabla h(\bar{y}^{0}(0),\bar{y}(0))\neq
0$, when, for instance, $\mathcal{T}_{0}=\mathcal{T}_{1}\times\mathcal{T}_{2}$
with $\mathcal{T}_{1}$, $\mathcal{T}_{2}$ closed subsets of
$\mathbb{R}^{1+n}$, $\mathcal{T}_{1}\subseteq\Omega$, and
$N_{\mathcal{T}_{1}}(\bar{y}^{0}(0),\bar{y}(0))$ is pointed.555A cone
${\mathcal{K}}\subseteq\mathbb{R}^{k}$ is pointed if it contains no line, i.e.
if $\xi$, $-\xi\in{\mathcal{K}}$ implies that $\xi=0$. In this case, indeed,
(36) can be derived by the following relations

$\begin{split}\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))&=\\{\nabla
h(\bar{y}^{0}(0),\bar{y}(0))\\}\subseteq
N^{P}_{\Omega}(\bar{y}^{0}(0),\bar{y}(0))\\\ &\subseteq
N^{P}_{\mathcal{T}_{1}}(\bar{y}^{0}(0),\bar{y}(0))\subseteq
N_{\mathcal{T}_{1}}(\bar{y}^{0}(0),\bar{y}(0)).\end{split}$

###### Remark 7

Consider the quite customary situation where initial and final time are fixed
and the state constraint is time independent, namely
$\mathcal{T}_{0}=\\{\bar{t}_{1}\\}\times\mathcal{T}^{1}_{0}\times\mathcal{T}^{2}_{0}$
with $\mathcal{T}^{1}_{0}\subseteq\mathbb{R}^{n}$,
$\mathcal{T}^{2}_{0}\subseteq\mathbb{R}^{1+n}$ closed subsets, and
$h(t,x)=\bar{h}(x)$. In this case,
$\partial^{>}h(t,x)=\\{0\\}\times\partial^{>}\bar{h}(x)$ and
$N_{\\{\bar{t}_{1}\\}\times\mathcal{T}^{1}_{0}}(t,x)=\mathbb{R}\times
N_{\mathcal{T}^{1}_{0}}(x)$ for all $(t,x)\in\mathbb{R}^{1+n}$. Hence,
condition (35) reduces to

$\partial^{>}\bar{h}(\bar{y}(0))\cap\left(-N_{\mathcal{T}^{1}_{0}}(\bar{y}(0))\right)=\emptyset.$

###### Proposition 2

Assume (H0)-(H2) and suppose that
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is a feasible extended sense process which is an abnormal extremal, namely,
there exist a set of multipliers $(p_{0},p,\pi,\lambda,\mu)$, and some
functions $(m_{0},m)$ as in Theorem 1.1, with $\lambda=0$. If condition (CNa)
is satisfied, then $(p_{0},p,\pi,0,\mu)$, $(m_{0},m)$ verify the strengthened
non-triviality condition (34).

Proof:  Assume that (CNa) is verified and suppose by contradiction that
$\|q_{0}\|_{L^{\infty}}+\|q\|_{L^{\infty}}+\mu(]0,\bar{S}])=0.$ Then, in view
of Theorem 1.1, we have

$(p_{0},p)\equiv-\mu(\\{0\\})(\xi_{0},\xi),\quad\mu(\\{0\\})\neq
0,\quad(\xi_{0},\xi)\in\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0)).$

In particular, $\mu(\\{0\\})\neq 0$ implies that
$(\bar{y}^{0}(0),\bar{y}(0))\in\partial\Omega$. By the transversality
condition (iii) of Theorem 1.1, it follows that

$(\xi_{0},\xi)\in\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))\cap\left(-\text{{\rm
proj}}_{(t_{1},x_{1})}(N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})))\right),$
(37)

in contradiction with (35).

To conclude it remains to show that $\|q\|_{L^{\infty}}+\mu(]0,\bar{S}])\neq
0$ whenever $\bar{y}_{0}(\bar{S})>\bar{y}_{0}(0)$. If we suppose by
contradiction $\mu(]0,\bar{S}])+\|q\|_{\infty}=0$, then Theorem 1.1, (ii),
(iv) and the first part of the proof yield that $0\neq q_{0}$ is a constant
and

$q_{0}\bar{w}_{0}(s)+\pi|\bar{w}(s)|=\max_{w_{0}\in[0,1]}\\{\pi+(q_{0}-\pi)w_{0}\\}=0\quad\text{a.e.
$s\in]0,\bar{S}[$.}$ (38)

Then $\pi<0$ leads easily to a contradiction. If $\pi=0$, since $q_{0}\neq 0$,
(38) yields $\bar{w}_{0}=0$ a.e., in contradiction with
$\int_{0}^{\bar{S}}\bar{w}_{0}(s)\,ds=\bar{y}^{0}(\bar{S})-\bar{y}^{0}(0)>0$.
∎

As a straightforward consequence of Proposition 2, we have:

###### Proposition 3

Assume (H0)-(H2). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be a feasible extended sense process verifying condition (CNa). Then
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is a nondegenerate normal extremal if and only if it is a normal extremal.

######  Remark 8 (Multiple state constraints)

The result of Proposition 2 can be easily extended to the case of multiple
state constraints $h_{i}$ for $i=1,\dots,N$, by considering the nondegeneracy
condition (34) with $(q_{0},q)$ and $\mu$ as in Remark 4 and by replacing the
nondegenerate abnormality condition (35) with

$\begin{split}&\left(\sum_{i=1}^{N}[0,+\infty[\cdot\partial^{>}h_{i}(\bar{y}^{0}(0),\bar{y}(0))\right)\,\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}lskjdflsdkj}\cap\left(-\text{{\rm
proj}}_{(t_{1},x_{1})}(N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})))\right)=\emptyset.\end{split}$
(39)

We now provide some sufficient conditions in the form of constraint and
endpoint qualifications to guarantee normality. In the following, we will use
the notation

$\displaystyle
F(t,x,\omega_{0},\omega,a):=f(t,x,a)\omega^{0}+\sum_{i=1}^{m}g_{i}(t,x)\omega^{i}\quad\forall(t,x,\omega^{0},\omega,a)\in\mathbb{R}^{1+n}\times{\mathbb{C}}\times
A,$

and, given a feasible extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
for problem (Pe), we set

$\displaystyle\bar{F}(s):=F((\bar{y}^{0},\bar{y},\bar{\omega}^{0},\bar{\omega},\bar{\alpha})(s))\quad\forall
s\in[0,\bar{S}],\qquad\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu}).$

Constraint qualifications for normality (CQn)b, (CQn)f. Let $\bar{z}$ be a
feasible extended sense process for the extended optimization problem (Pe).

