# Nondegenerate abnormality, controllability, and gap phenomena in optimal
control with state constraints

Giovanni Fusco  G. Fusco, Dipartimento di Matematica, Università di Padova
Via Trieste 63, Padova 35121, Italy
email<EMAIL_ADDRESS>and  Monica Motta  M. Motta, Dipartimento di
Matematica, Università di Padova
Via Trieste 63, Padova 35121, Italy
Telefax: (+39) 049 827 1499, Telephone: (39)(49) 827 1368 email:
<EMAIL_ADDRESS>
###### Abstract.

In optimal control theory, infimum gap means that there is a gap between the
infimum values of a given minimum problem and an extended problem, obtained by
enlarging the set of original solutions and controls. The gap phenomenon is
somewhat “dual” to the problem of the controllability of the original control
system to an extended solution. In this paper we present sufficient conditions
for the absence of an infimum gap and for controllability for a wide class of
optimal control problems subject to endpoint and state constraints. These
conditions are based on a nondegenerate version of the nonsmooth constrained
maximum principle, expressed in terms of subdifferentials. In particular,
under some new constraint qualification conditions, we prove that: (i) if an
extended minimizer is a nondegenerate normal extremal, then no gap shows up;
(ii) given an extended solution verifying the constraints, either it is a
nondegenerate abnormal extremal, or the original system is controllable to it.
An application to the impulsive extension of a free end-time, non-convex
optimization problem with control-polynomial dynamics illustrates the results.

###### Key words and phrases:

Optimal control problems, Maximum Principle, State constraints, Gap phenomena,
Controllability, Nondegeneracy

###### 2020 Mathematics Subject Classification:

49K15, 34K45, 49N25

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.

##  1\. Introduction

In the Calculus of Variations and in the Theory of Optimal Control is a rather
common procedure to enlarge the space of solutions for those problems that do
not admit a solution in a, say, ordinary space. Of course, a fundamental
requirement for a good extension is that there is no gap between the infimum
of the original problem and that of the extended problem. However, even if the
set of ordinary solutions is $C^{0}$-dense in the set of extended
trajectories, in the presence of constraints an infimum gap does in general
occur, whenever all ordinary solutions in a $C^{0}$-neighborhood of a feasible
extended trajectory (a local extended minimizer, for instance) violate the
constraints. In this case, we will refer to the extended trajectory as
isolated. By defining the original control system controllable to an extended
trajectory whenever the trajectory is not isolated, we see that gap avoidance
and controllability are strictly related issues. Since Warga’s early works
[31, 32], it has emerged that the existence of an infimum gap, or better,
following our terminology, the fact that an extended trajectory is isolated,
is related to the validity of a maximum principle in abnormal form (as
customary, abnormality means that the scalar multiplier associated to the cost
is zero). In particular, results of this kind have been obtained for the
classical extension by relaxation (convex [25, 26] or in measure [32, 15])
and, more recently, for the impulsive extension of control-affine systems with
unbounded controls, with or without state constraints (see [13], [23],
respectively). Let us also mention [24], where a general extension is
considered, but in the absence of state constraints and for smooth data. These
results are obtained by different techniques, which essentially reflect two
different approaches to the maximum principle: approach (a), based on the
construction of approximating cones to reachable sets and on set separation
arguments [31, 32, 15, 24]; approach (b), which makes use of perturbation and
penalization techniques and of the Ekeland’s variational principle [25, 26,
23, 13]. For nonsmooth optimal control problems, methods (a), (b) are not
easily comparable, as they require different assumptions on dynamics and
target but, above all, lead to different abnormality conditions, which involve
the ‘derivative containers’ introduced in [32] or the ‘quasi-differential
quotients’ defined in [24] in case (a), while in case (b) one uses a by now
standard form of the nonsmooth constrained maximum principle due to Clarke,
expressed in terms of subdifferentials (see [7]).

The main purpose of this paper is to extend approach (b), applied so far only
to particular cases, to identify under which general assumptions for the
extension of an optimal control problem the following statement is valid: an
isolated extended trajectory is an abnormal extremal. Furthermore, we give
sufficient conditions for which we prove the stronger result: an isolated
extended trajectory is an abnormal extremal of a nondegenerate version of the
maximum principle.

Precisely, we consider the optimization problem

$(P)\left\\{\parbox[c][56.9055pt]{398.9296pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(y(S))\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
}\\\ &\text{over $(\omega,\alpha,y)\in\mathcal{V}(S)\times\mathcal{A}(S)\times
W^{1,1}([0,S];\mathbb{R}^{n})$, verifying}\\\
&\dot{y}(s)=\mathcal{F}(s,y(s),\omega(s),\alpha(s))\ \text{a.e.,}\quad
y(0)=\check{x}_{0},\\\ &h(s,y(s))\leq 0\quad\forall s\in[0,S],\qquad
y(S)\in\mathcal{T},}\right.$

where $\mathcal{V}(S):=L^{1}([0,S];V)$, $\mathcal{A}(S):=L^{1}([0,S];A)$, and
the extended optimization problem, say $(P_{e})$, which is obtained by $(P)$
replacing in the minimization the control set $\mathcal{V}(S)$ with the larger
set $\mathcal{W}(S):=L^{1}([0,S];W)$, where $W=\overline{V}$. The data
comprise the functions $\Psi:\mathbb{R}^{n}\to\mathbb{R}$,
$\mathcal{F}:\mathbb{R}\times\mathbb{R}^{n}\times W\times A\to\mathbb{R}^{n}$,
$h:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$, the bounded set
$V\subset\mathbb{R}^{m}$, the compact set $A\subset\mathbb{R}^{q}$, and the
closed set $\mathcal{T}\subset\mathbb{R}^{n}$. We refer to any triple
$(\omega,\alpha,y)$, where
$(\omega,\alpha)\in\mathcal{W}(S)\times\mathcal{A}(S)$ and $y$ solves the
Cauchy problem

$\dot{y}(s)=\mathcal{F}(s,y(s),\omega(s),\alpha(s))\quad\text{a.e.,}\qquad
y(0)=\check{x}_{0},$ (1.1)

as extended process or simply process. A process $(\omega,\alpha,y)$ is
feasible if $h(s,y(s))\leq 0$ for all $s\in[0,S]$ and $y(S)\in\mathcal{T}$.
The processes $(\omega,\alpha,y)$ of $(P)$, where $\omega\in\mathcal{V}(S)$,
will be called strict sense processes.

As further extension, we consider the convex relaxation of $(P_{e})$:

$(P_{r})\left\\{\parbox[c][79.6678pt]{396.75938pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(y(S))\\\
&\text{over
$(\underline{\omega},\underline{\alpha},\lambda,y)\in\mathcal{W}^{1+n}(S)\times\mathcal{A}^{1+n}(S)\times\Lambda_{n}(S)\times
W^{1,1}([0,S];\mathbb{R}^{n})$, verifying }\\\
&\dot{y}(s)=\sum_{k=0}^{n}\lambda^{k}(s)\mathcal{F}(s,y(s),\omega^{k}(s),\alpha^{k}(s))\
\text{a.e.,}\quad y(0)=\check{x}_{0},\\\ &h(s,y(s))\leq 0\quad\forall
s\in[0,S],\qquad y(S)\in\mathcal{T},}\right.$

where $\Lambda_{n}(S):=L^{1}([0,S];\Delta_{n})$ and $\Delta_{n}$ is the
$n$-dimensional simplex:

$\Delta_{n}:=\left\\{\lambda=(\lambda^{0},\dots,\lambda^{n}):\ \
\lambda^{k}\geq 0,\ k=0,\dots,n,\ \ \sum_{k=0}^{n}\lambda^{k}=1\right\\}.$

Problem $(P_{r})$ is briefly referred to as the relaxed problem and a process
$(\underline{\omega},\underline{\alpha},\lambda,y)$ for $(P_{r})$ is referred
to as relaxed process. We will identify a process $(\omega,\alpha,y)$ with any
relaxed process $(\underline{\omega},\underline{\alpha},\lambda,y)$, where
$\underline{\omega}:=(\omega,\dots,\omega)$,
$\underline{\alpha}:=(\alpha,\dots,\alpha)$.

The controls $a$ and $w$ play a different role, as only the control set
$\mathcal{V}(S)$ to which $w$ belongs is extended. This distinction is
reflected by the hypotheses on the dynamics $\mathcal{F}$. Referring to
Section 2 for details, we observe that while continuity of $\mathcal{F}$ in
$a$ will be enough, with respect to $w$ a form of uniform continuity will be
needed, both of $\mathcal{F}$ and of its Clarke subdifferential
$\partial^{c}_{x}\mathcal{F}$. Moreover, not only $\overline{V}=W$, but there
must also exist an increasing sequence of closed subsets $V_{i}\subseteq V$
such that $\cup_{i}V_{i}=V$ (in Remark 2.2 below we will discuss possible
extensions of this hypothesis). This formulation of the problem includes as
special cases both the extension by convex relaxation considered in [26] (if
$\mathcal{F}$ does not depend on $w$), and the impulsive, in general non-
convex, extension investigated in [23, 13]. In fact, if the isolated process
that we show to be abnormal extremal belongs to a subclass of relaxed
processes (for example, it is an extended process), as a corollary, the
normality guarantees that there is no gap between $\inf\Psi$ over feasible
strict sense processes and on that subclass. In Theorem 2.1 of Section 2, we
state our first main result, that any isolated feasible relaxed process is an
abnormal extremal. The relevance of this result lies, in fact, in its
consequences, which are: (i) a ‘normality test’ for no gap, namely, if for a
(local) minimizer $\bar{z}$ of $(P_{r})$ or $(P_{e})$ the cost multiplier is
$\neq 0$ for any set of multipliers in the maximum principle, at $\bar{z}$
there is no (local) infimum gap; (ii) the original control system (1.1) is
controllable to any feasible relaxed process which is not an abnormal extremal
(see Theorems 2.2, 2.3 below).

When the state constraint is active at the initial point $(0,\check{x}_{0})$,
it is well-known that sets of degenerate multipliers such that any feasible
relaxed trajectory is abnormal may exist, making the above results (i), (ii)
in fact useless. This ‘degeneracy question’ seems to have been disregarded in
the literature on the relationship between gap and normality, apart from [13],
where, however, conditions are introduced which are never met in the case of a
fixed initial point.

Based on the above considerations, in Section 3 we provide a condition
inspired by the nondegeneracy conditions proposed in [9, 10] ((H4) below),
under which we refine the results of Section 2. In particular, we establish
that any feasible relaxed process which is isolated, is an abnormal extremal
for a nondegenerate maximum principle, and derive as corollaries a
‘nondegenerate normality test’ for no gap and a ‘nondegenerate controllability
condition’ (see Theorems 3.1, 3.2, 3.3 below).

The ‘normality’ and the ‘nondegenerate normality test’ are useful especially
because in certain situations they allow to deduce the absence of gap from
easily verifiable conditions, in the form of constraint and endpoint
qualification conditions for normality, on which there is a wide literature
(see e.g. [11, 12, 16, 2] and references therein). As shown in [23, 22, 13],
where some explicit normality sufficient conditions for the control-affine
impulsive extension are provided, these conditions are in general weaker than
those previously obtained to get the absence of gap directly, as in [1, 18].

In Section 4 we extend the previous results to free end-time optimal control
problems. We limit ourselves to considering the case of Lipschitz continuous
time dependence, leaving the case of measurable time dependence to future
investigations. Actually, Lipschitz continuous time dependence always arises
in the impulsive extension of nonlinear problems with unbounded controls under
the graph-completion approach, to which we apply our results in Section 5.
Impulsive optimal control problems have been extensively studied together with
their applications, mostly in the case of control-affine systems, starting
from [28, 30, 5, 17, 19]. We focus instead on the less investigated case of
control-polynomial dynamics [27, 21]. Among applications for which the
polynomial dependence is relevant let us mention Lagrangian mechanical
systems, possibly with friction forces, where inputs are identified with the
derivatives of some coordinates. In this case, the degree of the polynomial is
2, as a consequence of the fact that the kinetic energy is a quadratic form of
the velocity (see, e.g. [4, 6]).

###  1.1. Notations and preliminaries

Given an interval $I\subseteq\mathbb{R}$ and a set $X\subseteq\mathbb{R}^{k}$,
we write $W^{1,1}(I;X)$ $L^{1}(I;X)$, $L^{\infty}(I;X)$, for the space of
absolutely continuous functions Lebesgue integrable functions, essentially
bounded functions defined on $I$ and with values in $X$, respectively. When
codomain or domain is clear, we will use $\|\cdot\|_{L^{1}(I)}$,
$\|\cdot\|_{L^{\infty}(I)}$, or also $\|\cdot\|_{L^{1}}$,
$\|\cdot\|_{L^{\infty}}$ to denote the $L^{1}$ and the ess-sup norm,
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 $\ell(X)$, ${\rm co}(X)$, Int$(X)$, $\overline{X}$, $\partial X$ the
Lebesgue measure, the convex hull, the interior, the closure, 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 $x\in\mathbb{R}^{k}\setminus X$. For any subset
$Y\subset X$, $\text{{\rm proj}}_{Y}X$ will denote the projection of $X$ on
$Y$. 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 $a\vee
b:=\max\\{a,b\\}$. We use $NBV^{+}([0,S];\mathbb{R})$ to denote 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 by
$\mu([0,S])$, and its support is spt$\\{\mu\\}$.

Some standard constructs from nonsmooth analysis are employed in this paper.
For background material we refer the reader for instance to [7, 29]. 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 limiting normal cone $N_{D}(\bar{x})$ of $D$ at $\bar{x}$
is given by

$N_{D}(\bar{x}):=\left\\{\eta\in\mathbb{R}^{k}\text{ : }\exists
x_{i}\stackrel{{\scriptstyle D}}{{\to}}\bar{x},\,\eta_{i}\to\eta\,\,\text{
such that }\,\,\limsup_{x\to
x_{i}}\frac{\eta_{i}\cdot(x-x_{i})}{|x-x_{i}|}\leq 0\ \ \forall i\right\\},$

in which the notation $x_{i}\stackrel{{\scriptstyle
D}}{{\longrightarrow}}\bar{x}$ is used to indicate that all points in the
converging sequence $(x_{i})_{i}$ lay in $D$. 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

$\partial G(\bar{x}):=\left\\{\xi\text{:
}\exists\xi_{i}\to\xi,\,x_{i}\to\bar{x}\text{ s.t. }\limsup_{x\to
x_{i}}\frac{\xi_{i}\cdot(x-x_{i})-G(x)+G(x_{i})}{|x-x_{i}|}\leq 0\ \ \forall
i\right\\}.$

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. When $G$ is differentiable, $\nabla G$ is the usual
gradient operator and $\nabla_{x}G$, $\nabla_{y}G$ denote the partial
derivatives of $G$. Given a locally Lipschitz continuous function
$G:\mathbb{R}^{k}\to\mathbb{R}$ and $\bar{x}\in\mathbb{R}^{k}$, the reachable
hybrid subdifferential of $G$ at $\bar{x}$ is

$\partial^{*>}G(\bar{x}):=\,\left\\{\xi\text{:
}\exists(x_{i})_{i}\subset\text{diff}(G)\setminus\\{\bar{x}\\}\text{ s.t.
}x_{i}\to\bar{x},\,G(x_{i})>0\,\,\forall i,\ \nabla G(x_{i})\to\xi\right\\}$

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

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

where diff($G$) denotes the set of differentiability points of $G$. 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$\,\partial
G(\bar{x})$. With a small abuse of notation, given a locally Lipschitz
continuous function $G:\mathbb{R}^{k}\to\mathbb{R}^{l}$ and
$\bar{x}\in\mathbb{R}^{k}$, we still write $\partial^{c}G(\bar{x})$ to denote
the Clarke generalized Jacobian, defined as

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

where now $\nabla G$ denotes the classical Jacobian matrix of $G$.

##  2\. Infimum gap, isolated processes and abnormality

In the following, when the final time $S>0$ is clear from the context, we
simply write $\mathcal{V}$, $\mathcal{W}$, $\mathcal{A}$, $\Lambda_{n}$,
instead of $\mathcal{V}(S)$, $\mathcal{W}(S)$, $\mathcal{A}(S)$,
$\Lambda_{n}(S)$, respectively.

###  2.1. Basic assumptions

We shall consider the following hypotheses, in which
$(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$ is
a feasible relaxed process, which we call the reference process and, for some
$\theta>0$, we set

$\Sigma_{\theta}:=\left\\{(s,x)\in\mathbb{R}\times\mathbb{R}^{n}:\ \
s\in[0,S],\ x\in\bar{y}(s)+\theta\,\mathbb{B}\right\\}.$

  * (H1)

There exists a sequence $(V_{i})_{i}$ of closed subsets of $V$ such that
$V_{i}\subseteq V_{i+1}$ for every $i$ and $\bigcup_{i=1}^{+\infty}V_{i}=V$.

  * (H2)

The constraint function $h$ is upper semicontinuous and $K_{{}_{h}}$-Lipschitz
continuous in $x$, uniformly w.r.t. $s$ in $\Sigma_{\theta}$.

  * (H3)

(i) For all $(x,w,a)\in\\{x\in\mathbb{R}^{n}:\ (s,x)\in\Sigma_{\theta}\ \text{
for some $s\in[0,S]$}\\}\times W\times A$, $\mathcal{F}(\cdot,x,w,a)$ is
Lebesgue measurable on $[0,S]$ and for any $(s,x)\in\Sigma_{\theta}$,
$\mathcal{F}(s,x,\cdot,\cdot)$ is continuous on $W\times A$. Moreover, there
exists $k\in L^{1}([0,S];\mathbb{R}_{\geq 0})$ such that, for all $(s,x,w,a)$,
$(s,x^{\prime},w,a)\in\Sigma_{\theta}\times W\times A$, we have

$|\mathcal{F}(s,x,w,a)|\leq
k(s),\qquad|\mathcal{F}(s,x^{\prime},w,a)-\mathcal{F}(s,x,w,a)|\leq
k(s)|x^{\prime}-x|.$

(ii) There exists some continuous increasing function
$\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ with $\varphi(0)=0$ such
that for any $(s,x,a)\in\Sigma_{\theta}\times A$, we have

$\begin{array}[]{l}|\mathcal{F}(s,x,w^{\prime},a)-\mathcal{F}(s,x,w,a)|\leq
k(s)\varphi(|w^{\prime}-w|)\qquad\forall w^{\prime},\,w\in W,\\\
\partial_{x}^{c}\mathcal{F}(s,x,w^{\prime},a)\subseteq\partial_{x}^{c}\mathcal{F}(s,x,w,a)+k(s)\varphi(|w^{\prime}-w|)\,\mathbb{B}\qquad\forall
w^{\prime},\,w\in W.\end{array}$

When hypothesis (H3) is valid for $k\equiv K_{{}_{\mathcal{F}}}$ for some
constant $K_{{}_{\mathcal{F}}}>0$, we will refer to (H3) as (H3)′.

######  Remark 2.1.

Condition (H1) implies (and is in general stronger than) the density of
$\mathcal{V}$ in $\mathcal{W}$ in the $L^{1}$-norm. Indeed, given
$\omega\in\mathcal{W}$, from (H1) it follows that for any $\delta>0$ there
exists some index $i_{\delta}$, such that the Hausdorff distance
$d_{H}(V_{i},W)<\delta/S$ for every $i\geq i_{\delta}$. Hence, by the
selection theorem [3, Theorem 2, p. 91] for such $i$ there is some measurable
function $\omega_{i}(s)\in\text{proj}_{V_{i}}(\omega(s))$ for a.e. $s$, which
thus verifies $\|\omega_{i}-\omega\|_{L^{1}}\leq
S\|\omega_{i}-\omega\|_{L^{\infty}}\leq Sd_{H}(V_{i},W)<\delta$. In
particular, when $\mathcal{V}=L^{1}([0,S];V)$ for some subset $V\subset W$
such that int$(W)\subseteq V\subseteq W$ and $W=\overline{\text{int}(W)}$, the
validity of (H1) follows by elementary properties of closed and open subsets
of $\mathbb{R}^{n}$.

######  Remark 2.2.

As one can easily deduce from the proofs in Section 6 below, condition (H1)
could be replaced by the hypothesis that there exists a subset
$\mathcal{V}\subset\mathcal{W}:=L^{1}([0,S];W)$ which is closed by finite
concatenation and verifies:

(i) there exists an increasing sequence of closed subsets
$(\mathcal{V}_{i})_{i}\subseteq\mathcal{V}$ such that
$\cup_{i}\mathcal{V}_{i}=\mathcal{V}$ and, for any $w\in\mathcal{W}$ and
$\delta>0$, there are $i_{\delta}$ and
$w_{\delta}\in\mathcal{V}_{i_{\delta}}$, such that
$\|w_{\delta}-w\|_{L^{1}}\leq\delta$;

(ii) for every $i$, for the optimization problem obtained from $(P)$ by
replacing $\mathcal{V}$ with $\mathcal{V}_{i}$, a nonsmooth constrained
maximum principle is valid.

For example, from [14] a condition sufficient for (ii) to hold true is the
$C^{0}$-closure of the set of the solutions to (1.1) as
$(\omega,\alpha)\in\mathcal{V}_{i}\times\mathcal{A}$, for every $i$.

######  Remark 2.3.

Condition (H3)(ii) is satisfied, for instance, when
$\mathcal{F}(s,x,w,a)=\mathcal{F}_{1}(s,x,a)+\mathcal{F}_{2}(s,x,w,a)$, where
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$ verify hypothesis (H3)(i) and, in
addition, the function $\mathcal{F}_{2}(s,\cdot,w,a)$ is $C^{1}$ and
$\nabla_{x}\mathcal{F}_{2}$ is continuous on the compact set
$\Sigma_{\theta}\times W\times A$.

###  2.2. Infimum gap and isolated processes

Let us write $\Gamma$, $\Gamma_{e}$, $\Gamma_{r}$, to denote the sets of
strict sense, extended, and relaxed processes which are feasible,
respectively. As observed in the Introduction, $\Gamma$, $\Gamma_{e}$, can be
identified with subsets of $\Gamma_{r}$.

######  Definition 2.1 (Minimizer).

A process
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})\in\Gamma_{g}$,
$g\in\\{r,e\\}$, is called a local $\Psi$-minimizer for problem $(P_{g})$ if,
for some $\delta>0$, one has

$\Psi(\bar{y}(S))=\inf\left\\{\Psi(y(S)):\ \
(\underline{\omega},\underline{\alpha},\lambda,y)\in\Gamma_{g},\ \
\|y-\bar{y}\|_{L^{\infty}}<\delta\right\\}.$

The process $\bar{z}$ is a $\Psi$-minimizer for problem $(P_{g})$ if
$\displaystyle\Psi(\bar{y}(S))=\inf_{\Gamma_{g}}\Psi(y(S))$.

######  Definition 2.2 (Infimum gap).

Fix
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})\in\Gamma_{r}$.

(i) Let $\Psi:\mathbb{R}^{n}\to\mathbb{R}$ be a continuous function. When
there is some $\delta>0$ such that

$\displaystyle\Psi(\bar{y}(S))<\,\inf\left\\{\Psi(y(S)):\ \
(\omega,\alpha,y)\in\Gamma,\ \ \|y-\bar{y}\|_{L^{\infty}}<\delta\right\\}$

(as customary, when the set is empty we set the infimum $=+\infty$), we say
that at $\bar{z}$ there is a local $\Psi$-infimum gap. We say that there is a
$\Psi$-infimum gap with problem $(P_{g})$, $g\in\\{r,e\\}$, if
$\displaystyle\inf_{\Gamma_{g}}\Psi(y(S))<\,\inf_{\Gamma}\Psi(y(S)).$

(ii) When at $\bar{z}$ there is a local $\Psi$-infimum gap or if there is a
$\Psi$-infimum gap (with $(P_{e})$ or $(P_{r})$) for some $\Psi$ as above, we
say that at $\bar{z}$ there is a local infimum gap or that there is an infimum
gap with $(P_{e})$ or $(P_{r})$, respectively.

Obviously, a $\Psi$-infimum gap with $(P_{e})$ implies a $\Psi$-infimum gap
with $(P_{r})$, and it may happen that
$\displaystyle\inf_{\Gamma_{r}}\Psi(y(S))<\,\inf_{\Gamma_{e}}\Psi(y(S))<\,\inf_{\Gamma}\Psi(y(S)).$

The notion of local infimum gap at $\bar{z}$ is equivalent to the notion of
isolated process, first introduced in [23], which is independent of any
optimization problem.

######  Definition 2.3 (Isolated process and controllability).

We say that a process
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})\in\Gamma_{r}$
is isolated (in $\Gamma$) if, for some $\delta>0$, one has

$\left\\{(\omega,\alpha,y)\in\Gamma:\ \
\|y-\bar{y}\|_{L^{\infty}}<\delta\right\\}=\emptyset.$

The control system (1.1) is said controllable to $\bar{z}$ if $\bar{z}$ is not
isolated.

######  Proposition 2.1.

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})\in\Gamma_{r}$.
Then

  * (i)

if $\bar{z}$ is isolated, then at $\bar{z}$ there is a local infimum gap and,
for some $\delta>0$, one has $\inf\big{\\{}\Psi(y(S)):$
$(\omega,\alpha,y)\in\Gamma,\ \
\|y-\bar{y}\|_{L^{\infty}}<\delta\big{\\}}=+\infty$ for every continuous
$\Psi$;

  * (ii)

if at $\bar{z}$ there is a local infimum gap, then $\bar{z}$ is isolated.

As a consequence, $\bar{z}$ is isolated if and only if at $\bar{z}$ there is a
local infimum gap.

###### Proof.

The proof of (i) is trivial, hence we limit ourselves to prove (ii). Suppose
that at $\bar{z}$ there is a $\Psi$-local infimum gap for some continuous
$\Psi$ and some $\delta>0$, but $\bar{z}$ is not isolated. Then, for every
$i\in\mathbb{N}$, $i\geq\frac{1}{\delta}$, there exists a feasible strict
sense process $z_{i}=(\omega_{i},\alpha_{i},y_{i})\in\Gamma$ such that
$\|y_{i}-\bar{y}\|_{L^{\infty}}<\frac{1}{i}$, and, by the continuity of
$\Psi$,

$\Psi(\bar{y}(S))<\inf\left\\{\Psi(y(S)):\,(\omega,\alpha,y)\in\Gamma,\,\|y-\bar{y}\|_{L^{\infty}}<\delta\right\\}\leq\lim_{i}\Psi(y_{i}(S)),=\Psi(\bar{y}(S))$

which gives the desired contradiction. ∎

Incidentally, if at $\bar{z}$ there is a local $\Psi$-infimum gap for some
$\Psi$, for some $\delta>0$ there is in fact a local $\tilde{\Psi}$-infimum
gap and $\inf\big{\\{}\tilde{\Psi}(y(S)):\ (\omega,\alpha,y)\in\Gamma,\
\|y-\bar{y}\|_{L^{\infty}}<\delta\big{\\}}=+\infty$ for every continuous
function $\tilde{\Psi}$.

###  2.3. Abnormality and infimum gap

We introduce a notion of normal and abnormal extremal for the relaxed
optimization problem.

######  Definition 2.4 (Normal and abnormal extremal).

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
be a feasible relaxed process. Given a function
$\Psi:\mathbb{R}^{n}\to\mathbb{R}$ which is Lipschitz continuous on a
neighborhood of $\bar{y}(S)$, we say that $\bar{z}$ is a $\Psi$-extremal if
there exist a path $p\in W^{1,1}([0,S];\mathbb{R}^{n})$, $\gamma\geq 0$,
$\mu\in NBV^{+}([0,S];\mathbb{R})$, $m:[0,S]\to\mathbb{R}^{n}$ Borel
measurable and $\mu$-integrable function, verifying the following conditions:

$\displaystyle\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\gamma\neq 0;$ (2.2)
$\displaystyle-\dot{p}(s)\in\sum_{k=0}^{n}\bar{\lambda}^{k}(s)\,{\rm
co}\,\partial_{x}\left\\{q(s)\cdot\mathcal{F}(s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))\right\\}\
\text{ a.e.;}$ (2.3)
$\displaystyle-q(S)\in\gamma\partial\Psi\left(\bar{y}(S)\right)+N_{\mathcal{T}}(\bar{y}(S));$
(2.4) for every $k=0,\dots,n$, for a.e. $s\in[0,S]$, one has $\displaystyle
q(s)\cdot\mathcal{F}\big{(}s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s)\big{)}=\max_{(w,a)\in
W\times A}q(s)\cdot\mathcal{F}\big{(}s,\bar{y}(s),w,a\big{)};$ (2.5)
$\displaystyle
m(s)\in\partial_{x}^{>}\,h\left(s,\bar{y}(s)\right)\qquad\text{$\mu$-a.e.;}$
(2.6) $\displaystyle spt(\mu)\subseteq\\{s\in[0,S]\text{ :
}h\left(s,\bar{y}(s)\right)=0\\},$ (2.7)

where

$q(s):=\begin{cases}p(s)+\int_{[0,s[}m(\sigma)\mu(d\sigma)\qquad\,\,s\in[0,S[,\\\
p(S)+\int_{[0,S]}m(\sigma)\mu(d\sigma)\qquad s=S.\end{cases}$

We will call a $\Psi$-extremal normal if all possible choices of
$(p,\gamma,\mu,m)$ as above have $\gamma>0$, and abnormal when it is not
normal. Since the notion of abnormal $\Psi$-extremal is actually independent
of $\Psi$, in the following abnormal $\Psi$-extremals will be simply called
abnormal extremals.

######  Theorem 2.1.

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
be a feasible relaxed process. Assume that hypotheses (H1)–(H3) are verified.
If at $\bar{z}$ there is a local infimum gap, then $\bar{z}$ is an abnormal
extremal.

A first remarkable immediate consequence of Theorem 2.1 is the following
sufficient condition for the absence of an infimum gap.

######  Theorem 2.2.

Suppose that there exists a [local] $\Psi$-minimizer
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
for $(P_{e})$ or $(P_{r})$, for which hypotheses (H1)–(H3) are verified,
$\Psi$ is Lipschitz continuous in a neighborhood of $\bar{y}(S)$, and which is
a normal $\Psi$-extremal. Then, [at $\bar{z}$] there is no [local]
$\Psi$-infimum gap with $(P_{e})$ or $(P_{r})$, respectively.

######  Remark 2.4.

By a well known constrained maximum principle (see [29, Ch. 9]), local
$\Psi$-minimizers of $(P_{r})$ are $\Psi$-extremal in a stronger form than in
Definition 2.4, in which the costate differential inclusion (2.3) is replaced
by

$-\dot{p}(s)\in\,{\rm
co}\,\partial_{x}\left\\{\sum_{k=0}^{n}\bar{\lambda}^{k}(s)\,q(s)\cdot\mathcal{F}(s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))\right\\}\quad\text{for
a.e. $s\in[0,S]$.}$ (2.8)

The need to consider (2.3) derives from the perturbation technique used in the
proof of Theorem 2.1 (see also [25]). In fact, (2.3) may differ from (2.8)
only in case of nonsmooth dynamics. Precisely, if
$\mathcal{F}(s,\cdot,\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))$ is continuously
differentiable at $\bar{y}(s)$, for all $k=0,\dots,n$ and a.e. $s\in[0,S]$,
then both differential inclusions reduce to the adjoint equation

$-\dot{p}(s)=\sum_{k=0}^{n}\bar{\lambda}^{k}(s)\,q(s)\cdot\mathcal{F}_{x}(s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))\quad\text{for
a.e. $s\in[0,S]$.}$

Thanks to Proposition 2.1, from Theorem 2.1 we can also deduce the following
sufficient condition for controllability to the reference trajectory.

######  Theorem 2.3.

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
be a feasible relaxed process and assume that hypotheses (H1)–(H3) are
verified. Then, either

  * (i)

$\bar{z}$ is not isolated in $\Gamma$, namely, there exists a sequence of
feasible processes $(\omega_{i},\alpha_{i},y_{i})\in\Gamma$ such that
$\|y_{i}-\bar{y}\|_{L^{\infty}}\to 0$ as $i\to+\infty$; or

  * (ii)

$\bar{z}$ is an abnormal extremal.

###### Proof.

Suppose by contradiction statements (i), (ii) false, namely, let $\bar{z}$ be
an isolated process which is not an abnormal extremal. Then, by Proposition
2.1 at $\bar{z}$ there is a local infimum gap and $\bar{z}$ should be an
abnormal extremal by Theorem 2.1. ∎

The proof of Theorem 2.1 is given in Section 6.

##  3\. Infimum gap and nondegenerate abnormality

Disregarding the endpoint constraint, when the state constraint is active at
the initial time there always exist sets of degenerate multipliers, as, for
instance, $\gamma=0$, $\mu=\delta_{\\{0\\}}$,
$p(s)=-m(0)\in\partial^{>}h(0,\check{x}_{0})$ for all $s\in[0,S]$.111For any
$r\in\mathbb{R}$, $\delta_{\\{r\\}}$ is the Dirac unit measure concentrated at
$r$. With the degenerate multipliers, the maximum principle is clearly
useless, not only to select minimizers, but also to identify no-gap conditions
in the form of a ‘normality test’.

######  Definition 3.1 (Nondegenerate normal and abnormal extremal).

Assume that
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
is a feasible relaxed process. Given a function
$\Psi:\mathbb{R}^{n}\to\mathbb{R}$ which is Lipschitz continuous on a
neighborhood of $\bar{y}(S)$, let $\bar{z}$ be a $\Psi$-extremal. We call
nondegenerate multiplier any set of multipliers $(p,\gamma,\mu,m)$ that meets
the conditions of Definition 2.4, and also verifies the following strengthened
nontriviality condition

$\mu(]0,S])+\|q\|_{L^{\infty}}+\gamma\neq 0,$ (3.9)

where $q$ is as in Definition 2.4. We call $\bar{z}$ a nondegenerate normal
$\Psi$-extremal if all possible choices of nondegenerate multipliers have
$\gamma>0$, and a nondegenerate abnormal $\Psi$-extremal when there exists at
least one nondegenerate multiplier with $\gamma=0$. Since nondegenerate
abnormal $\Psi$-extremals do not depend on $\Psi$, in the following we will
call them simply nondegenerate abnormal extremals.

As it is easy to see, a nondegenerate abnormal extremal is always an abnormal
extremal, and, on the contrary, any normal $\Psi$-extremal is also
nondegenerate normal. However, we may have situations where a nondegenerate
normal $\Psi$-extremal, is not a normal $\Psi$-extremal, as illustrated in
Example 5.1 below.