  * (CQn)b We say that $\bar{z}$ meets condition (CQn)b if for every $s\in]0,\bar{S}]$ where $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$ there exist $\varepsilon$, $\delta>0$ such that

$\inf_{a\in
A}\,\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))}\left[\xi\cdot\left(f(\bar{y}^{0}(s),\bar{y}(s),a)-f(\bar{y}^{0}(s),\bar{y}(s),\bar{\alpha}(\sigma))\right)\right]<-\delta,$
$\inf_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\,\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))}\Big{[}\xi\cdot\big{(}\sum_{j=1}^{m}g_{j}(\bar{y}^{0}(s),\bar{y}(s))(\omega^{j}-\frac{{\bar{\omega}}^{j}(\sigma)}{|{\bar{\omega}}^{j}(\sigma)|})\big{)}\Big{]}<-\delta\,$

for a.e. $\sigma\in E(s,\varepsilon)$, defined as follows

$E(s,\varepsilon):=\\{r\in[s-\varepsilon,s]\cap[0,\bar{S}]:\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(r),\bar{y}(r))}(\xi_{0}{\bar{\omega}}^{0}+\xi\cdot\bar{F})(r)\geq
0\\};$

  * (CQn)f We say that $\bar{z}$ meets condition (CQn)f if for every $s\in[0,\bar{S}[$ where $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$ there exist $\varepsilon$, $\delta>0$ such that

$\sup_{a\in
A}\,\min_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))}\left[\xi\cdot\left(f(\bar{y}^{0}(s),\bar{y}(s),a)-f(\bar{y}^{0}(s),\bar{y}(s),\bar{\alpha}(\sigma))\right)\right]>\delta,$
$\sup_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\min_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))}\left[\xi\cdot(\sum_{j=1}^{m}g_{j}(\bar{y}^{0}(s),\bar{y}(s))(\omega^{j}-\frac{{\bar{\omega}}^{j}(\sigma)}{|{\bar{\omega}}^{j}(\sigma)|}))\right]>\delta$

for a.e. $\sigma\in\Gamma(s,\varepsilon)$, defined as follows

$\Gamma(s,\varepsilon):=\\{r\in[s,s+\varepsilon]\cap[0,\bar{S}]:\min_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(r),\bar{y}(r))}(\xi_{0}{\bar{\omega}}^{0}+\xi\cdot\bar{F})(r)\leq
0\\}.$

###### Remark 9

The ‘forward’ constraint qualification (CQn)f is at our knowledge new, while a
version of the ‘backward’ constraint qualification (CQn)b was first introduced
in MS20 , as an adaptation to impulsive optimal control of a condition due to
FoFr15 . In particular, (CQn)b, (CQn)f prescribe that drift and fast dynamics
satisfy separately outward or inward pointing conditions, respectively. This
is a stronger requirement than the existence of an inward/outward pointing
velocity. In fact, as discussed in (MS20, , Remark 4.4), the statement of
Theorem 3.3 below holds true even if (CQn)b, (CQn)f are replaced,
respectively, with the (weaker) conditions:

  * (CQn)${}^{\prime}_{b}$ for every $s\in]0,\bar{S}]$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$ there exist $\varepsilon$, $\delta>0$ and a measurable control $(\tilde{\omega},\hat{\alpha})$ taking values in $(\mathcal{C}\cap\partial\mathbb{B})\times A$, satisfying for all $\sigma\in]s-\varepsilon,s[\cap[0,\bar{S}]$:

$\displaystyle\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(\sigma),\bar{y}(\sigma))}\xi\cdot\Big{(}F(\bar{y}^{0},\bar{y},\bar{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime})-\bar{F}(s^{\prime})\Big{)}<-\delta,$
(40)

where $\hat{\omega}:=(1-\bar{\omega}^{0})\tilde{\omega}$, for a.e.
$s^{\prime}\in E(s,\varepsilon)$, defined as in (CQn)b;

  * (CQn)${}^{\prime}_{f}$ for every $s\in[0,\bar{S}[$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$ there exist $\varepsilon$, $\delta>0$ and a measurable control $(\tilde{\omega},\hat{\alpha})$ taking values in $(\mathcal{C}\cap\partial\mathbb{B})\times A$, satisfying for all $\sigma\in]s,s+\varepsilon[\cap[0,\bar{S}]$:

$\min_{(\xi^{0},\xi)\in\partial^{*}h(\bar{y}^{0}(\sigma),\bar{y}(\sigma))}\xi\cdot\Big{(}F(\bar{y}^{0},\bar{y},\bar{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime})-\bar{F}(s^{\prime})\Big{)}>\delta,$
(41)

where $\hat{\omega}:=(1-\bar{\omega}^{0})\tilde{w}$, for a.e.
$s^{\prime}\in\Gamma(s,\varepsilon)$, defined as in (CQn)f.

Notice that conditions (CQn)${}^{\prime}_{b}$, (CQn)${}^{\prime}_{f}$ also
cover situations in which the drift does not depend on the ordinary control
$a$, i.e. $f=f(t,x)$, unlike (CQn)b, (CQn)f, where in this case the hypotheses
involving $f$ are clearly never fulfilled.

###### Remark 10

The constraint qualifications (CQn)b, (CQn)f are respectively inward/outward
pointing conditions at the boundary which involve the minimizer but have to be
satisfied just on a subset of instants at which the optimal trajectory has an
outward/inward pointing velocity. As discussed in detail in (MS20, , Lemma
4.1), when the constraint function $h$ is smooth, (CQn)b, (CQn)f could be
replaced by the following simpler conditions:

  * (IPFCn)b for every $s\in]0,\bar{S}]$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$, one has $h\in C^{1}$ on a neighborhood of $(\bar{y}^{0}(s),\bar{y}(s))$ and there exists $\delta>0$ satisfying

$\begin{split}&\inf_{a\in A}\nabla_{x}h(\bar{y}^{0}(s),\bar{y}(s))\cdot
f(\bar{y}^{0}(s),\bar{y}(s),a)<-\delta,\\\
&\inf_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\nabla_{x}h(\bar{y}^{0}(s),\bar{y}(s))\cdot\sum_{i=1}^{m}g_{i}(\bar{y}^{0}(s),\bar{y}(s))\omega^{i}<-\delta;\end{split}$

  * (IPFCn)f for every $s\in[0,\bar{S}[$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\partial\Omega$, one has $h\in C^{1}$ on a neighborhood of $(\bar{y}^{0}(s),\bar{y}(s))$ and there exists $\delta>0$ satisfying

$\begin{split}&\sup_{a\in A}\nabla_{x}h(\bar{y}^{0}(s),\bar{y}(s))\cdot
f(\bar{y}^{0}(s),\bar{y}(s),a)>\delta,\\\
&\sup_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\nabla_{x}h(\bar{y}^{0}(s),\bar{y}(s))\cdot\sum_{i=1}^{m}g_{i}(\bar{y}^{0}(s),\bar{y}(s))\omega^{i}>\delta,\end{split}$

respectively. Here $\nabla_{x}$ denotes the classical partial gradient
operator w.r.t. the variable $x$.