To introduce our constraint qualification conditions, we define
$\Lambda_{n}^{1}\equiv\Lambda_{n}^{1}(S)$, as

$\displaystyle\Lambda_{n}^{1}:=L^{1}([0,S];\Delta^{1}_{n}),\
\Delta^{1}_{n}:=\cup_{k=0}^{n}\,\\{e_{k}\\}\ \ ({e}_{0},\dots,{e}_{n}\text{
canonical basis of }\mathbb{R}^{1+n}),$ (3.10)

and extend the relaxed control system by introducing a new variable, $\xi$.
Precisely, for any
$(\underline{\omega},\underline{\alpha},\lambda)\in\mathcal{W}^{1+n}\times\mathcal{A}^{1+n}\times\Lambda_{n}$,
we denote by $(\xi,y)[\underline{\omega},\underline{\alpha},\lambda]$ the
solution to

$\left\\{\begin{array}[]{l}(\dot{\xi},\dot{y})(s)=\Big{(}\lambda(s),\,\sum_{k=0}^{n}\lambda^{k}(s)\mathcal{F}(s,y(s),\omega^{k}(s),\alpha^{k}(s))\Big{)}\
\text{a.e.,}\\\ (\xi,y)(0)=(0,\check{x}_{0}),\end{array}\right.$ (3.11)

and with a small abuse of notation, in the following we call
$(\underline{\omega},\underline{\alpha},\lambda,\xi,y)$ with
$(\xi,y):=(\xi,y)[\underline{\omega},\underline{\alpha},\lambda]$, a relaxed
process. Observe that, when
$(\underline{\omega},\underline{\alpha})=(\omega,\dots,\omega,\alpha,\dots,\alpha)$
for some $(\omega,\alpha)\in\mathcal{W}\times\mathcal{A}$, the $y$-component
of the solution $(\xi,y)[\underline{\omega},\underline{\alpha},\lambda]$ is in
fact an extended trajectory, namely, it solves the original problem (1.1).

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
be a feasible relaxed process. Define the constraint set

$\Omega:=\\{(s,x)\in\mathbb{R}^{1+n}:\ \ h(s,x)\leq 0\\}.$

We shall consider the following hypotheses.

  * (H4)

If $(0,\check{x}_{0})\in\partial\Omega$, there exist some $\tilde{\delta}>0$,
$\bar{s}\in]0,S]$, sequences
$(\tilde{\omega}_{i},\tilde{\alpha}_{i},{\tilde{\lambda}}_{i})_{i}\subset(\mathcal{W}\cap\mathcal{V}(\bar{s}))\times\mathcal{A}\times\Lambda_{n}^{1}$,
$(\hat{\omega}_{i},\hat{\alpha}_{i})_{i}\subset\mathcal{V}(\bar{s})\times\mathcal{A}(\bar{s})$,
and $(\tilde{r}_{i})_{i}\subset L^{1}([0,S];\mathbb{R}_{\geq 0})$ with
$\lim_{i\to+\infty}\|\tilde{r}_{i}\|_{L^{1}}=0$, such that the following
properties (i)–(iv), where
$(\tilde{\xi}_{i},\tilde{y}_{i}):=(\xi,y)[\tilde{\omega}_{i},\dots,\tilde{\omega}_{i},\tilde{\alpha}_{i},\dots,\tilde{\alpha}_{i},{\tilde{\lambda}}_{i}]$,
are verified.

    * (i)

One has

$\lim_{i\to+\infty}\|(\tilde{\xi}_{i},\tilde{y}_{i})-(\bar{\xi},\bar{y})\|_{L^{\infty}}=0;$
(3.12)

    * (ii)

for every $i$, one has

$h(s,\tilde{y}_{i}(s))\leq 0\qquad\forall s\in[0,\bar{s}];$ (3.13)

    * (iii)

for every $i$, there is a Lebesgue measurable subset
$\tilde{E}_{i}\subset[0,S]$ such that

$\begin{array}[]{c}(\tilde{\omega}_{i},\tilde{\alpha}_{i},\tilde{\lambda}_{i})(s)\in\bigcup_{k=0}^{n}\\{(\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s),e^{k})\\}+(\tilde{r}_{i}(s),0,0)\mathbb{B},\
\text{a.e. $s\in\tilde{E}_{i}$;}\\\\[6.45831pt]
\lim_{i\to+\infty}\ell(\tilde{E}_{i})=S;\end{array}$ (3.14)

    * (iv)

for every $i$ and for all
$(\zeta_{0},\zeta)\in\partial^{*}h(0,\check{x}_{0})$, for a.e.
$s\in[0,\bar{s}]$ one has

$\begin{array}[]{l}\displaystyle\zeta\cdot\big{[}\mathcal{F}(s,\check{x}_{0},(\hat{\omega}_{i},\hat{\alpha}_{i})(s))-\mathcal{F}(s,\check{x}_{0},(\tilde{\omega}_{i},\tilde{\alpha}_{i})(s))\big{]}\leq-\tilde{\delta}.\end{array}$
(3.15)

######  Remark 3.1.

Some comments on hypothesis (H4) are in order.

  * (1)

It prescribes additional conditions to assumptions (H1)–(H3) only when the
initial point $(0,\check{x}_{0})$ lies on the boundary of the constraint set
$\Omega$. Incidentally, this is not equivalent to having
$h(0,\check{x}_{0})=0$, as it may clearly happen that $h(0,\check{x}_{0})=0$
but $(0,\check{x}_{0})\in$Int$(\Omega)$.

  * (2)

When $(0,\check{x}_{0})\in\partial\Omega$, the first part of hypothesis (H4)
substantially requires the existence of strict sense processes that
approximate the reference process and satisfy the state constraint on some
(small) interval $[0,\bar{s}]$, with controls which are close to controls
$(\bar{\omega}_{i},\bar{\alpha}_{i},\bar{\lambda}_{i})$ belonging to
$\bigcup_{k=0}^{n}\\{(\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s),e^{k})\\}$ for
a.e. $s\in[0,S]$. Let us point out that, disregarding the state constraint
(3.13), the existence of approximating controls that satisfy the remaining
conditions (3.12), (3.14) follows by the relaxation Theorem together with
hypothesis (H1), as we will see in the proof of Theorem 2.1 below, in
Subsection 6.2. Relation (3.15), on the other hand, is an adaptation of known
constraint qualification conditions (see e.g. [9, 10]), in which the reference
(relaxed) control is replaced by approximating strict sense controls
$(\tilde{\omega}_{i},\tilde{\alpha}_{i})\in\mathcal{V}(\bar{s})\times\mathcal{A}(\bar{s})$.

  * (3)

If hypotheses (H1), (H2), and (H3)′ with reference to $\bar{z}$ are verified,
then in hypothesis (H4) one can assume that the control sequence
$(\hat{\omega}_{i},\hat{\alpha}_{i})_{i}$ belongs to the extended control set
$\mathcal{W}(\bar{s})\times\mathcal{A}(\bar{s})$ rather than
$\mathcal{V}(\bar{s})\times\mathcal{A}(\bar{s})$. Indeed, using the notation
of (H1)–(H3)′, let us choose some $\rho>0$ such that
$K_{h}\,K_{\mathcal{F}}\varphi(\rho)\leq\frac{\tilde{\delta}}{2}$, and let
$j\in\mathbb{N}$ verify $d_{H}(V_{j},W)\leq\rho$. Hence, for every
$i\in\mathbb{N}$ there exists a measurable selection
$\hat{\omega}_{i}^{*}(s)\in\text{proj}_{V_{j}}(\hat{\omega}_{i}(s))$ for a.e.
$s\in[0,S]$, such that
$\|\hat{\omega}_{i}^{*}-\hat{\omega}_{i}\|_{L^{\infty}}\leq\rho$ (see also
Remark 2.1), and, for all $(\zeta_{0},\zeta)\in\partial^{*}h(0,\check{x}_{0})$
(by adding and subtracting
$\zeta\cdot\mathcal{F}(s,\check{x}_{0},(\hat{\omega}_{i}^{*},\hat{\alpha}_{i})(s))$),
one has

$\begin{array}[]{l}\zeta\cdot\big{[}\mathcal{F}(s,\check{x}_{0},(\hat{\omega}_{i}^{*},\hat{\alpha}_{i})(s))-\mathcal{F}(s,\check{x}_{0},(\tilde{\omega}_{i},\tilde{\alpha}_{i})(s))\big{]}\leq-\frac{\tilde{\delta}}{2},\quad\text{a.e.
$s\in[0,\bar{s}]$},\end{array}$

as soon as $(\hat{\omega}_{i},\hat{\alpha}_{i})$ satisfies (3.15).

  * (4)

When hypothesis (H3)′ is verified, then the upper semicontinuity of the set
valued map $\partial^{*}h(\cdot,\cdot)$ and (3.15) in (H4) imply that there
exist $\delta$, $\varepsilon>0$ such that for any
$(\zeta_{0},\zeta)\in\partial^{*}h(\sigma,x)$ with $\sigma\in[0,\varepsilon]$
and $x\in\\{\check{x}_{0}\\}+\varepsilon\mathbb{B}$, for any $s\leq\bar{s}$,
for any continuous path $y:[0,s]\to\\{\check{x}_{0}\\}+\varepsilon\mathbb{B}$
and for any measurable map $\eta:[0,s]\to\\{0,1\\}$, the following integral
condition holds:

$\int_{0}^{s}\eta(\sigma)\,\zeta\cdot\big{[}\mathcal{F}(\sigma,y,\hat{\omega}_{i},\hat{\alpha}_{i})(\sigma)-\mathcal{F}(\sigma,y,\tilde{\omega}_{i},\tilde{\alpha}_{i})(\sigma)\big{]}d\sigma\leq-\delta\,\ell(s,\eta(\cdot)),$
(3.16)

where

$\ell(s,\eta(\cdot)):=\ell(\\{\sigma\in[0,s]\,:\,\eta(\sigma)=1\\}).$ (3.17)

In particular, relation (3.16) holds for any
$(\zeta_{0},\zeta)\in\partial^{c}h(\sigma,x)$, as the scalar product is
bilinear, and for all $\zeta\in\partial_{x}^{>}h(\sigma,x)$, since (see e.g.
[29, Th. 5.3.1]):

$\partial_{x}^{>}h(\sigma,x)\subseteq\partial_{x}^{c}h(\sigma,x)\subseteq\\{\zeta\text{
: }\exists\zeta_{0}\text{ s.t.
}(\zeta_{0},\zeta)\in\partial^{c}h(\sigma,x)\\}\qquad\forall(\sigma,x)\in\mathbb{R}^{1+n}.$

Relation (3.16) is in fact the condition used in the proof of Theorem 3.1
below.

  * (5)

When hypothesis (H3)′ is verified, it is not difficult to verify that
condition (3.16) still holds if we replace (H4), (iv) with the following
assumption:

    * (iv)′

there exists $\tilde{\varepsilon}>0$ such that, for every $i$, for all
$\zeta\in\partial_{x}^{c}h(\sigma,x)$ with $\sigma\in[0,\tilde{\varepsilon}]$
and $x\in\\{\check{x}_{0}\\}+\tilde{\varepsilon}\mathbb{B}$, for a.e.
$s\in[0,\bar{s}]$ one has

$\begin{array}[]{l}\displaystyle\zeta\cdot\big{[}\mathcal{F}(s,\check{x}_{0},(\hat{\omega}_{i},\hat{\alpha}_{i})(s))-\mathcal{F}(s,\check{x}_{0},(\tilde{\omega}_{i},\tilde{\alpha}_{i})(s))\big{]}\leq-\tilde{\delta}.\end{array}$
(3.18)

Theorem 2.1 can be refined as follows:

######  Theorem 3.1.

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
be a feasible relaxed process. Assume that hypotheses (H1)-(H2)-(H3)′-(H4) are
verified. If at $\bar{z}$ there is a local infimum gap, then $\bar{z}$ is a
nondegenerate abnormal extremal.

As in the previous section, from Theorem 3.1 one can derive the following
results.

######  Theorem 3.2.

Suppose that there exists a [local] $\Psi$-minimizer
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
for $(P_{e})$ or $(P_{r})$, for which (H1)-(H2)-(H3)′-(H4) are verified and
$\Psi$ is Lipschitz continuous in a neighborhood of $\bar{y}(S)$. If $\bar{z}$
is a nondegenerate normal $\Psi$-extremal, then, [at $\bar{z}$] there is no
[local] infimum gap with $(P_{e})$ or $(P_{r})$, respectively.

######  Theorem 3.3.

Let
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
be a feasible relaxed process and assume that hypotheses (H1)-(H2)-(H3)′-(H4)
are verified. Then, either

  * (i)

$\bar{z}$ is not isolated in $\Gamma$, namely, there exists a sequence of
feasible processes $(\omega_{i},\alpha_{i},y_{i})\in\Gamma$ such that
$\|y_{i}-\bar{y}\|_{L^{\infty}}\to 0$ as $i\to+\infty$; or

  * (ii)

$\bar{z}$ is a nondegenerate abnormal extremal.

The proof of Theorem 3.1 is given in Section 6.

######  Remark 3.2.

As it will be clear from the proofs in Section 6, Theorems 3.1, 3.2, 3.3 –as
well as Theorems 2.1, 2.2, 2.3– remain true if we replace the fixed set of
control values $A$ with a compact Borel measurable multifunction
$A:[0,S]\rightsquigarrow\mathbb{R}^{q}$.

##  4\. Free end-time problems with Lipschitz time dependence

We consider the optimization problem

$(P^{*})\left\\{\parbox[c][56.9055pt]{390.25534pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(S,y(S))\\\
&\text{over the set of $S>0$ and
$(\omega,\alpha,y)\in\mathcal{V}(S)\times\mathcal{A}(S)\times
W^{1,1}([0,S];\mathbb{R}^{n})$},\\\ &\text{verifying the Cauchy problem
\eqref{E} and the constraints}\\\ &h(s,y(s))\leq 0\quad\forall
s\in[0,S],\qquad(S,y(S))\in\mathcal{T}^{*},}\right.$

where $\mathcal{T}^{*}$ is a closed subset of $\mathbb{R}^{1+n}$ and
$\Psi:\mathbb{R}^{1+n}\to\mathbb{R}$. An extended process or process is an
element $(S,\omega,\alpha,y)$, where $S>0$,
$(\omega,\alpha)\in\mathcal{W}(S)\times\mathcal{A}(S)$, and $y$ solves the
Cauchy problem (1.1). When $\omega\in\mathcal{V}(S)$, the process is called a
strict sense process. A process $(S,\omega,\alpha,y)$ is feasible if
$h(s,y(s))\leq 0$ for all $s\in[0,S]$ and $(S,y(S))\in\mathcal{T}^{*}$. We
call extended problem, and write $(P^{*}_{e})$, the problem of minimizing
$\Psi(S,y(S))$ over the set of feasible extended processes.

The associated relaxed problem is

$(P^{*}_{r})\left\\{\parbox[c][76.82234pt]{390.25534pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(S,y(S))\\\
&\text{over $S>0$,
$(\underline{\omega},\underline{\alpha},\lambda,y)\in\mathcal{W}^{1+n}(S)\times\mathcal{A}^{1+n}(S)\times\Lambda_{n}(S)\times
W^{1,1}([0,S];\mathbb{R}^{n})$ s.t.}\\\
&\dot{y}(s)=\sum_{k=0}^{n}\lambda^{k}(s)\mathcal{F}(s,y(s),\omega^{k}(s),\alpha^{k}(s))\
\text{a.e. $s\in[0,S]$,}\quad y(0)=\check{x}_{0},\\\ &h(s,y(s))\leq
0\quad\forall s\in[0,S],\qquad(S,y(S))\in\mathcal{T}^{*},}\right.$

A process $(S,\underline{\omega},\underline{\alpha},\lambda,y)$ for
$(P^{*}_{r})$ is referred to as relaxed process. As in the previous sections,
we can identify the set of strict sense processes [extended processes] with
the subset of relaxed processes
$(S,\underline{\omega},\underline{\alpha},\lambda,y)$ with
$(\underline{\omega},\underline{\alpha},\lambda)\in\mathcal{V}^{1+n}(S)\times\mathcal{A}^{1+n}(S)\times\Lambda_{n}^{1}(S)$
[$(\underline{\omega},\underline{\alpha},\lambda)\in\mathcal{W}^{1+n}(S)\times\mathcal{A}^{1+n}(S)\times\Lambda_{n}^{1}(S)$].
We will use $\Gamma^{*}$, $\Gamma^{*}_{e}$, $\Gamma^{*}_{r}$ to denote the
sets of feasible strict sense, feasible extended, and feasible relaxed
processes, respectively.

Throughout this section, we strengthen hypotheses (H2)-(H3) treating time as a
state variable. As in Section 3, we add to $(P^{*}_{r})$ the variable
$\xi(s)=\int_{0}^{s}\lambda(s^{\prime})\,ds^{\prime}$, $s\in[0,S]$ and call
relaxed process any element
$(S,\underline{\omega},\underline{\alpha},\lambda,\xi,y)$, where
$(\underline{\omega},\underline{\alpha},\lambda)\in\mathcal{W}^{1+n}(S)\times\mathcal{A}^{1+n}(S)\times\Lambda_{n}(S)$
and $(\xi,y):=(\xi,y)[\underline{\omega},\underline{\alpha},\lambda]$ on
$[0,S]$.

We shall consider the following hypotheses, in which
$(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
is a given feasible relaxed process for $(P^{*}_{r})$ and, for some
$\theta>0$, we set

$\Sigma^{*}_{\theta}:=\left\\{(t,x)\in\mathbb{R}\times\mathbb{R}^{n}:\ \
(t,x)\in(s,\bar{y}(s))+\theta\,\mathbb{B},\ \ s\in[0,\bar{S}]\right\\}.$

  * (H2)∗

The constraint function $h$ is $K_{{}_{h}}$-Lipschitz continuous in
$\Sigma^{*}_{\theta}$.

  * (H3)∗

(i) The function $\mathcal{F}$ is continuous on $\Sigma^{*}_{\theta}\times
W\times A$. Furthermore, there is some constant $K_{\mathcal{F}}>0$ such that,
for all $(s,x,w,a)$, $(s^{\prime},x^{\prime},w,a)\in\Sigma^{*}_{\theta}\times
W\times A$:

$|\mathcal{F}(s^{\prime},x^{\prime},w,a)-\mathcal{F}(s,x,w,a)|\leq
K_{\mathcal{F}}|(s^{\prime},x^{\prime})-(s,x)|.$

(ii) There exists some continuous increasing function
$\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ with $\varphi(0)=0$ such
that for any $(s,x,a)\in\Sigma^{*}_{\theta}\times A$, we have

$\partial_{t,x}^{c}\mathcal{F}(s,x,w^{\prime},a)\subseteq\partial_{t,x}^{c}\mathcal{F}(s,x,w,a)+\varphi(|w^{\prime}-w|)\,\mathbb{B}\qquad\forall
w^{\prime},\,w\in W.$

Identify a continuous function $z:[0,\tau]\to\mathbb{R}^{k}$ with its
extension to $\mathbb{R}$, by constant extrapolation of the left and right
endpoint values. Then, for all $\tau_{1}$, $\tau_{2}>0$, and $(z_{1},z_{2})\in
C^{0}([0,\tau_{1}];\mathbb{R}^{k})\times C^{0}([0,\tau_{2}];\mathbb{R}^{k})$,
we define the distance

$d_{\infty}\big{(}(\tau_{1},z_{1}),(\tau_{2},z_{2})\big{)}:=|\tau_{2}-\tau_{1}|+\|z_{2}-z_{1}\|_{\infty}\qquad(\|\cdot\|_{\infty}:=\|\cdot\|_{L^{\infty}(\mathbb{R})}).$
(4.19)

We can now extend the concepts of local minimizer, local infimum gap, and
isolated process to free end-time problems, by ‘formally replacing
trajectories $y$ and $L^{\infty}$-norm over trajectories with pairs $(S,y)$
endowed with the distance $d_{\infty}$’. For instance, if
$\bar{z}:=(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
is a feasible relaxed process, at $\bar{z}$ there is a local infimum gap if
there is some $\delta>0$ such that

$\displaystyle\Psi(\bar{S},\bar{y}(S))<\,\inf\left\\{\Psi(S,y(S)):\ \
(S,\omega,\alpha,y)\in\Gamma^{*},\ \
d_{\infty}\big{(}(S,y),(\bar{S},\bar{y})\big{)}<\delta\right\\},$

for some continuous function $\Psi:\mathbb{R}^{1+n}\to\mathbb{R}$, while
$\bar{z}$ is an isolated process if

$\left\\{(S,\omega,\alpha,y)\in\Gamma^{*}:\ \
d_{\infty}\big{(}(S,y),(\bar{S},\bar{y})\big{)}<\delta\right\\}=\emptyset.$

for some $\delta>0$. As in the case with fixed end-time, at $\bar{z}$ there is
a local infimum gap if and only if $\bar{z}$ is isolated.

######  Definition 4.1 (Extremal and nondegenerate extremal).

Let
$\bar{z}:=(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{y})$
be a feasible relaxed process and assume that hypotheses (H2)∗, (H3)∗ are
verified. Given a function $\Psi:\mathbb{R}^{1+n}\to\mathbb{R}$ which is
Lipschitz continuous on a neighborhood of $(\bar{S},\bar{y}(\bar{S}))$, we say
that $\bar{z}$ is a $\Psi$-extremal if there exist a pair of paths
$(p_{*},p)\in W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n})$, $\gamma\geq 0$, $\mu\in
NBV^{+}([0,\bar{S}];\mathbb{R})$, $(m_{*},m):[0,\bar{S}]\to\mathbb{R}^{1+n}$
Borel measurable and $\mu$-integrable functions, verifying the following
conditions:

$\displaystyle\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\gamma\neq 0;$ (4.20)
$\displaystyle\big{(}\dot{p}_{*},-\dot{p}\big{)}(s)\in\sum_{k=0}^{n}\bar{\lambda}^{k}(s)\,{\rm
co}\,\partial_{t,x}\left\\{q(s)\cdot\mathcal{F}(s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))\right\\}\
\text{a.e. $s\in[0,\bar{S}]$;}$
$\displaystyle\big{(}q_{*}(\bar{S}),-q(\bar{S})\big{)}\in\gamma\partial\Psi\left(S,\bar{y}(S)\right)+N_{\mathcal{T}}(\bar{S},\bar{y}(\bar{S}));$
for every $k=0,\dots,n$, for a.e. $s\in[0,\bar{S}]$, one has $\displaystyle
q(s)\cdot\mathcal{F}\Big{(}s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s)\Big{)}=\max_{(w,a)\in
W\times A}q(s)\cdot\mathcal{F}\Big{(}s,\bar{y}(s),w,a\Big{)};$ (4.21)
$\displaystyle\sum_{k=0}^{n}\bar{\lambda}^{k}(s)q(s)\cdot\mathcal{F}(s,\bar{y}(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s))=q_{*}(s)\
\text{a.e. $s\in[0,\bar{S}]$};$ (4.22)
$\displaystyle(m_{*},m)(s)\in\partial_{t,x}^{>}\,h\left(s,\bar{y}(s)\right)\text{
$\mu$-a.e. $s\in[0,\bar{S}]$;}$ $\displaystyle
spt(\mu)\subseteq\\{s\in[0,\bar{S}]\text{ : }h\left(s,\bar{y}(s)\right)=0\\},$

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

A $\Psi$-extremal is normal if all possible choices of
$(p_{*},p,\gamma,\mu,m_{*},m)$ as above have $\gamma>0$, and abnormal when it
is not normal. Given a $\Psi$-extremal $\bar{z}$, we call nondegenerate
multiplier any set of multipliers $(p_{*},p,\gamma,\mu,m_{*},m)$ and
$(q_{*},q)$ as above, that also verify

$\mu(]0,S])+\|q\|_{L^{\infty}}+\gamma\neq 0.$ (4.23)

A $\Psi$-extremal is nondegenerate normal if all the choices of nondegenerate
multipliers have $\gamma>0$, and it is nondegenerate abnormal when there
exists a nondegenerate multiplier with $\gamma=0$. In the following, abnormal
[nondegenerate abnormal] $\Psi$-extremals will be simply called abnormal
[nondegenerate abnormal] extremals.

Theorems 2.1, 3.1 extend to free end-time optimization problems as follows.

######  Theorem 4.1.

Let
$\bar{z}:=(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
be a feasible relaxed process for $(P_{r}^{*})$, and suppose that at $\bar{z}$
there is a local infimum gap.

  * (i)

If hypotheses (H1)-(H2)∗-(H3)∗ hold, then $\bar{z}$ is an abnormal extremal.

  * (ii)

If, in addition, also hypothesis (H4) for $S=\bar{S}$ is verified, then
$\bar{z}$ is a nondegenerate abnormal extremal.

###### Proof.

Let
$\bar{z}:=(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
be a feasible relaxed process at which there is a local infimum gap. By the
above considerations, this is equivalent to suppose that $\bar{z}$ is an
isolated process. Adapting a standard time-rescaling procedure (see e.g. [29,
Thm. 8.7.1]), we transform problems $(P^{*})$, $(P_{e}^{*})$, and
$(P_{r}^{*})$ into fixed end-time problems, and show that $\bar{z}$ is an
isolated process also with respect to feasible strict sense processes of a
rescaled, fixed end-time problem. At this point, the thesis follows by
applying Theorems 2.1, 3.1 to the rescaled problem.

From the fact that $\bar{z}$ is isolated, it follows that there exists some
$\delta>0$ such that

$\left\\{(S,\omega,\alpha,y)\in\Gamma^{*}:\ \
d_{\infty}\big{(}(S,y),(\bar{S},\bar{y})\big{)}<3\delta\right\\}=\emptyset.$
(4.24)

Set
$\bar{\delta}:=\min\left\\{\frac{\delta}{3\bar{S}\,K_{\mathcal{F}}},\frac{1}{2}\right\\}$.

We define the rescaled optimization problem $(\hat{P}^{*})$, as

$(\hat{P}^{*})\left\\{\parbox[c][56.9055pt]{390.25534pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(y^{*}(\bar{S}),y(\bar{S}))\\\
&\text{over
$(\omega,\alpha,\zeta,y^{*},y)\in\mathcal{V}(\bar{S})\times\mathcal{A}(\bar{S})\times
L^{1}([0,\bar{S}];[-\bar{\delta},\bar{\delta}])\times
W^{1,1}([0,\bar{S}];\mathbb{R}^{1+n})$ s.t. }\\\
&(\dot{y}^{*},\dot{y})(s)=(1+\zeta(s))\big{(}1,\mathcal{F}(y^{*}(s),y(s),\omega(s),\alpha(s))\big{)}\
\text{a.e.},\quad(y^{*},y)(0)=(0,\check{x}_{0})\\\ &h(y^{*}(s),y(s))\leq
0\quad\forall
s\in[0,\bar{S}],\qquad(y^{*}(\bar{S}),y(\bar{S}))\in\mathcal{T}^{*},}\right.$

A process $(\omega,\alpha,\zeta,y^{*},y)$ for the fixed end-time problem
$(\hat{P}^{*})$ is referred to as a rescaled strict sense process. Let
$\hat{\Gamma}^{*}$ denote the set of the feasible rescaled strict sense
processes, that is, the set rescaled strict sense processes that verify
$h(y^{*}(s),y(s))\leq 0$ for all $s\in[0,\bar{S}]$ and
$(y^{*}(S),y(S))\in\mathcal{T}^{*}.$ We call rescaled extended processes the
processes $(\omega,\alpha,\zeta,y^{*},y)$ with $\omega\in\mathcal{W}(\bar{S})$
and write $(\hat{P}_{e}^{*})$ to denote the rescaled extended problem
associated to $(\hat{P}^{*})$.

We can identify
$\bar{z}=(\bar{S},\underline{\bar{\omega}},\underline{\bar{\alpha}},\bar{\lambda},\bar{\xi},\bar{y})$
with a process
$\check{z}:=(\underline{\check{\omega}},\underline{\check{\alpha}},\underline{\check{\zeta}},\check{\lambda},\check{\xi},\check{y}^{*},\check{y})\in\mathcal{W}^{2+n}(\bar{S})\times\mathcal{A}^{2+n}(\bar{S})\times(L^{1}[0,\bar{S}];[-\bar{\delta},\bar{\delta}]))^{2+n}\times\Lambda_{n+1}(\bar{S})\times
W^{1,1}([0,\bar{S}];\mathbb{R}^{2+n}\times\mathbb{R}^{1+n})$ of the relaxed
problem associated to $(\hat{P}^{*}_{e})$, by setting

$\begin{array}[]{l}\underline{\check{\omega}}:=(w,\underline{\bar{\omega}}),\
\ \underline{\check{\alpha}}:=(a,\underline{\bar{\alpha}}),\ \
\underline{\check{\zeta}}:=0,\ \check{\lambda}:=(0,\bar{\lambda}),\ \
\check{\xi}=(0,\bar{\xi}),\ \ \check{y}^{*}:=id,\ \
\check{y}:=\bar{y},\end{array}$

for arbitrary $w\in W$ and $a\in A$. Since $\bar{z}$ is an isolated feasible
relaxed process for the free end-time problem, $\check{z}$ is feasible and
isolated for the relaxed rescaled problem. In particular, we claim that

$\left\\{(\omega,\alpha,\zeta,y^{*},y)\in\hat{\Gamma}^{*}:\ \
\|(y^{*},y)-(\check{y}^{*},\check{y})\|_{L^{\infty}([0,\bar{S}])}<\delta\right\\}=\emptyset.$
(4.25)

Indeed, let $(\omega,\alpha,\zeta,y^{*},y)$ be an arbitrary feasible, rescaled
strict sense process verifying
$\|(y^{*},y)-(\check{y}^{*},\check{y})\|_{L^{\infty}([0,\bar{S}])}<\delta$.
Consider the time-transformation $y^{*}:[0,\bar{S}]\to[0,S]$, where
$S:=y^{*}(\bar{S}).$ Observe that $y^{*}$ is a strictly increasing, Lipschitz
continuous function, with Lipschitz continuous inverse, $(y^{*})^{-1}$. It can
be deduced that the process $(S,\hat{\omega},\hat{\alpha},\hat{y})$, where

$(\hat{\omega},\hat{\alpha},\hat{y}):=(\omega,\alpha,y)\circ(y^{*})^{-1}\quad\text{in
$[0,S]$,}$

is a feasible strict sense process for the free end-time problem $(P^{*})$,
i.e. $(S,\hat{\omega},\hat{\alpha},\hat{y})\in\Gamma^{*}$. Indeed, recalling
the definitions of $\bar{\delta}$ and $d_{\infty}$,222By definition, $\hat{y}$
and $\check{y}$ are replaced with their constant, continuous extensions to
$\mathbb{R}$. after some calculations, we get

$\begin{array}[]{l}\displaystyle
d_{\infty}\big{(}(S,\hat{y}),(\bar{S},\bar{y})\big{)}=|S-\bar{S}|+\|\hat{y}-\bar{y}\|_{\infty}\leq|y^{*}(\bar{S})-\check{y}^{*}(\bar{S})|\\\\[6.45831pt]
\displaystyle\ +\sup_{s\in[0,S\vee\bar{S}]}\left[|y((y^{*})^{-1}(s\land
S))-\check{y}((y^{*})^{-1}(s\land S))|+|\check{y}((y^{*})^{-1}(s\land
S))-\check{y}(s\land\bar{S})|\right]\\\
\qquad\qquad\leq\|y^{*}-\check{y}^{*}\|_{L^{\infty}([0,\bar{S}])}+\|y-\check{y}\|_{L^{\infty}([0,\bar{S}])}+\delta<3\delta,\end{array}$

since, in particular,

$\sup_{s\in[0,S\vee\bar{S}]}|\check{y}((y^{*})^{-1}(s\land
S))-\check{y}(s\land\bar{S})|\leq
3K_{\mathcal{F}}\bar{S}\,\bar{\delta}\leq\delta.$

Therefore, (4.24) yields (4.25), and the feasible rescaled relaxed process
$\check{z}$ is isolated in $\hat{\Gamma}^{*}$, as claimed. In order to apply
the results of Theorems 2.1, 3.1 to $(\hat{P}^{*})$ with reference to the
process $\check{z}$, it remains to show that, if we consider
$(w,a,\lambda,\zeta)$ as control variables and $\tilde{x}:=(s,x)$ as the state
variable for the (now, time-independent) problem $(\hat{P}^{*})$, all the
hypotheses assumed in their statements are fulfilled. To this aim, observe
that (H2) trivially follows from (H2)∗, while (H3)∗ easily implies (H3)′. In
particular, the compactness of $\Sigma^{*}_{\theta}\times W\times
A\times[-\bar{\delta},\bar{\delta}]$ and the continuity of
$(1+\zeta)\mathcal{F}$ on it, guarantee the existence of a constant
$M_{\mathcal{F}}>0$ and of a continuous increasing function
$\varphi:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$ with $\varphi(0)=0$ such
that

$|(1+\zeta)\mathcal{F}(t,x,w,a)|\leq
M_{\mathcal{F}},\quad|(1+\zeta)\mathcal{F}(t,x,w^{\prime},a)-(1+\zeta)\mathcal{F}(t,x,w,a)|\leq\varphi(|w^{\prime}-w|),$

for all $(t,x,w^{\prime},a,\zeta)$,
$(t,x,w,a,\zeta)\in\Sigma^{*}_{\theta}\times W\times
A\times[-\bar{\delta},\bar{\delta}]$. Finally, recalling that
$\underline{\check{\zeta}}\equiv 0$ and $\check{y}^{*}(s)=s$ for all
$s\in[0,\bar{S}]$, hypothesis (H4) can be trivially reformulated as an
hypothesis on the rescaled process $\check{z}$. At this point, from Theorems
2.1, 3.1 we can derive that $\bar{z}$ is an abnormal extremal, nondegenerate
when hypothesis (H4) is verified. In particular, the only nontrivial results,
namely conditions (4.21), (4.22), and the nontriviality conditions (4.20),
(3.9) can be obtained through routine arguments (see e.g. the proof of [29,
Thm. 8.2.1]). ∎

Again, from Theorem 4.1 we can get normality tests for gap avoidance and
sufficient controllability conditions for the free end-time problem completely
analogous to Theorems 2.2\- 2.3 and 3.2-3.3, respectively.

##  5\. An application to non-convex, control-polynomial impulsive problems

We consider the free end-time optimal control problem:

$(\mathcal{P}^{*})\left\\{\parbox[c][88.2037pt]{390.25534pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(T,x(T),v(T))\\\
&\text{over }T>0,\,(u,a,x,v)\in L^{d}([0,T];U)\times L^{1}([0,T];A)\times
W^{1,1}([0,T];\mathbb{R}^{n+1})\ \text{ s.t.}\\\
&(\dot{x},\dot{v})(t)=\Bigg{(}f(t,x,a)+\sum_{k=1}^{d}\Bigg{(}\sum_{1\leq
j_{1}\leq\dots\leq j_{k}\leq
m}g^{k}_{j_{1},\dots,j_{k}}(t,x)\,\,u^{j_{1}}\cdots
u^{j_{k}}\Bigg{)},\left|u\right|^{d}\Bigg{)}\,\text{a.e.,}\\\
&(x,v)(0)=(\check{x}_{0},0)\\\ &h(t,x(t))\leq 0\qquad\text{ for all
}t\in[0,T],\quad v(T)\leq K,\quad(T,x(T))\in\mathcal{T}^{*}.}\right.$

Here, $U\subseteq\mathbb{R}^{m}$ is a closed cone, $A\subseteq\mathbb{R}^{q}$
is a compact subset, $K>0$ is a fixed constant, possibly equal to $+\infty$,
and the target set $\mathcal{T}^{*}\subseteq\mathbb{R}^{1+n}$ is closed.
Notice that $v(t)$ is simply the $L^{d}$-norm to the power $d$ of the control
function $u$ on $[0,t]$. The variable $v$ is sometimes called fuel or energy
and $v\mapsto\Psi(t,x,\cdot)$ is usually assumed monotone nondecreasing for
every $(t,x)$ (see e.g. [20, 21]). The integer $d\geq 1$ will be called the
degree of the control system. Problem $(\mathcal{P}^{*})$ is referred to as
the original problem and we call a process $(T,u,a,x,v)$ for
$(\mathcal{P}^{*})$ an original process. We say that $(T,u,a,x,v)$ is feasible
if $h(t,x(t))\leq 0$ for all $t\in[0,T]$, $v(T)\leq K$, and
$(T,x(T))\in\mathcal{T}^{*}$.