###### Remark 11

Note that for a feasible process
$\bar{z}=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
with $h(\bar{y}^{0}(\bar{s}),\bar{y}(\bar{s}))=0$ for some
$\bar{s}\in]0,\bar{S}]$, we can assume that there exists some
$\bar{\varepsilon}_{1}>0$ sufficiently small such that
$\ell(E(\bar{s},\varepsilon))>0$ for all
$\varepsilon\in]0,\bar{\varepsilon}_{1}]$. Indeed, if $\exists\varepsilon>0$
such that $\ell(E(\bar{s},\varepsilon))=0$, then
$\displaystyle\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(r),\bar{y}(r))}\left[\xi_{0}{\bar{\omega}}^{0}(r)+\xi\cdot\bar{F}(r)\right]<0$
for a.e. $r\in[\bar{s}-\varepsilon,\bar{s}].$ But then, the function
${\mathcal{H}}:=h\circ(\bar{y}^{0},\bar{y})$ is differentiable a.e. in
$[\bar{s}-\varepsilon,\bar{s}]$ and verifies

$\begin{split}\frac{d{\mathcal{H}}}{ds}(s)&\leq\max_{(\xi_{0},\xi)\in\partial^{C}h(\bar{y}^{0}(s),\bar{y}(s))}\left[\xi_{0}\frac{d{\bar{y}}^{0}}{ds}(s)+\xi\cdot\frac{d{\bar{y}}}{ds}(s)\right]\\\
&=\max_{(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))}\left[\xi_{0}{\bar{\omega}}^{0}(s)+\xi\cdot\bar{F}(s)\right]<0,\end{split}$

since the scalar product is bilinear. Thus, for all
$s\in[\bar{s}-\varepsilon,\bar{s}[$ one has

$-h(\bar{y}^{0}(s),\bar{y}(s))=h(\bar{y}^{0}(\bar{s}),\bar{y}(\bar{s}))-h(\bar{y}^{0}(s),\bar{y}(s))=\int_{s}^{\bar{s}}\frac{d{\mathcal{H}}}{ds}(\sigma)\,d\sigma<0,$

so that $h(\bar{y}^{0}(s),\bar{y}(s))>0$, in contradiction with the
feasibility of $\bar{z}$. In an analogous way, one can derive that, if
$h(\bar{y}^{0}(\bar{s}),\bar{y}(\bar{s}))=0$ at $\bar{s}\in[0,\bar{S}[$, then
$\ell(\Gamma(\bar{s},\varepsilon))>0$ for all $\varepsilon>0$ small enough.

Furthermore, by the bilinearity of the scalar product, in (CQn)b, (CQn)f, and
(CQn)${}^{\prime}_{b}$, (CQn)${}^{\prime}_{f}$, one can replace
$\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))$ with
$\partial^{C}h(\bar{y}^{0}(s),\bar{y}(s))$. Hence, in particular, all these
conditions are verified for any $(\xi_{0},\xi)\in$
$\partial^{>}h(\bar{y}^{0}(s),\bar{y}(s))$, since
$\partial^{>}h(\bar{y}^{0}(s),\bar{y}(s))\subseteq\partial^{C}h(\bar{y}^{0}(s),\bar{y}(s))$.

We can now establish the following normal Maximum Principle, which extends
(MS20, , Th. 4.2).

###### Theorem 3.2

Assume (H0)-(H2). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense local minimizer for (Pe). Assume that
$(p_{0},p,\lambda,\pi,\mu)$, $(m^{0},m)$ meet the conditions of Theorem 1.1
and verify the strenghtened non-triviality condition (34). Then,

  * (i)

if hypothesis (CQn)b is satisfied, one has

$\displaystyle|q(\bar{S})|+\lambda\neq 0\qquad$ $\displaystyle\text{if
}\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0),$ (42)
$\displaystyle|q_{0}(\bar{S})|+|q(\bar{S})|+\lambda\neq 0\qquad$
$\displaystyle\text{if }\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0).$

In particular, if
proj${}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))=\\{(\xi_{t_{2}},0)\\}$
and $\xi_{t_{2}}=0$ whenever $\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0)$, then
$\lambda\neq 0$;

  * (ii)

if hypothesis (CQn)f is satisfied, one has

$\displaystyle|q(0)|+\lambda\neq 0\qquad$ $\displaystyle\text{if
}\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0),$ (43)
$\displaystyle|q_{0}(0)|+|q(0)|+\lambda\neq 0\qquad$ $\displaystyle\text{if
}\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0).$

In particular, if
proj${}_{(t_{1},x_{1})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))=\\{(\xi_{t_{1}},0)\\}$
and $\xi_{t_{1}}=0$ whenever $\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0)$, then
$\lambda\neq 0$.

The proof of this result is given in the Appendix.

######  Remark 12 (Multiple state constraints)

The normal Maximum Principle in Theorem 3.2 can be extended to the case with
multiple state constraints $h_{i}$ for $i=1,\dots,N$, by modifying condition
(CQn)${}^{\prime}_{b}$ as follows: for every $s\in]0,\bar{S}]$ such that
$h_{i_{j}}(\bar{y}^{0}(s),\bar{y}(s))=0$ for some indexes $1\leq
i_{1}<\dots<i_{k}\leq N$, $j=1,\dots,k$, there exist $\varepsilon$, $\delta>0$
and a measurable control $(\tilde{\omega},\hat{\alpha})$ taking values in
$(\mathcal{C}\cap\partial\mathbb{B})\times A$, satisfying for all
$\sigma\in]s-\varepsilon,s[\cap[0,\bar{S}]$ and for all $j=1,\dots,k$:

$\displaystyle\max_{(\xi_{0},\xi)\in\partial^{*}h_{i_{j}}(\bar{y}^{0}(\sigma),\bar{y}(\sigma))}\xi\cdot\Big{(}F(\bar{y}^{0},\bar{y},\bar{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime})-\bar{F}(s^{\prime})\Big{)}<-\delta,$
(44)

where $\hat{\omega}:=(1-\bar{\omega}^{0})\tilde{\omega}$, for a.e.
$s^{\prime}\in\mathcal{E}(s,\varepsilon)$, defined as follows

$\begin{split}&\mathcal{E}(s,\varepsilon):=\Bigl{\\{}r\in[s-\varepsilon,s]\cap[0,\bar{S}]\text{
: }\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fdsfsdfsd}\max_{(\xi_{0},\xi)\in\partial^{*}h_{i_{j}}(\bar{y}^{0}(r),\bar{y}(r))}[\xi_{0}\bar{\omega}^{0}(r)+\xi\cdot\bar{F}(r)]\geq
0\qquad\forall j=1,\dots,k\Bigr{\\}}.\end{split}$

Notice that, arguing as in Remark 11, one can deduce that
$\ell(\mathcal{E}(s,\varepsilon))$ for all $\varepsilon>0$ sufficiently small.
Similar modifications are to be made for condition (CQn)${}^{\prime}_{f}$.

Theorem 3.2 implies nondegenerate normality when essentially the endpoint
constraint either at the final or at the initial position is inactive. We
provide below some sufficient conditions to guarantee normality even in some
situations where initial and final positions lay on the boundary of the
endpoint constraint. Endpoint qualifications for normality (TQn)b, (TQn)f. Let
us consider
$\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$,
a feasible extended sense process for the extended optimization problem (Pe).

  * (TQn)b We say that $\bar{z}$ meets condition (TQn)b if there exists $\varepsilon>0$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\text{{\rm Int}}(\Omega)$ for each $s\in[\bar{S}-\varepsilon,\bar{S}[$ and one among the following conditions (a), (b) holds true:

$\begin{split}\text{{\em(a)}}\quad&\left(-\text{{\rm
proj}}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\setminus\\{0_{1+n}\\}\right)\,{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}llllllllll}\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fssdsdfsdfdf}\cap\,\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))=\emptyset\end{split}$
(45)

and for any $(\zeta_{t_{2}},\zeta_{x_{2}})\in\text{{\rm
proj}}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$$+[0,+\infty[\cdot\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$,888As
in Remark 8, we adopt the convention that $X+\emptyset=X$. one has

$\min_{a\in A}\left[\zeta_{x_{2}}\cdot
f(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),a)+\zeta_{t_{2}}\right]<0\qquad\text{if
}(\zeta_{t_{2}},\zeta_{x_{2}})\neq(0,0);$ (46)
$\begin{split}\text{{\em(b)}}\quad&\Big{(}-\text{{\rm
proj}}_{x_{2}}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\setminus\\{0_{n}\\}\Big{)}\,{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}lllllllllllllllll}\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fssdsdfs}\cap\,\text{{\rm
proj}}_{x}\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))=\emptyset\end{split}$
(47)

and for any $(\zeta_{t_{2}},\zeta_{x_{2}})\in\text{{\rm
proj}}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$$+[0,+\infty[\cdot\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$
with $\zeta_{x_{2}}\neq 0$, one has

$\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0),\,\,\,\bar{\nu}(\bar{S})<K,\,\,\,\min_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\left[\zeta_{x_{2}}\cdot\left(\sum_{j=1}^{m}g_{j}(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\omega^{j}\right)\right]<0.$
(48)

  * (TQn)f We say that $\bar{z}$ meets condition (TQn)f if there exists $\varepsilon>0$ such that $(\bar{y}^{0}(s),\bar{y}(s))\in\text{{\rm Int}}(\Omega)$ for each $s\in]0,\varepsilon]$ and one among the following conditions (a), (b) holds true:

$\begin{split}\text{{\em(a)}}\quad&\left(-\text{{\rm
proj}}_{(t_{1},x_{1})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\setminus\\{0_{1+n}\\}\right)\,{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}llllllllll}\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fssdsdfsdfdf}\cap\,\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))=\emptyset\end{split}$
(49)

and for any $(\zeta_{t_{1}},\zeta_{x_{1}})\in\text{{\rm
proj}}_{(t_{1},x_{1})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$
$+[0,+\infty[\cdot\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))$ one has

$\max_{a\in A}\left[\zeta_{x_{1}}\cdot
f(\bar{y}^{0}(0),\bar{y}(0),a)+\xi_{t_{1}}\right]>0\qquad\text{if
}(\zeta_{t_{1}},\zeta_{x_{1}})\neq(0,0);$ (50)
$\begin{split}\text{{\em(b)}}\quad&\Big{(}-\text{{\rm
proj}}_{x_{1}}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\setminus\\{0_{n}\\}\Big{)}\,{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}lllllllllllllllll}\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fssdsdfs}\cap\,\text{{\rm
proj}}_{x}\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))=\emptyset\end{split}$ (51)

and for any $(\zeta_{t_{1}},\zeta_{x_{1}})\in\text{{\rm
proj}}_{(t_{1},x_{1})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$
$+[0,+\infty[\cdot\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))$ with
$\zeta_{x_{1}}\neq 0$, one has

$\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0),\ \ \bar{\nu}(\bar{S})<K,\ \
\max_{\omega\in\mathcal{C}\cap\partial\mathbb{B}}\zeta_{x_{1}}\cdot\sum_{j=1}^{m}g_{j}(\bar{y}^{0}(0),\bar{y}(0))\omega^{j}>0.$
(52)

Condition (TQn)b generalizes the endpoint constraint qualifications considered
in MS20  for the case with fixed initial point, which were in turn inspired by
no gap conditions in MRV ; AMR15 . Notice that both conditions (45), (47)
[resp., (49), (51)] are trivially verified whenever
$(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\in\text{{\rm Int}}(\Omega)$ [resp.,
$(\bar{y}^{0}(0),\bar{y}(0))\in\text{{\rm Int}}(\Omega)$], since
$\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))=\emptyset$ [resp.,
$\partial^{>}h(\bar{y}^{0}(0),\bar{y}(0))=\emptyset$].

###### Proposition 4

Assume (H0)-(H2). Let
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be an extended sense local minimizer for (Pe). Assume that
$(p_{0},p,\lambda,\pi,\mu)$, $(m_{0},m)$ meet the conditions of Theorem 1.1.
Then, when either (i) or (ii) below holds true, one has $\lambda\neq 0$:

  * (i)

hypothesis (TQn)b is satisfied and the multipliers $(p_{0},p,\lambda,\pi,\mu)$
verify the strengthened non-triviality condition (42);

  * (ii)

hypothesis (TQn)f is satisfied and the multipliers $(p_{0},p,\lambda,\pi,\mu)$
verify the strengthened non-triviality condition (43).