Throughout this section, we shall consider the following structural
hypotheses:

  * (H5)

the functions $f:\mathbb{R}^{1+n}\times A\to\mathbb{R}^{n}$,
$g^{k}_{j_{1}\dots j_{k}}:\mathbb{R}^{1+n}\to\mathbb{R}^{n}$ are continuous,
all $g^{k}_{j_{1}\dots j_{k}}$ are locally Lipschitz continuous, and
$f(\cdot,\cdot,a)$ is locally Lipschitz continuous uniformly w.r.t. $a\in A$.
Furthermore, the constraint function $h:\mathbb{R}^{1+n}\to\mathbb{R}$ is
locally Lipschitz continuous.

In order to apply the theory developed in the previous sections, we
reformulate problem $(\mathcal{P}^{*})$ and embed it into a free end-time
extended problem with bounded controls. To do this, we use a compactification
procedure based on a reparameterization technique, commonly adopted to obtain
an impulsive extension of unbounded control problems, as generalized to
polynomial systems (see e.g. [27, 21]). Let us choose

$W:=\left\\{(w^{0},w)\in\mathbb{R}_{\geq 0}\times U:\ \
(w^{0})^{d}+|w|^{d}=1\right\\},\quad V:=\left\\{(w^{0},w)\in W:\ \
w^{0}>0\right\\}.$

For every $S>0$, we set $\mathcal{W}(S):=L^{1}([0,S];W)$,333The controls
$(\omega^{0},\omega)\in\mathcal{W}(S)$ actually belong to $L^{\infty}\cap
L^{1}$, since $W$ is compact. $\mathcal{V}(S):=L^{1}([0,S];V)$, and
$\mathcal{A}(S):=L^{1}([0,S];A)$, and introduce the space-time or extended
problem: 444The original time $t$ coincides now with the state component
$y^{0}$, while $s$ is the new ‘pseudo-time’ variable.

$(P^{*}_{e})\left\\{\parbox[c][79.6678pt]{390.25534pt}{\@@amsalign&\qquad\qquad\text{minimize}\,\,\,\Psi(y^{0}(S),y(S),\nu(S))\\\
&\text{over $S>0$,
}(\omega^{0},\omega,\alpha,y^{0},y,\nu)\in\mathcal{W}(S)\times\mathcal{A}(S)\times
W^{1,1}([0,S];\mathbb{R}^{1+n+1})\text{ s.t.}\\\
&(\dot{y}^{0},\dot{y},\dot{\nu})(s)=\Big{(}(\omega^{0})^{d}(s),\mathcal{F}(y^{0}(s),y(s),\omega^{0}(s),\omega(s),\alpha(s)),\left|\omega(s)\right|^{d}\Big{)}\
\text{a.e.,}\\\ &(y^{0},y,\nu)(0)=(0,\check{x}_{0},0),\\\
&h(y^{0}(s),y(s))\leq 0\ \text{ for all
}s\in[0,S],\quad\left(y^{0}(S),y(S),\nu(S)\right)\in\mathcal{T}^{*}\times]-\infty,K],}\right.$

where, for any
$(t,x,w^{0},w,a)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}_{\geq
0}\times U\times A$, we have set

$\mathcal{F}(t,x,w^{0},w,a):=f(t,x,a)(w^{0})^{d}+\sum_{k=1}^{d}\Bigg{(}\sum_{1\leq
j_{1}\leq\dots\leq j_{k}\leq
m}g^{k}_{j_{1},\dots,j_{k}}(t,x)\,\,w^{j_{1}}\cdots
w^{j_{k}}\,(w^{0})^{d-k}\Bigg{)}.$

Adopting notation and terminology of Section 4, a process
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ of problem $(P^{*}_{e})$ is
referred to as an extended process and it is feasible if $h(y^{0}(s),y(s))\leq
0$ for all $s\in[0,S]$ and
$\left(y^{0}(S),y(S),\nu(S)\right)\in\mathcal{T}^{*}\times]-\infty,K]$. When
$\omega^{0}>0$ almost everywhere, namely
$(\omega^{0},\omega)\in\mathcal{V}(S)$,
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$ is said a strict sense process. The
problem of minimizing $\Psi(y^{0}(S),y(S))$ over feasible strict sense
processes is still denoted by $(P^{*})$, and the sets of feasible extended
processes and feasible strict sense processes are $\Gamma$ and $\Gamma^{*}$,
respectively.

The original problem $(\mathcal{P}^{*})$ can be identified with problem
$(P^{*})$, as established by the following lemma, immediate consequence of the
chain rule.

######  Lemma 5.1 (Embedding).

Assume hypothesis (H5). Then the map

${\mathcal{I}}:\\{(T,u,a,x,v),\ \text{original
processes}\\}\to\\{(S,\omega^{0},\omega,\alpha,y^{0},y,\nu),\ \text{extended
processes}\\}$

defined as

${\mathcal{I}}(T,u,a,x,v):=(S,\omega^{0},\omega,\alpha,\zeta,y^{0},y,\nu),$

where, setting $\sigma(t):=t+v(t)$ for all $t\in[0,T]$,

$\begin{array}[]{l}\displaystyle
S:=\sigma(T),\quad(y^{0},y,\nu)(s):=(id,x,v)\circ\sigma^{-1}(s)\ \ \forall
s\in[0,S],\\\
\displaystyle(\omega^{0},\omega)(s):=(1+|u|^{d})^{-\frac{1}{d}}\,\,(1,u)\circ\sigma^{-1}(s),\quad\alpha(s):=a\circ\sigma^{-1}(s)\
\ \text{a.e. $s\in[0,S]$,}\end{array}$

is injective and has as image the subset of strict sense processes. Moreover,
${\mathcal{I}}$ maps any feasible original process into a feasible strict
sense process, with the same cost.

The extended problem $(P^{*}_{e})$ consists thus in considering processes
$(S,\omega^{0},\omega,\alpha,y^{0},y,\nu)$, where $\omega^{0}$ may be zero on
nondegenerate subintervals of $[0,S]$. On these intervals, the time variable
$t=y^{0}$ is constant –i.e. the time stops–, while the state variable $y$
evolves, according to $\mathcal{F}(t,y,0,w,\alpha)=$$\sum_{1\leq
j_{1}\leq\dots\leq j_{d}\leq
m}g^{d}_{j_{1},\dots,j_{d}}(t,y)\,\,w^{j_{1}}\cdots w^{j_{d}}$, which can be
called fast dynamics. For this reason, problem $(P^{*}_{e})$ is often referred
to as the impulsive extension of the original problem $(\mathcal{P}^{*})$
(more details on polynomial impulsive problems can be found in [27, 21] and
references therein).

Let us introduce the unmaximized Hamiltonian $H$, defined by

$\begin{array}[]{l}H(t,x,p_{0},p,\pi,\omega^{0},\omega,a):=p_{0}(\omega^{0})^{d}+p\cdot\mathcal{F}(t,x,w^{0},w,a)+\pi|\omega|^{d},\end{array}$

for all $(t,x,p_{0},p,\pi,\omega^{0},\omega,a)\in\mathbb{R}^{1+n+1+n+1}\times
W\times A$. The concepts of extremal and nondegenerate extremal read now as
follows:

######  Definition 5.1 (Extremal and nondegenerate extremal).

Assume (H5) and let
$\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be a feasible extended process. Given a cost function $\Psi$ which is
Lipschitz continuous on a neighborhood of
$(\bar{y}^{0}(\bar{S}),\bar{y}(S),\bar{\nu}(\bar{S}))$, we say that $\bar{z}$
is a $\Psi$-extremal if there exist a path $(p_{0},p)\in
W^{1,1}([0,\bar{S}];\mathbb{R}\times\mathbb{R}^{n})$, $\gamma\geq 0$, $\pi\leq
0$, $\mu\in NBV^{+}([0,\bar{S}];\mathbb{R})$,
$(m_{0},m):[0,\bar{S}]\to\mathbb{R}^{1+n}$ Borel measurable and
$\mu$-integrable functions, verifying the following conditions:

$\displaystyle\|p_{0}\|_{L^{\infty}}+\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\gamma\neq
0;$ (5.26) $\displaystyle(-\dot{p}_{0},-\dot{p})(s)\in{\rm co}\
\partial_{t,x}\
H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}\
\text{a.e.;}$
$\displaystyle\left(-q_{0}(\bar{S}),-q(\bar{S}),-\pi\right)\in\gamma\partial\Psi\left(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+N_{\mathcal{T}^{*}\times]-\infty,K]}\left(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right);$
(5.27) for a.e. $s\in[0,S]$, one has $\displaystyle
H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\bar{\omega}^{0}(s),\bar{\omega}(s),\bar{\alpha}(s)\Big{)}$
$\displaystyle\qquad\qquad=\max_{(\omega^{0},\omega,a)\in W\times
A}H\Big{(}\bar{y}^{0}(s),\bar{y}(s),q_{0}(s),q(s),\pi,\omega^{0},\omega,a\Big{)}=0;$
(5.28)
$\displaystyle(m_{0},m)(s)\in\partial_{t,x}^{>}\,h\left(\bar{y}^{0}(s),\bar{y}(s)\right)\qquad\text{$\mu$-a.e.;}$
$\displaystyle spt(\mu)\subseteq\\{s\in[0,\bar{S}]\text{ :
}h\left(\bar{y}^{0}(s),\bar{y}(s)\right)=0\\},$

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

Given a $\Psi$-extremal $\bar{z}$ we call nondegenerate multipliers all
$(p_{0},p,\pi,\gamma,\mu,m_{0},m)$ and $(q_{0},q)$ as above, that also verify

$\mu(]0,S])+\|q_{0}\|_{L^{\infty}}+\|q\|_{L^{\infty}}+\gamma\neq 0.$ (5.29)

If
$\gamma\partial_{v}\Psi\left(\bar{y}^{0}(\bar{S}),\bar{y}(\bar{S}),\bar{\nu}(\bar{S})\right)=0$
and $\bar{\nu}(\bar{S})<K$, then $\pi=0$. Furthermore, if
$\bar{y}^{0}(0)<\bar{y}^{0}(\bar{S})$, (5.26) [(5.29)] can be strengthened to

$\|p\|_{L^{\infty}}+\|\mu\|_{TV}+\gamma\neq
0\quad[\mu(]0,S])+\|q\|_{L^{\infty}}+\gamma\neq 0].$

In the special case of the impulsive extension considered in this section,
given a feasible extended process
$(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
the constraint qualification condition (H4) for nondegeneracy can be replaced
by the following weaker assumptions:666We recall that $\Omega=\\{(t,x):\
h(t,x)\leq 0\\}$.

  * (H6)

If $(0,\check{x}_{0})\in\partial\Omega$, there are some $\tilde{\delta}>0$,
$\bar{s}\in]0,\bar{S}]$, some sequence of strict sense processes
$(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i},\tilde{y}^{0}_{i},\tilde{y}_{i},\tilde{\nu}_{i})_{i}\subset\mathcal{V}(\bar{s})\times\mathcal{A}(\bar{s})\times
W^{1,1}([0,\bar{s}];\mathbb{R}^{1+n+1})$, some sequences
$(\hat{\omega}_{i}^{0},\hat{\omega}_{i},\hat{\alpha}_{i})_{i}\subset\mathcal{W}(\bar{s})\times\mathcal{A}(\bar{s})$,
and $(\tilde{r}_{i})_{i}\subset L^{1}([0,\bar{s}];\mathbb{R}_{\geq 0})$ with
$\displaystyle\lim_{i\to+\infty}\|\tilde{r}_{i}\|_{L^{1}([0,\bar{s}])}=0$,
such that the following properties (i)–(iii) are verified.

    * (i)

For every $i$, one has

$h(\tilde{y}^{0}_{i}(s),\tilde{y}_{i}(s))\leq 0\qquad\forall s\in[0,\bar{s}];$

    * (ii)

for every $i$, there is a Lebesgue measurable subset
$\tilde{E}_{i}\subset[0,\bar{s}]$ such that

$\begin{array}[]{c}(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i})(s)\in(\bar{\omega}^{0},\bar{\omega})(s)+\tilde{r}_{i}(s)\mathbb{B},\quad\tilde{\alpha}_{i}(s)=\bar{\alpha}(s),\quad\text{a.e.
$s\in\tilde{E}_{i}$;}\\\\[6.45831pt]
\lim_{i\to+\infty}\ell(\tilde{E}_{i})=\bar{s};\end{array}$

    * (iii)

for every $i$, for all $(\zeta_{0},\zeta)\in\partial^{*}h(0,\check{x}_{0})$,
and for a.e. $s\in[0,\bar{s}]$, one has

$\begin{array}[]{l}\displaystyle\zeta_{0}\cdot[(\hat{\omega}_{i}^{0}(s))^{d}-(\tilde{\omega}_{i}^{0}(s))^{d}]\\\
\qquad+\zeta\cdot\big{[}\mathcal{F}(0,\check{x}_{0},(\hat{\omega}_{i}^{0},\hat{\omega}_{i},\hat{\alpha}_{i})(s))-\mathcal{F}(0,\check{x}_{0},(\tilde{\omega}_{i}^{0},\tilde{\omega}_{i},\tilde{\alpha}_{i})(s))\big{]}\leq-\tilde{\delta}.\end{array}$

In some situations, hypothesis (H6) simplifies considerably.

######  Lemma 5.2.

Assume (H5). Let $(0,\check{x}_{0})\in\partial\Omega$ and let
$\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be a feasible extended process. If there are some $\tilde{\delta}>0$,
$\bar{s}\in]0,\bar{S}]$, and an extend control
$(\hat{\omega}^{0},\hat{\omega},\hat{\alpha})\in\mathcal{W}(\bar{s})\times\mathcal{A}(\bar{s})$
such that, for all $(\zeta_{0},\zeta)\in\partial^{*}h(0,\check{x}_{0})$ and
for a.e. $s\in[0,\bar{s}]$,

$\begin{array}[]{l}\displaystyle\zeta_{0}\cdot[(\hat{\omega}^{0}(s))^{d}-(\bar{\omega}^{0}(s))^{d}]\\\
\ \
+\zeta\cdot\big{[}\mathcal{F}(0,\check{x}_{0},(\hat{\omega}^{0},\hat{\omega},\hat{\alpha})(s))-\mathcal{F}(0,\check{x}_{0},(\bar{\omega}^{0},\bar{\omega},\bar{\alpha})(s))\big{]}\leq-\tilde{\delta}.\end{array}$
(5.30)

and either $\bar{\omega}^{0}>0$ a.e. in $[0,\bar{s}]$, or there is some
$\tilde{\delta}_{1}>0$ such that, for a.e. $s\in[0,\bar{s}]$,

$\begin{array}[]{l}\displaystyle\sup_{(\zeta_{0},\zeta)\in\partial^{*}h(0,\check{x}_{0})}\left[\zeta_{0}\cdot(\bar{\omega}^{0}(s))^{d}+\zeta\cdot\mathcal{F}(0,\check{x}_{0},((\bar{\omega}^{0},\bar{\omega}),\bar{\alpha})(s))\right]\leq-\tilde{\delta}_{1},\end{array}$
(5.31)

then condition (H6) is satisfied.

###### Proof of Lemma 5.2.

Let us first suppose that $\bar{\omega}^{0}>0$ a.e. in $[0,\bar{s}]$. Then,
conditions (H6),(i),(ii) are verified by choosing, for every $i$,
$(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})=(\bar{w}^{0},\bar{w},\bar{\alpha})$,
while (H6),(iii) follows directly from (5.30), by taking
$(\hat{\omega}_{i}^{0},\hat{\omega}_{i},\hat{\alpha}_{i})\equiv(\hat{\omega}^{0},\hat{\omega},\hat{\alpha})$
for every $i$. If instead (5.31) is assumed, let us consider a sequence
$\delta_{i}\downarrow 0$ and for every $i$, let us set

$(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})(s):=\begin{cases}(\bar{\omega}^{0},\bar{\omega},\bar{\alpha})(s)\
\ \ \qquad\qquad\text{if $\bar{\omega}^{0}(s)>0$,}\\\
(\delta_{i},\sqrt[d]{1-\delta_{i}^{d}}\,\bar{\omega}(s),\bar{\alpha}(s))\ \
\text{if $\bar{\omega}^{0}(s)=0$,}\end{cases}$

for a.e. $s\in[0,\bar{s}]$. Then,
$(\tilde{\omega}_{i}^{0},\tilde{\omega}_{i},\tilde{\alpha}_{i})\in\mathcal{V}(\bar{s})\times\mathcal{A}(\bar{s})$,
it verifies (H6),(ii) with $\tilde{E}_{i}=[0,\bar{s}]$ and
$\tilde{r}_{i}\equiv\delta_{i}$. Let
$(\tilde{y}_{i}^{0},\tilde{y}_{i},\tilde{\nu}_{i})$ be the corresponding
solution of the extended control system in $(P^{*}_{e})$ with initial
condition
$(\tilde{y}^{0}_{i},\tilde{y}_{i},\tilde{\nu}_{i})(0)=(0,\check{x}_{0},0)$.
From condition (5.31), using the Lebourgh Mean Value Theorem [29, Th. 4.5.3]
to estimate ‘$h(\tilde{y}_{i}^{0}(s),\tilde{y}_{i}(s))-h(0,\check{x}_{0})$’,
one can derive that $h(\tilde{y}^{0}_{i}(s),\tilde{y}_{i}(s))\leq 0$ for all
$s\in[0,\bar{s}]$, for every $i$ large enough, so proving the validity of
(H6),(i). Finally, from condition (5.30) (by adding and subtracting
‘$\zeta_{0}\cdot(\tilde{\omega}^{0}_{i}(s))^{d}+\zeta\cdot\mathcal{F}(0,\check{x}_{0},(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})(s))$’
and by taking
$(\hat{\omega}^{0}_{i},\hat{\omega}_{i},\hat{\alpha}_{i})=(\hat{w}^{0},\hat{w},\hat{\alpha})$)
we get condition (H6),(iii), possibly reducing $\tilde{\delta}$, for all $i$
large enough. ∎

######  Theorem 5.1.

Let
$\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{\alpha},\bar{y}^{0},\bar{y},\bar{\nu})$
be a feasible process for the impulsive extension $(P^{*})$, and suppose that
at $\bar{z}$ there is a local infimum gap. If hypothesis (H5) is verified,
then $\bar{z}$ is an abnormal extremal. If hypothesis (H6) is also satisfied,
then $\bar{z}$ is a nondegenerate abnormal extremal.