The proof of this result is postponed to the Appendix.

######  Remark 13 (Multiple state constraints)

Proposition 4 can be easily adapted to the case with multiple state
constraints $h_{i}$ for $i=1,\dots,N$ by simply choosing
$h:=h_{1}\vee\dots\vee h_{N}$. In particular, by applying the max-rule for
subdifferentials, in (TQn)b, (b), condition (47) can be replaced with

$\begin{split}&\left(\sum_{i=1}^{N}\,\text{{\rm
proj}}_{x}[0,+\infty[\cdot\partial^{>}h_{i}(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\right)\,\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}dfsfsd}\cap\,\left(-\text{{\rm
proj}}_{x_{2}}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\setminus\\{0_{n}\\}\right)=\emptyset,\end{split}$

and one can require that condition (48) is satisfied for all
$(\zeta_{t},\zeta_{x})$ in the set $\text{{\rm
proj}}_{(t_{1},x_{1})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))$
$+\sum_{i=1}^{N}\,[0,+\infty[\cdot\partial^{>}h_{i}(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})).$
The other conditions can be adapted in a similar way.

From Propositions 3, 4, and Theorem 3.2 we deduce as a corollary the main
result of this section.

###### Theorem 3.3

Assume (H0)-(H2). Consider the optimal control problem (P) and its extended
sense formulation (Pe). Assume that there exists a local or global extended
sense minimizer
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
such that condition (CNa) and either (CQn)b-(TQn)b or (CQn)f-(TQn)f are
verified. Then
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
is a normal extremal and, in consequence of Theorem 3.1, there is no local or
global infimum gap, respectively.

##  4 Some examples

Let us illustrate the preceding theory through some examples.

###### Example 1

In this example the absence of an infimum gap can be easily deduced from the
sufficient conditions introduced in Section 3. Consider the problem

$\left\\{\begin{array}[]{l}\mbox{minimize }\ -x(1)\\\\[6.45831pt] \mbox{over
}(x,v,u)=(x^{1},x^{2},x^{3},v,u^{1},u^{2})\in
W^{1,1}([0,1];\mathbb{R}^{3}\times\mathbb{R}\times\mathbb{R}^{2})\quad\mbox{satisfying
}\\\\[6.45831pt]
\begin{array}[]{l}\displaystyle\frac{dx}{dt}(t)=f(x(t))+g_{1}(x(t))\,\frac{du^{1}}{dt}(t)+g_{2}(x(t))\,\frac{du^{2}}{dt}(t)\\\\[6.45831pt]
\displaystyle\frac{dv}{dt}(t)=\,\left|\frac{du}{dt}(t)\right|\\\\[6.45831pt]
\displaystyle\frac{du}{dt}(t)\in\mathcal{C}:=\mathbb{R}^{2}\qquad\mbox{ a.e.
}t\in[0,1],\\\\[6.45831pt] x(t)\in\Omega(t)\qquad\forall
t\in[0,1],\\\\[6.45831pt] \displaystyle v(0)=0,\ \ v(1)\leq 2,\quad
x(0)=\mathcal{T}^{1}_{0},\quad
x(1)\in\mathcal{T}^{2}_{0}.\end{array}\end{array}\right.$ (53)

in which

$\begin{array}[]{l}\Omega(t):=\\{(t,x)\in\mathbb{R}^{4}:\ -1\leq x^{1}\leq
1+t,\ -1\leq x^{2}\leq 1,\ -1\leq x^{3}\leq 1\\},\\\\[6.45831pt] \
\mathcal{T}^{1}_{0}:=\\{x\in\mathbb{R}^{3}:\
(x^{1}-1)^{2}+(x^{2})^{2}+(x^{3})^{2}\leq 1/9,\ x^{1}\leq 1\\},\\\\[6.45831pt]
\ \mathcal{T}^{2}_{0}:=\\{x\in\mathbb{R}^{3}:\
(x^{1}+1)^{2}+(x^{2})^{2}+(x^{3})^{2}\leq 1,\ x^{1}\geq-1\\},\end{array}$

and

$\displaystyle g_{1}(x):=\left(\begin{array}[]{l}1\\\ 0\\\
0\end{array}\right),\quad\displaystyle g_{2}(x):=\left(\begin{array}[]{l}\ \
0\\\ -1\\\ -x^{1}\end{array}\right),\quad\displaystyle
f(x):=\left(\begin{array}[]{l}\ 0\\\ x^{2}x^{3}\\\ \
0\end{array}\right)\quad\forall x\in\mathbb{R}^{3}\,.$

The extended problem is

$\left\\{\begin{array}[]{l}\mbox{minimize }\ -y^{1}(S)\\\\[6.45831pt]
\mbox{over }S>0,\
(y^{0},y^{1},y^{2},y^{3},\nu,\omega^{0},\omega^{1},\omega^{2})\in
W^{1,1}([0,S])\quad\mbox{satisfying }\\\\[6.45831pt]
\begin{array}[]{l}\displaystyle\frac{dy^{0}}{ds}(s)=\omega^{0}(s)\\\\[6.45831pt]
\displaystyle\frac{dy}{ds}(s)=f(y(s))\omega^{0}(s)+g_{1}(y(s))\,\omega^{1}(s)+g_{2}(y(s))\,\omega^{2}(s)\\\\[6.45831pt]
\displaystyle\frac{d\nu}{ds}(s)=\,\left|\omega(s)\right|\\\\[6.45831pt]
\displaystyle(\omega^{0},\omega)(s)\in{\mathbb{C}}\qquad\mbox{ a.e.
}s\in[0,S],\\\\[6.45831pt] y(s)\in\Omega(y^{0}(s))\qquad\forall
s\in[0,S],\\\\[6.45831pt] \displaystyle\nu(0)=0,\,\,\nu(S)\leq
2,\,\,(y^{0}(0),y(0))=\\{0\\}\times\mathcal{T}^{1}_{0},\,\,(y^{0}(S),y(S))\in\\{1\\}\times\mathcal{T}^{2}_{0}.\end{array}\end{array}\right.$
(54)

An extended sense minimizer is clearly given by the feasible extended sense
process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$, where

$\bar{S}=2,\qquad(\bar{\omega}^{0},\bar{\omega})=(\bar{\omega}^{0},\bar{\omega}^{1},\bar{\omega}^{2})=(1,0,0)\chi_{{}_{[0,1]}}+(0,-1,0)\chi_{{}_{]1,2]}}\,,$
(55)

and one considers the corresponding trajectory with initial state condition
$y(0)=(1,0,0)$, namely,

$(\bar{y}^{0},\bar{y},\bar{\nu})=(\bar{y}^{0},\bar{y}^{1},\bar{y}^{2},\bar{y}^{3},\bar{\nu})=(s,1,0,0,0)\chi_{[0,1]}+(1,2-s,0,0,s-1)\chi_{[1,2]}\,.$
(56)

It is not difficult to check that this process verifies conditions (CNa),
(CQn)b, and (TQn)b. In consequence, the absence of a gap between the infima of
problems (53), (54) follows directly from Theorem 3.3.

Next example shows how the criterion of normality can guarantee the absence of
the infimum gap in situations where other sufficient conditions fail.

###### Example 2

Consider again the minimization problem (53) and its extended version (54),
where $\mathcal{T}^{1}_{0}$ is as above, while the time-dependent state
constraint $\Omega(t)$ and the final-point constraint $\mathcal{T}^{2}_{0}$
are replaced with

$\begin{array}[]{l}\Omega:=\\{(x^{1},x^{2},x^{3}):\ -1\leq x^{1}\leq 1,\
-1\leq x^{2}\leq 1,\ -1\leq x^{3}\leq 1\\},\\\\[6.45831pt]
\mathcal{T}^{2}_{0}:=\\{(x^{1},x^{2},x^{3}):\ -1\leq x^{1}\leq 0,\ 0\leq
x^{2}\leq 1,\ 0\leq x^{3}\leq 1\\},\end{array}$

respectively. Then the extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$ given
by (55), (56) is still a (feasible) minimizer for the extended problem (54).
However, as it is easy to check, now the nondegeneracy condition (CNa) is met,
but none of the conditions (CQn)b, (TQn)b, (CQn)f, and (TQn)f is satisfied.

From Theorem 1.1 there exist a set of multipliers $(p_{0},p,\pi,\lambda,\mu)$
and functions $(m_{0},m)$ with $\pi=0$, since $\nabla_{v}\Psi\equiv 0$ and
$\bar{\nu}(2)=1<2$. Also, $m_{0}\equiv 0$ as the state constraint does not
depend on time, $\mu([0,2])=\mu([0,1])$, and
$m(s)\in\partial^{>}_{x}h(\bar{y}(0))$ $\mu$-a.e. yields $m(s)=(1,0,0)$
$\mu$-a.e. in $[0,1]$. By the adjoint equation it follows that the path
$(p_{0},p)=(p_{0},p_{1},p_{2},p_{3})\equiv(\bar{p}_{0},\bar{p}_{1},\bar{p}_{2},\bar{p}_{3})$
is constant. From the transversality conditions

$\begin{split}&(p_{0},p_{1},p_{2},p_{3})(0)\in\mathbb{R}\times
N_{\mathcal{T}^{1}_{0}}(1,0,0),\\\
&-(q_{0},q_{1},q_{2},q_{3})(2)\in\lambda\\{(0,-1,0,0)\\}+\mathbb{R}\times
N_{\mathcal{T}^{2}_{0}}(0,0,0),\end{split}$ (57)

where $q_{0}\equiv\bar{p}_{0}$, and
$q(s)=(\bar{p}_{1}+\mu([0,1]),\bar{p}_{2},\bar{p}_{3})$ for all $s\in]1,2]$,
we derive that $\bar{p}_{0}\in\mathbb{R}$, $\bar{p}_{1}\geq 0$,
$\bar{p}_{2}=\bar{p}_{3}=0$,
$q_{1}(2)=\bar{p}_{1}+\mu([0,1])=\lambda-\alpha_{1}$ with $\alpha_{1}\geq 0$.
The maximality condition implies the relations

$\begin{array}[]{l}\bar{p}_{0}\chi_{[0,1]}(s)=0,\quad-
q_{1}(s)\chi_{]1,2]}(s)=0,\end{array}$ (58)

from which we deduce that $\bar{p}_{0}=0$ and
$\bar{p}_{1}+\mu([0,1])=\lambda-\alpha_{1}=0$. Hence, recalling that
$\bar{p}_{1}\geq 0$, we get $\bar{p}_{1}=\mu([0,1])=0$,
$\lambda=\alpha_{1}\geq 0$. So, the strengthened non-triviality condition
$\|p\|_{L^{\infty}}+\mu([0,2])+\lambda\neq 0$ implies that $\lambda\neq 0$ and
this shows that
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$ is a
normal extremal. Consequently, there is no infimum gap for the ‘normality
test’ established in Theorem 3.1.

However, normality itself is only a sufficient condition to avoid the gap
(even for systems with drift, $f$).

###### Example 3

Let us consider the problem in Example 2 where we only modify the initial-
point target $\mathcal{T}^{1}_{0}$, replacing it with
$\mathcal{T}^{1}_{0}:=\\{(1,0,0)\\}$. Then the extended sense process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$ of
before is obviously still admissible and minimizing, but it is easy to see
that the set of degenerate multipliers $(p_{0},p,\lambda,\mu)$ with
$p_{0}=p_{2}=p_{3}=0$, $p_{1}=-1$, $\mu=\delta_{\\{0\\}}$, and $\lambda=0$
meets all the conditions of Theorem 1.1. So,
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$ is an
abnormal extremal. But there is no gap, because, for any $n\in\mathbb{N}$, the
strict sense process
$(\bar{S},\omega^{0}_{n},\omega_{n},y^{0}_{n},y_{n},\nu_{n})$, where

$\bar{S}=2,\qquad(\omega^{0}_{n},\omega_{n})=\left(1-\frac{1}{n},-\frac{1}{n},0\right)\chi_{{}_{[0,1]}}+\left(\frac{1}{n},-1+\frac{1}{n},0\right)\chi_{{}_{]1,2]}}\,,$

and $(y^{0}_{n},y_{n},\nu_{n})$ is the corresponding trajectory of the control
system in (54) with initial condition $(y^{0},y,\nu)(0)=(0,1,0,0,0)$, is
feasible and minimizing for the original problem, since it has cost equal to
zero.

##  Appendix A  Appendix

Proof of Theorem 3.2:  Thanks to Remark 11, when (CQn)b is in force the proof
of Theorem (3.2) is analogous to the proof of (MS20, , Th. 4.2), while under
assumption (CQn)f it requires some adaptation. For this reason, we limit
ourselves to give the proof in the last case.