###### Proof.

It is sufficient to show that hypothesis (H5) allows the application of
Theorem 4.1, (i), while assuming (H5)-(H6), Theorem 4.1, (ii) is applicable.
To this aim, we observe that hypothesis (H1) is trivially verified, by
choosing, e.g., $V_{i}:=\\{(w^{0},w)\in V:\ \ w^{0}\geq\frac{1}{i+1}\\}$ for
every $i\in\mathbb{N}$, while (H5) yields (H2)∗ directly. Condition (H3)∗
easily follows from (H5), taking into account the control-polynomial structure
of the dynamics as regards point (ii). To prove that (H5)-(H6) imply condition
(H4), let us consider a sequence $\delta_{i}\downarrow 0$ and for every $i$,
define the strict sense control

$(\check{\omega}^{0}_{i},\check{\omega}_{i},\check{\alpha}_{i})(s):=\begin{cases}(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})(s)\,\quad\qquad\qquad\qquad\text{if
$s\in[0,\bar{s}]$,}\\\ (\bar{\omega}^{0},\bar{\omega},\bar{\alpha})(s)\ \
\,\quad\qquad\qquad\qquad\text{if $s\in]\bar{s},\bar{S}]$ and
$\bar{\omega}^{0}(s)>0$,}\\\
(\delta_{i},\sqrt[d]{1-\delta_{i}^{d}}\,\bar{\omega}(s),\bar{\alpha}(s))\,\
\qquad\text{if $s\in]\bar{s},\bar{S}]$ and
$\bar{\omega}^{0}(s)=0$,}\end{cases}$

where $(\tilde{\omega}^{0}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})$ is as
in (H6). By identifying, for every $i$, the strict sense process, say
$(\check{\omega}^{0}_{i},\check{\omega}_{i},\check{\alpha}_{i},\check{y}_{i}^{0},\check{y}_{i},\check{\nu}_{i})$,
of $(P^{*})$ corresponding to
$(\check{\omega}^{0}_{i},\check{\omega}_{i},\check{\alpha}_{i})$ with a
relaxed process –as we have been doing since the Introduction–, we derive that
conditions (H4),(ii), and (H4),(iii) on
$\check{E}_{i}:=\tilde{E}_{i}\cup]\bar{s},\bar{S}]\subset[0,\bar{S}]$, are
verified. Condition (H4),(i) follows from well-known continuity properties of
the input-output map, associated to the control system in $(P^{*})$. Finally,
in view of Remark 3.1,(3), (H6),(iii) implies (H4),(iv), although the controls
$(\hat{\omega}_{i}^{0},\hat{\omega}_{i},\hat{\alpha}_{i})$ are extended, not
necessarily strict sense, controls. ∎

As corollaries, we have:

######  Theorem 5.2.

Assume hypothesis (H5) and let $\Psi$ be locally Lipschitz continuous.

  * (i)

If a [local] $\Psi$-minimizer $\bar{z}$ for $(P^{*}_{e})$ is a normal
$\Psi$-extremal, then [at $\bar{z}$] there is no [local] infimum gap.

  * (ii)

If $\bar{z}$ is a [local] $\Psi$-minimizer for $(P^{*}_{e})$, at which
condition (H6) is verified and it is a nondegenerate normal $\Psi$-extremal,
then [at $\bar{z}$] there is no [local] infimum gap.

######  Theorem 5.3.

Assume hypothesis (H5). Then, either

  * (i)

$\bar{z}$ is not isolated in $\Gamma^{*}$; or

  * (ii)

$\bar{z}$ is an abnormal extremal [a nondegenerate abnormal extremal, if
condition (H6) is verified].

All the above results can be easily extended to the case when there is a local
infimum gap at some $\bar{z}$, which is a process of the relaxed problem
associated to $(P^{*}_{e})$. We limit ourselves to establish gap-abnormality
relations for the original problem with respect to its impulsive extension
$(P^{*}_{e})$, to illustrate how the results of the present paper apply to
general, possibly not convex, extensions.

In the following example there is no infimum gap but this fact cannot be
deduced from the normality criterion in Theorem 5.2,(i), since the extended
minimizer is abnormal. Instead, the absence of gap is detected by the Theorem
5.2,(ii), as the minimizer is nondegenerate normal.

######  Example 5.1.

Consider the problem

$\left\\{\begin{array}[]{l}\qquad\qquad\mbox{minimize }\ -x(1)\\\ \mbox{over
}(x,v,u)\in W^{1,1}([0,1];\mathbb{R}^{3}\times\mathbb{R})\times
L^{1}([0,1];\mathbb{R}^{2})\text{ satisfying}\\\
(\dot{x},\dot{v})(t)=\Big{(}f(x(t))+g_{1}(x(t))\,u^{1}(t)+g_{2}(x(t))\,u^{2}(t),\,\,\left|u(t)\right|\Big{)}\\\
(x,v)(0)=((1,0,0),0),\\\ x(t)\in\Omega\ \ \forall t\in[0,1],\ v(1)\leq 2,\
x(1)\in\mathcal{T},\end{array}\right.$ (5.32)

in which $\Omega:=[-1,1]^{3}$, $\mathcal{T}:=[-1,0]\times[0,1]^{2}$, and

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

Here, $W=\\{(\omega^{0},w)\in\mathbb{R}_{\geq 0}\times\mathbb{R}^{2}:\ \
w^{0}+|w|=1\\}$, $V=\\{(\omega^{0},w)\in W:\ \ w^{0}>0\\}$, and the associated
extended problem is

$\left\\{\begin{array}[]{l}\qquad\qquad\mbox{minimize }\ -y^{1}(S)\\\
\mbox{over }S>0,\ (y^{0},y,\nu,\omega^{0},\omega^{1},\omega^{2})\in
W^{1,1}([0,S];\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R})\times\mathcal{W}(S)\quad\mbox{satisfying
}\\\
(\dot{y}^{0},\dot{y},\dot{\nu})(s)=\Big{(}\omega^{0}(s),\,\,f(y(s))\omega^{0}(s)+g_{1}(y(s))\,\omega^{1}(s)+g_{2}(y(s))\,\omega^{2}(s),\,\,\left|\omega(s)\right|\Big{)}\\\
(y^{0},y,\nu)(0)=(0,(1,0,0),0)\\\ y(s)\in\Omega\ \ \forall s\in[0,S],\ \
(y^{0}(S),y(S),\nu(S))\in\\{1\\}\times\mathcal{T}\times]-\infty,2].\end{array}\right.$

As it is easy to see, an extended minimizer is given by the following feasible
extended process
$\bar{z}:=(\bar{S},\bar{\omega}^{0},\bar{\omega},\bar{y}^{0},\bar{y},\bar{\nu})$,
where

$\begin{array}[]{c}\bar{S}=2,\qquad(\bar{\omega}^{0},\bar{\omega})=(\bar{\omega}^{0},\bar{\omega}^{1},\bar{\omega}^{2})=(1,0,0)\chi_{{}_{[0,1]}}+(0,-1,0)\chi_{{}_{]1,2]}},\\\
(\bar{y}^{0},\bar{y},\bar{\nu})=(\bar{y}^{0},\bar{y}^{1},\bar{y}^{2},\bar{y}^{3},\bar{\nu})=(s,1,0,0,0)\chi_{[0,1]}+(1,2-s,0,0,s-1)\chi_{[1,2]}\,.\end{array}$

From the maximum principle [13, Thm. 1.1], $\bar{z}$ is a $\Psi$-extremal
accordingly to Definition 5.1. Hence, there exist a set of multipliers
$(p_{0},p,\pi,\gamma,\mu)$ and functions $(m_{0},m)$ with $\pi=0$, since
$\nabla_{v}\Psi\equiv 0$ and $\bar{\nu}(2)=1<2$, $m_{0}\equiv 0$, as the state
constraint does not depend on time, and $\mu([0,2])=\mu([0,1])$. Moreover, for
every $s\in[0,1]$ the fact that $\bar{y}(s)\in\Omega$ is equivalent to
$h(\bar{y}(s))\leq 0$, with $h(x^{1},x^{2},x^{3}):=x^{1}-1$, so that the
condition $m(s)\in\partial^{>}_{x}h(\bar{y}(0))$ $\mu$-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 condition

$-(q_{0},q_{1},q_{2},q_{3})(2)\in\gamma\\{(0,-1,0,0)\\}+\mathbb{R}\times
N_{\mathcal{T}}(0,0,0),$

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 $\bar{p}_{0}$, $\bar{p}_{1}\in\mathbb{R}$, $\bar{p}_{2}$,
$\bar{p}_{3}\geq 0$, and $q_{1}(2)=\bar{p}_{1}+\mu([0,1])=\gamma-\alpha_{1}$
with $\alpha_{1}\geq 0$. The maximality condition in $]1,2]$ implies that
$\bar{p}_{2}=\bar{p}_{3}=0$. In particular, from the relations

$\begin{array}[]{l}\displaystyle\max_{w^{1}\in[-1,1]}\left\\{q_{1}(s)w^{1}\right\\}\chi_{[0,1]}(s)=\bar{p}_{0}\chi_{[0,1]}(s)=0,\quad-
q_{1}(s)\chi_{]1,2]}(s)=0,\end{array}$

we also deduce that $\bar{p}_{0}=0$, $q_{1}(s)=\bar{p}_{1}+\mu([0,s[)=0$ for
a.e. $s\in[0,1[$, and $q_{1}(s)=\bar{p}_{1}+\mu([0,1])=\gamma-\alpha_{1}=0$
for every $s\in]1,2]$. In particular, $q(s)=0$ for a.e. $s\in[0,2]$,
$\mu([0,s[)=-\bar{p}_{1}$ for a.e. $s\in[0,1]$ implies that ($\bar{p}_{1}\leq
0$ and) $\mu=-\bar{p}_{1}\mu(\\{0\\})$, while the last relation yields that
$\gamma=\alpha_{1}$.

It is immediate to see that the set of degenerate multipliers
$(p_{0},p,\gamma,\mu)$ with $p_{0}=p_{2}=p_{3}=0$, $p_{1}=-1$,
$\mu=\delta_{\\{0\\}}$, and $\gamma=0$ meets all the conditions of the maximum
principle. So, $\bar{z}$ is an abnormal extremal. However, since
$\bar{w}^{0}>0$ for a.e. $s\in[0,1]$ and the control
$(\hat{\omega}^{0},\hat{\omega})=(\hat{\omega}^{0},\hat{\omega}^{1},\hat{\omega}^{2})\equiv(0,-1,0)$
verifies (5.30), from Lemma 5.2 it follows that condition (H6) is satisfied.
Therefore, in view of Theorem 5.2, (ii), to deduce that there is no infimum
gap it is enough to observe that $\bar{z}$ is nondegenerate normal, namely,
that $\gamma\neq 0$ for all sets of multipliers as above, which in addition
verify

$\mu(]0,2])+\|q\|_{L^{\infty}}+\gamma\neq 0.$

This is true, since the previous calculations imply that
$\|q\|_{L^{\infty}}=0$ and $\mu(]0,2])=0$.

##  6\. Proofs of Theorems 2.1, 3.1

Preliminarily, let us observe that, since the proofs involve only relaxed and
extended processes with trajectories close to the reference trajectory
$(\bar{\xi},\bar{y})$ and the controls assume values in compact sets, using
standard cut-off techniques we can assume that all hypotheses (H2)-(H3)′ are
satisfied not only in $\Sigma_{\theta}$, but in the whole space
$\mathbb{R}^{1+n}$. Hence, for any
$(\underline{\omega},\underline{\alpha},\lambda)\in\mathcal{W}^{1+n}\times\mathcal{A}^{1+n}\times\Lambda_{n}$
there is a unique solution
$(\xi,y)[\underline{\omega},\underline{\alpha},\lambda]$ to (3.11) defined on
$[0,S]$. Similarly, for any $(\omega,\alpha)\in\mathcal{W}\times\mathcal{A}$,
we will write $y[\omega,\alpha]$ to denote the corresponding solution to
(1.1).

###  6.1. Proof of Theorem 3.1

The proof is divided into several steps in which successive sequences of
optimization problems are introduced that have as eligible controls only
strict sense 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 a maximum principle to these approximate problems with reference to
the above mentioned minimizers, we obtain in the limit a set of multipliers
with $\gamma=0$ and verifying the strengthened non-triviality conditions (3.9)
for the relaxed problem with reference to the isolated process
$(\underline{\bar{\omega}},\underline{\bar{\alpha}},{\bar{\lambda}},\bar{y})$.

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

$\Phi\left(x,z\right):=d_{\mathcal{T}}(x)\vee z$

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

${\mathcal{J}}(y):=\Phi\Big{(}y(S),\,\max_{s\in[0,S]}h(s,y(s))\Big{)}.$

Fix a sequence $(\varepsilon_{i})_{i}$ such that $\varepsilon_{i}\downarrow
0$. Let $(\tilde{w}_{i},\tilde{\alpha}_{i},\tilde{\lambda}_{i})_{i}$ be a
control sequence as in hypothesis (H4), such that, eventually passing to a
subsequence, for every $i$, the corresponding trajectory
$(\tilde{\xi}_{i},\tilde{y}_{i})$ of (3.11) verifies

$\|(\tilde{\xi}_{i},\tilde{y}_{i})-(\bar{\xi},\bar{y})\|_{L^{\infty}}\leq\varepsilon_{i}.$
(6.33)

For every $i$, let $\rho_{i}\geq 0$ verify

$\rho_{i}^{4}=\sup\left\\{\begin{array}[]{l}{\mathcal{J}}(y):\ \
(w,\alpha,y)\in\Gamma,\ \ \|y-\bar{y}\|_{L^{\infty}}\leq
2\varepsilon_{i}\end{array}\right\\}.$

By the Lipschitz continuity of $\Phi$, it follows that
$\lim_{i\to+\infty}\rho_{i}^{4}=0$. Moreover, $\rho_{i}>0$ for every $i$ large
enough, since $\bar{z}$ is an isolated process by Proposition 2.1.

In the following, as it is clearly not restrictive, we will always assume that
the properties valid from a certain index onwards, apply to each index
$i\in\mathbb{N}$. By well-known continuity properties of the input-output map
$(\omega,\alpha)\mapsto y[\omega,\alpha]$, for every $\varepsilon_{i}$ there
exists $\delta_{i}>0$ such that, if
$\|\omega-\tilde{\omega}_{i}\|_{L^{1}}\leq\delta_{i}$, then
$\|y[\omega,\tilde{\alpha}_{i}]-\tilde{y}_{i}\|_{L^{\infty}}\leq\varepsilon_{i}$.
According to hypothesis (H1) and Remark 2.1, for any $i$ there exist an
element of the sequence $(V_{j})_{j}$, which we denote by $V_{\delta_{i}}$,
and some
$\mathring{\omega}_{i}\in\mathcal{V}_{\delta_{i}}:=L^{1}([0,S];V_{\delta_{i}})$
such that
$\|\mathring{\omega}_{i}-\tilde{\omega}_{i}\|_{L^{1}}\leq\delta_{i}$. In
particular, if we define

$\check{\omega}_{i}(s):=\begin{cases}\tilde{\omega}_{i}(s)\ \ \text{a.e.
}s\in[0,\rho_{i}]\\\ \mathring{\omega}_{i}(s)\ \ \text{a.e.
}s\in]\rho_{i},S],\end{cases}(\check{\alpha}_{i}(s),\check{\lambda}_{i}(s)):=(\tilde{\alpha}_{i}(s),\tilde{\lambda}_{i}(s))\
\ \text{a.e. }s\in[0,S],$

and
$(\check{\xi}_{i},\check{y}_{i}):=(\xi,y)[\check{\omega}_{i},\dots,\check{\omega}_{i},\check{\alpha}_{i},\dots,\check{\alpha}_{i},\check{\lambda}_{i}],$
then $\|\check{\omega}_{i}-\tilde{\omega}_{i}\|_{L^{1}}\leq\delta_{i}$ and
$\check{y}_{i}$ is a strict sense trajectory such that
$\|(\check{\xi}_{i},\check{y}_{i})-(\tilde{\xi}_{i},\tilde{y}_{i})\|_{L^{\infty}}\leq\varepsilon_{i}.$
From (6.33) it follows that

$\|(\check{\xi}_{i},\check{y}_{i})-(\bar{\xi},\bar{y})\|_{L^{\infty}}\leq
2\varepsilon_{i}.$ (6.34)

Hence, by the very definition of $\rho_{i}$ we deduce that for any $i$ the
process
$\check{z}_{i}:=(\check{\omega}_{i},\check{\alpha}_{i},\check{\lambda}_{i},\check{\eta}_{i},\check{\xi}_{i},\check{y}_{i})$,
where $\check{\eta}_{i}\equiv 0$, is a $\rho_{i}^{4}$-minimizer for the
optimal control problem:

$\big{(}\hat{P}_{i}\big{)}\begin{cases}\qquad\qquad\qquad\qquad\qquad\text{Minimize
}\,\,\,{\mathcal{J}}(y)\\\ \text{over the set of control
}(\omega,\alpha,\lambda,\eta)\in\mathcal{V}_{\delta_{i}}\times\mathcal{A}\times\Lambda_{n}^{1}\times
L^{1}([0,S];\\{0,1\\}),\\\ \text{and trajectories }\ (\xi,y)\in
W^{1,1}([0,S];\mathbb{R}^{1+n}\times\mathbb{R}^{n}),\text{ satisfying}\\\
\dot{\xi}(s)=\lambda(s)\qquad\text{ a.e. }s\in[0,S]\\\
\dot{y}(s)=\mathcal{F}(s,y,\tilde{\omega}_{i},\tilde{\alpha}_{i})+\eta(s)[\mathcal{F}(s,y,\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y,\tilde{\omega}_{i},\tilde{\alpha}_{i})]\,\,\text{a.e.}\,s\in[0,\rho_{i}]\\\
\dot{y}(s)=\mathcal{F}(s,y(s),\omega(s),\alpha(s))\quad\text{ a.e.
}s\in]\rho_{i},S]\\\ (\xi,y)(0)=(0,\check{x}_{0}),\end{cases}$

where $(\hat{\omega}_{i},\hat{\alpha}_{i})$ is as in hypothesis (H4). We call
an element $(\omega,\alpha,\lambda,\eta,\xi,y)$ verifying the constraints in
$(\hat{P}_{i})$ a process for problem $(\hat{P}_{i})$ and use ${\Gamma}_{i}$
to denote the set of such processes. By introducing, for every
$(\omega^{\prime},\alpha^{\prime},\lambda^{\prime},\eta^{\prime},\xi^{\prime},y^{\prime})$,
$(\omega,\alpha,\lambda,\eta,\xi,y)\in\Gamma_{i}$, the distance

$\begin{array}[]{l}\mathbf{d}((\omega^{\prime},\alpha^{\prime},\lambda^{\prime},\eta^{\prime},\xi^{\prime},y^{\prime}),(\omega,\alpha,\lambda,\eta,\xi,y))\\\
\qquad\qquad:=\|\omega^{\prime}-\omega\|_{L^{1}([0,S])}+\ell\\{s\in[0,S]:\
(\alpha^{\prime},\lambda^{\prime},\eta^{\prime})(s)\neq(\alpha,\lambda,\eta)(s)\\},\end{array}$
(6.35)

we can make $({\Gamma}_{i},\mathbf{d})$ a complete metric space. Then, from
Ekeland’s Principle it follows that there exists a process
$z_{i}:=(\omega_{i},\alpha_{i},\lambda_{i},\eta_{i},\xi_{i},y_{i})\in{\Gamma}_{i}$,
which is a minimizer of the optimization problem

$\left(P_{i}\right)\begin{cases}\text{Minimize}\,\,\,\,{\mathcal{J}}(y)+\rho^{2}_{i}\,\int_{0}^{S}\left[|\omega(s)-\omega_{i}(s)|+\ell_{i}(s,\alpha(s),\lambda(s),\eta(s))\right]\,ds\\\
\text{over }\ (\omega,\alpha,\lambda,\eta,\xi,y)\in\Gamma_{i},\end{cases}$

where
$\ell_{i}(s,a,\lambda,\eta):=\chi_{\\{(a,\lambda,\eta)\neq(\alpha_{i}(s),\lambda_{i}(s),\eta_{i}(s))\\}}$
for any $(s,a,\lambda,\eta)\in[0,S]\times
A\times\Lambda_{n}^{1}\times\\{0,1\\}$, and verifies

$\mathbf{d}\big{(}(\omega_{i},\alpha_{i},\lambda_{i},\eta_{i},\xi_{i},y_{i}),(\check{\omega}_{i},\check{\alpha}_{i},\check{\lambda}_{i},\check{\eta}_{i},\check{\xi}_{i},\check{y}_{i})\big{)}\leq\rho^{2}_{i}.$
(6.36)

Thus, by (6.34) and the continuity of the input-output map associated to the
control system (3.11), it follows that, eventually passing to a subsequence,
as $i\to+\infty$,

$\left\|\left(\xi_{i},y_{i}\right)-\left(\bar{\xi},\bar{y}\right)\right\|_{L^{\infty}}\to
0,\quad\big{(}\dot{\xi}_{i},\dot{y}_{i}\big{)}\rightharpoonup\big{(}\dot{\bar{\xi}},\dot{\bar{y}}\big{)}\quad\text{
weakly in $L^{1}$.}$ (6.37)

Furthermore, hypothesis (H4) and (6.36) imply that, for every $i$, there exist
some nonempty subset $E_{i}\subseteq\tilde{E}_{i}\subseteq[0,S]$ and some
$r_{i}\in L^{1}([0,S];\mathbb{R}_{\geq 0})$ with $r_{i}\geq\tilde{r}_{i}$
($\tilde{E}_{i}$, $\tilde{r}_{i}$ as in (H4)) such that, as $i\to+\infty$,
$\ell(E_{i})\to S$, $\|r_{i}\|_{L^{1}}\to 0$, and

$(\omega_{i},\alpha_{i},\lambda_{i})(s)\in\bigcup_{k=0}^{n}\\{(\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s),e^{k})\\}+(r_{i}(s),0,0)\mathbb{B}\quad\text{for
a.e. $s\in E_{i}$}.$ (6.38)

From (6.37) and the fact that $\bar{z}$ is isolated, it follows that
${\mathcal{J}}(y_{i})>0$ for all $i$, namely, at least one of the following
inequalities holds true: 777Notice that, to any process
$(\omega,\alpha,\lambda,\eta,\xi,y)\in\Gamma_{i}$ it corresponds a strict
sense process $(\breve{\omega},\breve{\alpha},\breve{y})$, where
$\breve{y}\equiv y$ and
$(\breve{\omega},\breve{\alpha})(s)=(\tilde{\omega}_{i},\tilde{\alpha}_{i})(s)+\eta(s)(\hat{\omega}_{i}-\tilde{\omega}_{i},\hat{\alpha}_{i}-\tilde{\alpha}_{i})(s)$
a.e. $s\in[0,\rho_{i}]$,
$(\breve{\omega},\breve{\alpha})(s)=(\omega,\alpha)(s)$ a.e.
$s\in]\rho_{i},S]$.

$d_{\mathcal{T}}(y_{i}(S))>0,\qquad c_{i}:=\max_{s\in[0,S]}\,h(s,y_{i}(s))>0.$
(6.39)

Step 2. For each $i\in\mathbb{N}$, set

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

The process
$(z_{i},c_{i})=(\omega_{i},\alpha_{i},\lambda_{i},\eta_{i},\xi_{i},y_{i},c_{i})$
turns out to be a minimizer for

$\left(Q_{i}\right)\begin{cases}\text{Minimize}\,\,\,\Phi(y(S),c(S))+\rho^{2}_{i}\,\int_{0}^{S}\left[|\omega(s)-\omega_{i}(s)|+\ell_{i}(s,\alpha(s),\lambda(s),\eta(s))\right]\,ds\\\
\text{over }\ (\omega,\alpha,\lambda,\eta,\xi,y)\in\Gamma_{i},\ c\in
W^{1,1}([0,S];\mathbb{R}),\ \text{verifying}\\\
\dot{c}(s)=0,\qquad\tilde{h}(s,y(s),c(s))\leq 0\quad\forall
s\in[0,S].\end{cases}$

Passing eventually to a subsequence, we may suppose that either (a) ‘$c_{i}>0$
for each $i\in\mathbb{N}$’, or (b) ‘$c_{i}\leq 0$ for each $i\in\mathbb{N}$’.

Case (a). Preliminarily, we show that, for every $i$, one has
$h(s,y_{i}(s))<c_{i}$ for all $s\in[0,\rho_{i}]$. This result is a
straightforward consequence of the following lemma:

######  Lemma 6.1.

For every $i\in\mathbb{N}$, one has $h(s,y_{i}(s))\leq 0$ for all
$s\in[0,\rho_{i}]$.

###### Proof.

From a standard application of the Gronwall’s Lemma one can deduce that there
is $\bar{C}>0$ such that, for every $i$, one has

$|y_{i}(s)-\tilde{y}_{i}(s)|\leq\bar{C}\,\ell(s,\eta_{i}(\cdot))\qquad\forall
s\in]0,\rho_{i}],$ (6.40)

where the nondecreasing map $s\mapsto\ell(s,\eta_{i}(\cdot))$ is as in (3.17).
Fix now $i\in\mathbb{N}$. By the Lebourgh Mean Value Theorem [29, Th. 4.5.3],
for every $s\in[0,\rho_{i}]$ there exists
$(\zeta_{0_{i}}^{s},\zeta_{{}_{i}}^{s})\in\partial^{c}h(s,x_{i}(s))$ for some
$x_{i}(s)$ in the segment
$[y_{i}(s)\,,\,\tilde{y}_{i}(s)]\subseteq\mathbb{R}^{n}$, such that 888Notice
that by the boundedness of the dynamics, both $\tilde{y}_{i}(s)$ and
$y_{i}(s)$ lay on $\check{x}_{0}+sK_{{}_{\mathcal{F}}}\mathbb{B}$. Hence, for
$i$ sufficiently large, $s\in[0,\varepsilon]$ and
$x_{i}(s)\in\check{x}_{0}+\varepsilon\mathbb{B}$, where $\varepsilon>0$ is as
in Remark 3.1,(4).

$\begin{array}[]{l}h(s,y_{i}(s))-h(s,\tilde{y}_{i}(s))=\zeta_{i}^{s}\cdot(y_{i}(s)-\tilde{y}_{i}(s))\\\
\,\,\,=\int_{0}^{s}\zeta_{i}^{s}\cdot[\mathcal{F}(\sigma,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})-\mathcal{F}(\sigma,\tilde{y}_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})]\,d\sigma\\\
\qquad\qquad\qquad+\int_{0}^{s}\eta(\sigma)\zeta_{i}^{s}\cdot[\mathcal{F}(\sigma,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(\sigma,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})]\,d\sigma\\\
\,\,\,\leq\int_{0}^{s}\bar{C}K_{{}_{h}}K_{{}_{\mathcal{F}}}\,\ell(\sigma,\eta_{i}(\cdot))d\sigma-\delta\,\ell(s,\eta_{i}(\cdot))\leq\ell(s,\eta_{i}(\cdot))\left(-\delta+\bar{C}K_{{}_{h}}K_{{}_{\mathcal{F}}}\,s\right)\leq
0,\end{array}$

where the last relations follow from (3.16), (6.40), and the fact that
$s\leq\rho_{i}\downarrow 0$. Finally, condition (3.13) implies the thesis. ∎

Our aim is now to apply the Pontryagin Maximum Principle to problem $(Q_{i})$
with reference to the minimizer $(z_{i},c_{i})$, for which, thanks to Lemma
6.1, the constraint is inactive on $[0,\rho_{i}]$. By standard arguments (see
the proof of [13, Th. 2.2]) we deduce that
$\partial_{t,x,c}^{{}^{>}}\tilde{h}(s,y_{i}(s),c_{i})=\partial_{t,x}^{{}^{>}}h(s,y_{i}(s))\times\\{-1\\},$
and that, if
$(\beta_{y_{i}},\beta_{c_{i}})\in\partial\Phi(y_{i}(S),c_{i}(S))$, then there
is some $\sigma_{i}^{1},\sigma_{i}^{2}\geq 0$ with
$\sigma_{i}^{1}+\sigma_{i}^{2}=1$, such that
$\beta_{y_{i}}\in\sigma_{i}^{1}\,\left(\partial
d_{\mathcal{T}}(y_{i}(S))\cap\partial\mathbb{B}_{n}\right),$
$\beta_{c_{i}}=\sigma_{i}^{2},$ and $\sigma_{i}^{k}=0$, $k\in\\{1,2\\}$, when
the maximum in $d_{\mathcal{T}}(y_{i}(S))\vee c_{i}(S)$ is strictly greater
than the $k$-th term in the maximization. Thus, the Maximum Principle [29, Th.
9.3.1] yields the existence of some multipliers $(p_{i},\pi_{i})\in
W^{1,1}([0,S];\mathbb{R}^{n+1})$ associated with $(y_{i},c_{i})$, $\mu_{i}\in
NBV^{+}([0,S];\mathbb{R})$, $\gamma_{i}\geq 0$, $\sigma^{1}_{i}$,
$\sigma_{i}^{2}\geq 0$ with $\sum_{k=1}^{2}\sigma_{i}^{k}=1$, and Borel-
measurable, $\mu_{i}$-integrable functions $m_{i}:[0,S]\to\mathbb{R}^{n}$,
such that

  * (i)′

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

  * (ii)′

$-\dot{p}_{i}(s)\in{\rm co}\text{
}\partial_{x}\Big{\\{}q_{i}(s)\cdot\mathcal{F}(s,y_{i},\omega_{i},\alpha_{i})(s)\Big{\\}}$
for a.e. $s\in[\rho_{i},S]$,
and $\dot{\pi}_{i}(s)=0$ for a.e. $s\in[0,S]$;

  * (iii)′

$-q_{i}(S)\in\gamma_{i}\,\sigma_{i}^{1}\,\left(\partial
d_{\mathcal{T}}(y_{i}(S))\cap\partial\mathbb{B}_{n}\right)$, $\pi_{i}(0)=0$,
$-\pi_{i}(S)+\int_{[0,S]}\mu_{i}(d\sigma)=\gamma_{i}\sigma_{i}^{2}$;

  * (iv)′

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

  * (v)′

$spt(\mu_{i})\subseteq\\{s\text{ :
}h\left(s,y_{i}(s)\right)-c_{i}=0\\}\subset[\rho_{i},S]$,

  * (vi)${}_{1}^{\prime}$

$\int_{0}^{\rho_{i}}\eta_{i}\,p_{i}\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]\,ds$
$\geq\int_{0}^{\rho_{i}}\left\\{(1-\eta_{i})\,p_{i}\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]-2\gamma_{i}\rho_{i}^{2}\right\\}ds$;
999By (v)′ it follows that $q_{i}\equiv p_{i}$ on $[0,\rho_{i}]$. Notice also
that (vi)${}_{1}^{\prime}$ holds in a more general form, in fact we can
replace $1-\eta_{i}(\cdot)$ in the right hand side with any measurable
function $\eta:[0,\rho_{i}]\to\\{0,1\\}$. Furthermore, we assume without loss
of generality diam$(W)=1$, since $W$ is supposed to be compact, so that
$|\omega(s)-\omega_{i}(s)|+\ell_{i}(s,\alpha(s),\lambda(s),\eta(s))\leq 2$ for
any $s\in[0,S]$.

  * (vi)${}_{2}^{\prime}$

$\int_{\rho_{i}}^{S}q_{i}\cdot\mathcal{F}(s,y_{i},\omega_{i},\alpha_{i})ds\geq\int_{\rho_{i}}^{S}\left\\{q_{i}\cdot\mathcal{F}(s,y_{i},\omega,\alpha)-2\gamma_{i}\rho_{i}^{2}\right\\}ds$
for all
$(\omega,\alpha,\lambda,\eta)\in\mathcal{V}_{\delta_{i}}\times\mathcal{A}\times\Lambda_{n}^{1}\times
L^{1}([0,S];\\{0,1\\})$;

where

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

Observe that, for each $i$, by (ii)′ and (iii)′ we derive
$\|\mu_{i}\|_{TV}=\int_{[0,S]}\mu_{i}(ds)=\gamma_{i}\sigma_{i}^{2}$ and
$\pi_{i}\equiv 0$. Furthermore, since $\|m_{i}\|_{L^{\infty}}\leq K_{{}_{h}}$,
then by (iii)′ we get

$\gamma_{i}\sigma_{i}^{1}=\left|q_{i}(S)\right|\leq\|p_{i}\|_{L^{\infty}}+K_{{}_{h}}\|\mu_{i}\|_{TV}.$

By summing up these estimates and the non-triviality condition (i)′, we get

$2\|p_{i}\|_{L^{\infty}}+(2+K_{{}_{h}})\|\mu_{i}\|_{TV}+\geq\gamma_{i}(\sigma_{i}^{1}+\sigma_{i}^{2}-1)+1=1.$

Hence, scaling the multipliers, we obtain
$\|p_{i}\|_{L^{\infty}}+\|\mu_{i}\|_{TV}=1$ and
$\gamma_{i}\leq\tilde{L}:=2+K_{{}_{h}}$.