By standard truncation and mollification arguments, we can assume $h$
Lipschitz continuous, with Lipschitz constant $L>0$, and $f$,
$g_{1},\dots,g_{m}$, and their limiting subdifferentials in $(t,x)$,
$L^{\infty}$-bounded by some constant $\tilde{M}>0$. Set $M:=(1+m)\tilde{M}$.

By assumption, the local minimizer
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
has a set of multipliers $(p_{0},p,\pi,\lambda,\mu)$ and some functions
$(m_{0},m)$ such that the conditions (i)-(vi) of Theorem 1.1 hold true, and
verifying the strengthened non-triviality condition (34). Let us first assume
that $\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0)$ and suppose by contradiction that

$q(0)=0,\qquad\lambda=0.$ (59)

Set

$\bar{s}:=\sup\\{s\in[0,\bar{S}]:\ \ \mu(]0,s])=0\\}.$

Observe that $\bar{s}<\bar{S}$. Indeed, if not, $\mu(]0,\bar{S}])=0$. But in
this case $q(s)=p(s)+\mu(\\{0\\})m(0)$, so that it is absolutely continuous
and by the adjoint equation with initial condition $q(0)=0$ it follows that
$q\equiv 0$. Precisely, by known properties of the convex hull of the limiting
subdifferential of locally Lipschitz continuous functions (see e.g. (OptV, ,
Ch. 6)), we have

$|q(s)|\leq\int_{0}^{s}\left|\frac{dq}{ds}(s)\right|\,ds\leq
M\int_{0}^{s}|q(s)|\,ds,$

which implies that $q\equiv 0$ by Gronwall’s Lemma. Since $\lambda=0$ by (59),
this is in contradiction with the first relation in (34). When
$\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0)$ and we assume by contradiction that

$q_{0}(0)=0,\qquad q(0)=0,\qquad\lambda=0,$ (60)

the value $\bar{s}$ defined as above is still strictly smaller than $\bar{S}$,
since otherwise $\mu(]0,\bar{S}])=0$, so that $(q_{0},q)\equiv 0$, again by
the adjoint equation. In view of (60), this yields contradiction with the
second relation in (34). Obviously,
$(\bar{y}^{0}(\bar{s}),\bar{y}(\bar{s}))\in\partial\Omega$.

From now on, the proof is the same for both cases. Introduce

$(z_{0},z)(s):=\left(p_{0}(s)+m_{0}(0)\mu(\\{0\\}),p(s)+m(0)\mu(\\{0\\})\right),$

so that, for any $s\in[0,\bar{S}[$,

$\begin{array}[]{l}\displaystyle(q_{0}(s),q(s))=\left(p_{0}(s)+\int_{[0,s[}m_{0}(\sigma)\mu(d\sigma)\,,\,p(s)+\int_{[0,s[}m(\sigma)\mu(d\sigma)\right)\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad=\left(z_{0}(s)+\int_{]0,s[}m_{0}(\sigma)\mu(d\sigma)\,,\,z(s)+\int_{]0,s[}m(\sigma)\mu(d\sigma)\right).\end{array}$
(61)

By the adjoint equation, $(z_{0},z)$ verifies

$\begin{cases}\displaystyle-\left(\frac{dz_{0}}{ds},\frac{dz}{ds}\right)(s)\in\text{co}\,\partial_{t,x}\\{q(s)\cdot\bar{F}(s)\\}=\text{co}\,\partial_{t,x}\left\\{z(s)\cdot\bar{F}(s)+\int_{]0,s[}m(\sigma)\mu(d\sigma)\cdot\bar{F}(s)\right\\}\\\
z(0)=0,\quad(\text{and $z_{0}(0)=0$, \, if
$\bar{y}^{0}(\bar{S})=\bar{y}^{0}(0)$}).\end{cases}$ (62)

Since the integral on the right hand side is identically zero in
$]0,\bar{s}[$, arguing as above we derive that $z(s)=0$ and therefore $q(s)=0$
for all $s\in[0,\bar{s}[$, by continuity. Moreover, Gronwall’s Lemma implies
that $|z(s)|\leq C\,\mu([\bar{s},s[)$ for all $s\in[\bar{s},\bar{S}[$, for
some $C>0$, so that

$|q(s)|\leq|z(s)|+L\mu([\bar{s},s[)\leq(C+L)\mu([\bar{s},s[)\qquad\forall
s\in[\bar{s},\bar{S}[.$ (63)

As a consequence of (61), for every $s\in[\bar{s},\bar{S}]$ one gets
$q(s)=z(s)+\int_{[\bar{s},s[}m(\sigma)\mu(d\sigma)$, and (62), (63) imply

$\left|q(s)-\int_{[\bar{s},s[}m(\sigma)\mu(d\sigma)\right|\leq
M\int_{\bar{s}}^{s}|q(\sigma)|\,d\sigma\leq\bar{C}\,\mu([\bar{s},s[)(s-\bar{s}),$
(64)

where $\bar{C}:=M(L+C)$. In view of (CQn)${}^{\prime}_{f}$ in Remark 9, there
exist $\varepsilon$, $\delta>0$ and a measurable control
$(\tilde{\omega},\hat{\alpha}):[0,\bar{S}]\to(\mathcal{C}\cap\partial\mathbb{B})\times
A$, verifying for all
$(\xi_{0},\xi)\in\partial^{*}h(\bar{y}^{0}(s),\bar{y}(s))$ with
$s\in]\bar{s},\bar{s}+\varepsilon[\cap[0,\bar{S}]$:

$\xi\cdot\left(F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime}))-\bar{F}(s^{\prime})\right)>\delta,\quad\text{for
a.e. $s^{\prime}\in\Gamma(\bar{s},\varepsilon)$,}$ (65)

where
$(\hat{\omega}^{0}(s),\hat{\omega}(s),\hat{\alpha}(s)):=\left(\bar{\omega}^{0}(s),(1-\bar{\omega}^{0}(s))\tilde{\omega}(s),\hat{\alpha}(s)\right)$
for a.e. $s\in[0,\bar{S}]$. Observe that, being
$\hat{\omega}^{0}\equiv\bar{\omega}^{0}$, one has
$|\hat{\omega}|=1-\bar{\omega}^{0}=|\bar{\omega}|$ a.e. As observed in Remark
11, $\ell(\Gamma(\bar{s},\varepsilon))$ is $>0$ for any $\varepsilon>0$
sufficiently small. Moreover, (65) is valid for any
$(\xi_{0},\xi)\in\partial^{C}h(\bar{y}^{0}(s),\bar{y}(s))$, so that it is
true, in particular, for
$(\xi_{0},\xi)\in\partial^{>}h(\bar{y}^{0}(s),\bar{y}(s))$. On the other hand,
by the maximization condition (11) of Theorem 1.1, it follows that, for a.e.
$s\in]\bar{s},\bar{s}+\varepsilon[\cap[0,\bar{S}]$,

$\begin{split}&q_{0}(s)\left(\hat{\omega}^{0}(s)-\bar{\omega}^{0}(s)\right)+\pi(|\hat{\omega}(s)|-|\bar{\omega}(s)|)+q(s)\left[F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s))-\bar{F}(s)\right]\\\
&{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}fdsjhfsdkhksdj}=q(s)\left[F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s))-\bar{F}(s)\right]\leq
0.\end{split}$ (66)

Putting together (64), (65), and (66) we get the desired contradiction.
Indeed, for $\varepsilon>0$ small enough, for any
$s^{\prime}\in\Gamma(\bar{s},\varepsilon)$, one has

$\begin{split}0&\geq
q(s^{\prime})\left[F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime}))-\bar{F}(s^{\prime})\right]\\\
&=\left(q(s^{\prime})-\int_{[\bar{s},s^{\prime}[}m(\sigma)\mu(d\sigma)+\int_{[\bar{s},s^{\prime}[}m(\sigma)\mu(d\sigma)\right)\left[F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime}))-\bar{F}(s^{\prime})\right]\\\
&\geq\int_{[\bar{s},s^{\prime}[}m(\sigma)\,\left[F((\bar{y}^{0},\bar{y},\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s^{\prime}))-\bar{F}(s^{\prime})\right]\mu(d\sigma)-2M\bar{C}\,\mu([\bar{s},s^{\prime}[)(s^{\prime}-\bar{s})\\\
&\geq\mu([\bar{s},s^{\prime}[)\left[\delta-2M\,\bar{C}\,(s^{\prime}-\bar{s})\right]>0\end{split}$

for $\varepsilon>0$ sufficiently small. This concludes the proof. ∎

Proof Proposition 4:  The proof follows the same lines of the proof of (MS20,
, Prop. 4.1), where however only condition (TQn)b for an implicit state
constraint $\Omega\subseteq\mathbb{R}^{n}$ is considered. Let us prove (i).
Assume by contradiction $\lambda=0$. Then the transversality condition (10)
implies that

$(-q_{0}(\bar{S}),-q(\bar{S}))=(\zeta_{t_{2}},\zeta_{x_{2}})\in\text{{\rm
proj}}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})),$

where $(\zeta_{t_{2}},\zeta_{x_{2}})\neq(0,0)$ and, in particular,
$\zeta_{x_{2}}\neq 0$ if $\bar{y}^{0}(\bar{S})>\bar{y}^{0}(0)$ by (42). By
hypothesis (TQn)b, there is some $\varepsilon>0$ such that
$(\bar{y}^{0}(s),\bar{y}(s))\in\,$Int$(\Omega)$ for all
$s\in[\bar{S}-\varepsilon,\bar{S}[$. Hence
$\mu([\bar{S}-\varepsilon,\bar{S}[)=0$, so that, for any
$s\in]\bar{S}-\varepsilon,\bar{S}[$, $(q_{0},q)$ is continuous at $s$ and

$(q_{0}(s),q(s))=\Big{(}p_{0}(s)+\int_{[0,\bar{S}-\varepsilon]}m_{0}(r)\mu(dr)\,,\,p(s)+\int_{[0,\bar{S}-\varepsilon]}m(r)\mu(dr)\Big{)}.$

Set
$(q_{0}(\bar{S}^{-}),q(\bar{S}^{-})):=\lim_{s\to\bar{S}^{-}}(q_{0}(s),q(s))=(p_{0},p)(\bar{S})+\int_{[0,\bar{S}-\varepsilon]}(m_{0},m)(r)\mu(dr).$
We get

$\begin{array}[]{l}(q_{0}(\bar{S}^{-}),q(\bar{S}^{-}))=\Big{(}q_{0}(\bar{S})-m_{0}(\bar{S})\mu(\\{\bar{S}\\})\,,\,q(\bar{S})-m(\bar{S})\mu(\\{\bar{S}\\})\Big{)}=(-\tilde{\zeta}_{t_{2}},-\tilde{\zeta}_{x_{2}}),\end{array}$

where
$(\tilde{\zeta}_{t_{2}},\tilde{\zeta}_{x_{2}}):=\Big{(}\zeta_{t_{2}}+\mu(\\{\bar{S}\\})m_{0}(\bar{S}),\zeta_{x_{2}}+\mu(\\{\bar{S}\\})m(\bar{S})\Big{)}$.
Thus, in particular, the pair $(\tilde{\zeta}_{t_{2}},\tilde{\zeta}_{x_{2}})$
verifies

$(\tilde{\zeta}_{t_{2}},\tilde{\zeta}_{x_{2}})\in\text{{\rm
proj}}_{(t_{2},x_{2})}N_{\mathcal{T}_{0}}(\bar{y}^{0}(0),\bar{y}(0),\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))+[0,+\infty[\cdot\partial^{>}h(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S})).$
(67)

The continuity of $(q_{0},q)$ on $]\bar{S}-\varepsilon,\bar{S}[$ also implies
that the equality (11) in the Maximum Principle is verified for all
$s\in]\bar{S}-\varepsilon,\bar{S}[$. Hence, passing to the limit in it as $s$
tends to $\bar{S}^{-}$, we obtain

$\left(\tilde{\zeta}_{x_{2}}\cdot
f(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),a)+\tilde{\zeta}_{t_{2}}\right)\omega^{0}+\tilde{\zeta}_{x_{2}}\cdot\sum_{j=i}^{m}g_{j}(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}))\omega^{j}-\pi|\omega|\geq
0.$ (68)

Suppose first that condition (a) in (TQn)b is satisfied. Then, from (45) we
deduce that $(\tilde{\zeta}_{t_{2}},\tilde{\zeta}_{x_{2}})\neq(0,0)$ and
choosing $\omega=0$ in (68) we obtain a contradiction to (46).

If instead condition (b) in (TQn)b is valid, $\pi=0$ and (42) implies that
$\zeta_{x_{2}}\neq 0$. In view of (67) and hypothesis (47), this yields
$\tilde{\zeta}_{x_{2}}\neq 0$. At this point, we get a contradiction to (48)
by choosing $\omega^{0}=0$ in (68).

The proof of (ii) is very similar, hence we omit it. ∎

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