Case (b). Now, $c_{i}\leq 0$ for each $i$, so that (6.39) implies
$d_{\mathcal{T}}(y_{i}(S))>0$. Thus, the process
$(\omega_{i},\alpha_{i},\lambda_{i},\eta_{i},y_{i},\xi_{i},\hat{c}_{i})$ with
$\hat{c}_{i}:=c_{i}+\hat{\varepsilon}$, for $\hat{\varepsilon}>0$ suitably
small is still a minimizer of problem $(Q_{i})$ and, in addition, it verifies
$h(s,y_{i}(s))-\hat{c}_{i}<0$ for all $s\in[0,S]$ (namely, the state
constraint is inactive on $[0,S]$). Hence, by applying the Maximum Principle
for problem $(Q_{i})$ with reference to this minimizer we deduce the existence
of multipliers $(p_{i},\pi_{i})\in W^{1,1}([0,S];\mathbb{R}^{n+1})$, which
satisfy conditions (i)′–(vi)′ with $\pi_{i}\equiv 0$, $\mu_{i}=0$,
$\sigma_{i}^{2}=0$, and $\gamma_{i}>0$. In this case, from (iii)′ we get
$0<\gamma_{i}=\left|q_{i}(S)\right|=\left|p_{i}(S)\right|\leq\|p_{i}\|_{L^{\infty}}$,
and, scaling the multipliers appropriately, we obtain
$\|p_{i}\|_{L^{\infty}}=1$ and $\gamma_{i}\leq 2$ ($\leq\tilde{L}$ as above).

Step 3. For either the case where $c_{i}>0$ for each $i$ or the case where
$c_{i}\leq 0$ for each $i$, passing to the limit as $i\to+\infty$ for suitable
subsequences and arguing as in the proof of [13, Thm. 2.2, Step. 4], we can
deduce the existence of a set of multipliers $p\in
W^{1,1}([0,S];\mathbb{R}^{n})$, $\mu\in NBV^{+}([0,S];\mathbb{R})$ and a
Borel-measurable, $\mu$-integrable function $m:[0,S]\to\mathbb{R}^{n}$, such
that

$\displaystyle\|p\|_{L^{\infty}}+\|\mu\|_{TV}=1,\qquad$ $\displaystyle
spt(\mu)\subseteq\\{s\in[0,S]:h(s,\bar{y}(s))=0\\},$ (6.41)
$\displaystyle-q(S)\in N_{\mathcal{T}}(\bar{y}(S))\qquad$ $\displaystyle
m(s)\in\partial_{x}^{>}h(s,\bar{y}(s))\,\,\text{$\mu$-a.e. $s\in[0,S]$}$

where

$q(s):=\begin{cases}p(s)+\int_{[0,s[}m(\sigma)\mu(d\sigma)\qquad s\in[0,S[\\\
p(S)+\int_{[0,S]}m(\sigma)\mu(d\sigma)\qquad s=S.\end{cases}$

Furthermore,

$q_{i}\to q\,\,\,\text{in $L^{1}$},\qquad p_{i}\to p\,\,\,\text{in
$L^{\infty}$},\qquad\dot{p}_{i}\rightharpoonup\dot{p}\,\,\,\text{weakly in
$L^{1}$}.$ (6.42)

Now, let $\Omega_{i}:=[\rho_{i},S]\setminus E_{i}$, where $E_{i}$ is as in
(6.38), so that $\ell(\Omega_{i})\to S$. Recalling
$\partial_{x}(q\cdot\mathcal{F})=q\cdot\partial_{x}^{c}\mathcal{F}$, by (ii)′,
(6.38) and (H3),(ii) we deduce, for a.e. $s\in\Omega_{i}$,

$\begin{array}[]{l}\displaystyle\big{(}-\dot{p}_{i},\dot{\xi}_{i},\dot{y}_{i}\big{)}(s)\in\bigcup_{k=0}^{n}\Big{[}\big{(}{\rm
co}\text{
}\partial_{x}\big{\\{}q_{i}(s)\cdot\mathcal{F}(s,y_{i},\bar{\omega}^{k},\bar{\alpha}^{k})(s)\big{\\}}\big{)}\\\
\qquad\qquad\,\,\times\\{(e^{k},\mathcal{F}(s,y_{i},\bar{\omega}^{k},\bar{\alpha}^{k})(s))\\}\Big{]}+\big{(}(1+K_{{}_{h}})\varphi(r_{i}(s)),\,0,\,\varphi(r_{i}(s))\big{)}\mathbb{B}\\\\[6.45831pt]
\displaystyle\subseteq\bigcup_{k=0}^{n}\Big{[}\big{(}{\rm co}\text{
}\partial_{x}\big{\\{}q(s)\cdot\mathcal{F}(s,y_{i},\bar{\omega}^{k},\bar{\alpha}^{k})(s)\big{\\}}\big{)}\times\\{(e^{k},\mathcal{F}(s,y_{i},\bar{\omega}^{k},\bar{\alpha}^{k})(s))\\}\Big{]}\\\
\qquad\qquad\,\,+\big{(}(1+K_{{}_{h}})\varphi(r_{i}(s))+K_{{}_{\mathcal{F}}}|q_{i}(s)-q(s)|,\,0,\,\varphi(r_{i}(s))\big{)}\mathbb{B}.\end{array}$

By the properties of $\varphi(\cdot)$, the compactness of $W$, and the
Dominated Convergence Theorem we have $\varphi(r_{i})\to 0$ in $L^{1}$ as
$i\to\infty$. Hence, all the hypotheses of the Compactness of Trajectories
Theorem [29, Th. 2.5.3] are satisfied, so that we can pass to the limit and
get

$\begin{array}[]{l}\displaystyle\Big{(}-\dot{p},\dot{\bar{\xi}},\dot{\bar{y}}\Big{)}(s)\in\text{{\rm
co}}\Big{(}\bigcup_{k=0}^{n}\big{[}\big{(}{\rm co}\text{
}\partial_{x}\big{\\{}q(s)\cdot\mathcal{F}(s,\bar{y},\bar{\omega}^{k},\bar{\alpha}^{k})(s)\big{\\}}\big{)}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\times\\{(e^{k},\mathcal{F}(s,\bar{y},\bar{\omega}^{k},\bar{\alpha}^{k})(s))\\}\big{]}\Big{)}\
\ \text{ for a.e. $s\in[0,S]$}.\end{array}$

By the Caratheodory Representation Theorem, there exists a measurable function
$\lambda=(\lambda^{0},\dots,\lambda^{n})\in\Lambda_{n}$ such that

$\begin{array}[]{l}\displaystyle\Big{(}-\dot{p},\dot{\bar{\xi}},\dot{\bar{y}}\Big{)}(s)\in\sum_{k=0}^{n}\lambda^{k}(s)\Big{(}{\rm
co}\text{
}\partial_{x}\left\\{q(s)\cdot\mathcal{F}(s,\bar{y},\bar{\omega}^{k},\bar{\alpha}^{k})(s)\right\\}\\\
\text{
}\qquad\qquad\qquad\qquad\qquad\times\\{(e^{k},\mathcal{F}(s,\bar{y},\bar{\omega}^{k},\bar{\alpha}^{k})(s))\\}\Big{)}\
\ \text{ for a.e. $s\in[0,S]$}.\end{array}$ (6.43)

But now

$\dot{\bar{\xi}}(s)=\sum_{k=0}^{n}\lambda^{k}(s)e^{k}=\sum_{k=0}^{n}\bar{\lambda}^{k}(s)e^{k}\qquad\text{a.e.
$s\in[0,S]$.}$

Therefore, for every $k=0,\dots,n+1$, $\lambda^{k}(s)=\bar{\lambda}^{k}(s)$
a.e. $s\in[0,S]$ and (2.3) is proved.

Let us prove (2.5). Take $(\omega,\alpha)\in\mathcal{W}\times\mathcal{A}$, by
(H1) and Remark 2.1 there exists a sequence $(v_{i})_{i}\in\mathcal{V}$ such
that $v_{i}\in\mathcal{V}_{\delta_{i}}$ for any $i$ and $\|\omega-
v_{i}\|_{L^{1}}\leq\delta_{i}\downarrow 0$. By (vi)${}_{2}^{\prime}$, we
deduce that, for any $i$, one has

$\int_{0}^{S}q_{i}(s)\cdot\dot{y}_{i}(s)\chi_{[\rho_{i},S]}(s)ds\geq\int_{0}^{S}\left\\{q_{i}(s)\cdot\mathcal{F}(s,y_{i},v_{i},\alpha)(s)-2\gamma_{i}\rho_{i}^{2}\right\\}\chi_{[\rho_{i},S]}(s)ds.$

Passing to the limit and using (6.37), (6.42) in the left hand side and the
Dominated Convergence Theorem in the right hand side, we obtain

$\int_{0}^{S}q(s)\cdot\dot{\bar{y}}(s)\,ds\geq\int_{0}^{S}q(s)\cdot\mathcal{F}(s,\bar{y}(s),\omega(s),\alpha(s))\,ds.$

Since this last relation holds for any selector
$(\omega,\alpha)\in\mathcal{W}\times\mathcal{A}$, by a measurable selection
theorem we can conclude that

$q(s)\cdot\dot{\bar{y}}(s)=\max_{(w,a)\in W\times
A}q(s)\cdot\mathcal{F}(s,\bar{y}(s),w,a)\qquad\text{a.e. $s\in[0,S]$.}$ (6.44)

Finally, (6.44) trivially implies (2.5). Thus $\bar{z}$ is an abnormal
extremal. To prove that it is in fact a nondegenerate abnormal extremal, it
remains to show that the above multipliers verify the strengthened non-
triviality condition

$\|q\|_{L^{\infty}}+\mu(]0,S])\neq 0.$ (6.45)

To this aim, assume by contradiction that $\|q\|_{L^{\infty}}+\mu(]0,S])=0.$
Then, the non-triviality condition (6.41) yields that $\mu(\\{0\\})\neq 0$ and
$p\equiv-\mu(\\{0\\})\zeta$ for some
$\zeta\in\partial_{x}^{>}h(0,\check{x}_{0})$. For every $i$, by the maximality
condition (vi)${}_{1}^{\prime}$ and condition (3.16), it follows that

$\begin{array}[]{l}\displaystyle
0\geq\int_{0}^{\rho_{i}}(1-2\eta_{i})\,p\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]\,ds\\\
\displaystyle\qquad+\int_{0}^{\rho_{i}}\left\\{(1-2\eta_{i})\,(p_{i}-p)\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]-2\gamma_{i}\rho_{i}^{2}\right\\}\,ds\\\\[6.45831pt]
\displaystyle\ \
\geq\int_{0}^{\rho_{i}}p\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]\chi_{\\{\sigma:\
\eta_{i}(\sigma)=0\\}}(s)\,ds\\\
\qquad\displaystyle-\int_{0}^{\rho_{i}}p\cdot\left[\mathcal{F}(s,y_{i},\hat{\omega}_{i},\hat{\alpha}_{i})-\mathcal{F}(s,y_{i},\tilde{\omega}_{i},\tilde{\alpha}_{i})\right]\chi_{\\{\sigma:\
\eta_{i}(\sigma)=1\\}}(s)\,ds\\\
\qquad\qquad-2\rho_{i}(K_{{}_{\mathcal{F}}}\|p_{i}-p\|_{L^{\infty}}+\tilde{L}\rho_{i}^{2})\\\\[6.45831pt]
\displaystyle\ \
\geq\mu(\\{0\\})\,\delta\,\ell(\rho_{i},1-\eta_{i}(\cdot))-2K_{{}_{\mathcal{F}}}K_{{}_{h}}\,\ell(\rho_{i},\eta_{i}(\cdot))-2\rho_{i}(K_{{}_{\mathcal{F}}}\|p_{i}-p\|_{L^{\infty}}+\tilde{L}\rho_{i}^{2})\\\\[6.45831pt]
\displaystyle\ \
\geq\rho_{i}\,\big{[}\mu(\\{0\\})\,\delta-\mu(\\{0\\})\,\delta\,\rho_{i}-2K_{{}_{\mathcal{F}}}K_{{}_{h}}\rho_{i}-2K_{{}_{\mathcal{F}}}\|p_{i}-p\|_{L^{\infty}}-2\tilde{L}\rho_{i}^{2}\big{]}>0,\end{array}$

where we use the facts that $\ell(\rho_{i},\eta_{i}(\cdot))\leq\rho_{i}^{2}$
and consequently $\ell(\rho_{i},1-\eta_{i}(\cdot))\geq\rho_{i}-\rho_{i}^{2}$,
which follow from (6.36). Thus, we obtain the desired contradiction.

###  6.2. Proof of Theorem 2.1

Preliminarily, observe that hypothesis (H3) can be reduced to (H3)′. We can
clearly take $k\geq 1$ in assumption (H3), but actually we may (and we do)
assume without loss of generality $k\equiv 1$. Indeed, reasoning as in [8,
Sec. 2], we can introduce the time change
$t=\sigma(s):=\int_{0}^{s}k(\tau)d\tau$, so that
$(\hat{\underline{\omega}},\hat{\underline{\alpha}},\hat{\lambda},\hat{y}):=(\underline{\omega},\underline{\alpha},\lambda,y)\circ\sigma^{-1}$
is a process for the transformed problem, with dynamics
$\hat{F}=\frac{1}{k}\sum_{j=0}^{n}\lambda^{j}\,\mathcal{F}(s,y,\omega^{j},\alpha^{j})$,
verifying (H3) for $k\equiv 1$, and interval $[0,\sigma(S)]$, if and only if
$(\underline{\omega},\underline{\alpha},\lambda,y)$ is a process for the
relaxed problem. Furthermore, the transformed process, say $\hat{z}$,
corresponding to
$\bar{z}:=(\underline{\bar{\omega}},\underline{\bar{\alpha}},{\bar{\lambda}},\bar{\xi},\bar{y})$
is isolated for the transformed problem, and if $\hat{z}$ is an abnormal
extremal for the transformed problem for some $(\hat{p},0,\hat{\mu},\hat{m})$
as in Definition 2.4, then $\bar{z}$ is an abnormal extremal with
$(p,\gamma,\mu,m)$ verifying $p=\hat{p}\circ\sigma$, $\gamma=0$,
$d\mu=k\,d\hat{\mu}$ and $m=\hat{m}\circ\sigma$. First of all, we notice that
$(\bar{\xi},\bar{y})$ is a solution of the differential inclusion

$\big{(}\dot{\xi},\dot{y}\big{)}(s)\in\text{{\rm co}
}\bigcup_{k=0}^{n}\\{(e^{k},\mathcal{F}(s,y(s),\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s)))\\}\qquad\text{a.e.
$s\in[0,S]$}.$

Let us fix a sequence $\varepsilon_{i}\downarrow 0$. By the Relaxation Theorem
[29, Th. 2.7.2], there exists a sequence of extended processes
$(\bar{\omega}_{i},\bar{\alpha}_{i},\bar{\lambda}_{i})(s)\in\bigcup_{k=0}^{n}\\{(\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s),e^{k})\\}$
for a.e. $s\in[0,S]$ such that, for any $i$, the corresponding trajectory
$(\bar{\xi}_{i},\bar{y}_{i}):=(\xi,y)[\bar{\omega}_{i},\dots,\bar{\omega}_{i},\bar{\alpha}_{i},\dots,\bar{\alpha}_{i},\bar{\lambda}_{i}]$
satisfies

$\|(\bar{\xi}_{i},\bar{y}_{i})-(\bar{\xi},\bar{y})\|_{L^{\infty}}\leq\varepsilon_{i}.$

Let $\mathcal{J}(\cdot)$, and $(\rho_{i})_{i}$ be as in the proof of Theorem
3.1 and $(\delta_{i})_{i}$ such that for every $\omega\in\mathcal{W}$ with
$\|\omega-\bar{\omega}_{i}\|_{L^{1}}\leq\delta_{i}$, then
$\|y[\omega,\bar{\alpha}_{i}]-\bar{y}_{i}\|_{L^{\infty}}\leq\varepsilon_{i}$.
Then, thanks to hypothesis (H1), for any $i$ there exists
$\check{\omega}_{i}\in\mathcal{V}_{\delta_{i}}$ such that
$\|\check{\omega}_{i}-\bar{\omega}_{i}\|_{L^{1}}\leq\delta_{i}$. As a
consequence, if we define $\check{\alpha}_{i}\equiv\bar{\alpha}_{i}$,
$\check{\lambda}_{i}\equiv\bar{\lambda}_{i}$ and
$\check{y}_{i}=y[\check{\omega}_{i},\check{\alpha}_{i}]$, then $\check{y}_{i}$
is a strict sense trajectory that satisfies
$\|\check{y}_{i}-\bar{y}_{i}\|_{L^{\infty}}\leq\varepsilon_{i}$ and

$(\check{\omega}_{i},\check{\alpha}_{i},\check{\lambda}_{i})(s)\in\bigcup_{k=0}^{n}\\{(\bar{\omega}^{k}(s),\bar{\alpha}^{k}(s),e^{k})\\}+(\check{r}_{i}(s),0,0)\mathbb{B}\qquad\text{a.e.
$s\in[0,S]$,}$

for some measurable sequence $\check{r}_{i}\to 0$ in $L^{1}$. Therefore, the
process
$(\check{\omega}_{i},\check{\alpha}_{i},\check{\lambda}_{i},\check{\xi}_{i},\check{y}_{i})$
is a $\rho_{i}^{4}$-minimizer for the optimal control problem

$\left(\check{P}_{i}\right)\begin{cases}\qquad\qquad\qquad\qquad\qquad\text{minimize
}\,\,\,{\mathcal{J}}(y)\\\ \text{over
}(\omega,\alpha,\lambda,y,\xi)\in\mathcal{V}_{\delta_{i}}\times\mathcal{A}\times\Lambda_{n}^{1}\times
W^{1,1}([0,S];\mathbb{R}^{n+n}),\text{ satisfying}\\\
\displaystyle(\dot{\xi},\dot{y})(s)=\big{(}\lambda(s),\,\,\mathcal{F}(s,y(s),\omega(s),\alpha(s))\big{)}\
\text{ a.e. }s\in[0,S]\\\ (\xi,y)(0)=(0,\check{x}_{0}).\end{cases}$

From now on, except for minor obvious changes, the proof proceeds as the proof
of Theorem 3.1 and is actually simpler, since we disregard the nondegeneracy
issue.

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