# Optimal impulsive control for time delay systems

Giovanni Fusco  G. Fusco, Department of Mathematics ‘Tullio Levi-Civita‘,
University of Padova, Padova, 35121, Italy
email<EMAIL_ADDRESS>,  Monica Motta  M. Motta, Department of
Mathematics ‘Tullio Levi-Civita‘, University of Padova, Padova, 35121, Italy
email<EMAIL_ADDRESS>and  Richard Vinter  R. Vinter, Department of
Electrical and Electronic Engineering, Imperial College London, Exhibition
Road, London SW7 2BT, UK
email<EMAIL_ADDRESS>
###### Abstract.

We introduce discontinuous solutions to nonlinear impulsive control systems
with state time delays in the dynamics and derive necessary optimality
conditions in the form of a Maximum Principle for associated optimal control
problems. In the case without delays, if the measure control is scalar valued,
the corresponding discontinuous state trajectory, understood as a limit of
classical state trajectories for absolutely continuous controls approximating
the measure, is unique. For vector valued measure controls however, the
limiting trajectory is not unique and a full description of the control must
include additional ‘attached’ controls affecting instantaneous state evolution
at a discontinuity. For impulsive control systems with time delays we reveal a
new phenomenon, namely that the limiting state trajectory resulting from
different approximations of a given measure control needs not to be unique,
even in the scalar case. Correspondingly, our framework allows for additional
attached controls, even though the measure control is scalar valued.

###### Key words and phrases:

Impulsive Control, Time Delay Systems, Optimal Control, Maximum Principle,
Bounded Variation.

###### 2020 Mathematics Subject Classification:

49N25, 93C43, 49K21, 26A45, 49J21

This research is partially supported by the INdAM-GNAMPA Project, codice
CUP-E53C22001930001.

##  1\. Introduction

This paper provides a well-posed notion of solution for a control system of
the form

${\rm(DIS)}\left\\{\begin{array}[]{l}\displaystyle
dx(t)=f(x(t),x(t-h))dt+g(x(t),x(t-h))d\mu(t),\;t\in[0,T]\\\\[4.30554pt]
\mbox{in which }\mu\mbox{ is a non-negative Borel measure on $[0,T]$, and
}\\\\[4.30554pt] x(0)=x_{0}\mbox{ and }x(t)=\xi_{0}\mbox{ for
}t\in[-h,0[\,,\end{array}\right.$

and necessary conditions of optimality for associated optimal impulsive
control problems. Here the data comprise a constant $T>0$, functions
$f:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and
$g:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, vectors
$x_{0}\in\mathbb{R}^{n}$ and $\xi_{0}\in\mathbb{R}^{n}$.

There is an extensive literature on control systems of this kind whenever $f$
and $g$ do not depend on the delayed state, both in the case of scalar and
vector valued control measures, which includes versions of the maximum
principle for optimal impulsive control problems. Representative papers
include [5, 6, 16, 15, 21, 22, 19, 25, 3, 13, 1, 18]. The novel feature of the
class of problems treated in this paper is that the dynamic constraint
contains a time delay. Since first order necessary optimality conditions are
available for conventional (‘non-impulsive’) optimal control problems with
time delay (see, e.g, [4] and the references therein), it might be thought
that broadening of the theory to allow for time delays in the state would be a
routine exercise. But, contrary to such expectations, the presence of a time
delay gives rise to interesting and unexpected phenomena which, to the
authors’ knowledge, have not been previously explored in the literature.
Furthermore, the derivation of necessary conditions poses significant
challenges, because the ‘change of independent variable’ techniques employed
in the delay free literature, to reduce impulsive optimal control problems to
conventional optimal control problems, cannot be adapted directly to the delay
setting, owing to the fact that the analogous change of variable does not
generate a time-delay optimal control problem of standard form.

Notice that when the ‘control’ $\mu$ in (DIS) is absolutely continuous w.r.t.
Lebesgue measure, i.e. $d\mu(t)=u(t)dt$ for some integrable function $u$, the
state trajectory is simply a solution to the delay differential equation

$\dot{x}(t)=f(x(t),x(t-h))+g(x(t),x(t-h))u(t),\mbox{ a.e. }t\in[0,T]$

satisfying the specified left end-point condition. But if the measure $\mu$
has an atom at $\bar{t}$, then according to the control system, the state
variable may jump at time $\bar{t}$.

The motivation behind the definition of ‘solution’ to the state equation, in
the case without delays, was first provided by aerospace applications, in
which (as in (DIS)) the right side of the dynamic relation is affine in the
control variable and where an ‘impulsive control’ is an idealization of high
intensity control action over a time interval of short duration. The
appropriate definition of state trajectory is then that it should be the limit
of some sequence of state trajectories associated with a convergent sequence
of absolutely continuous controls [11].

The delay-free impulsive control literature, where the dynamic constraint
takes the form

$\left\\{\begin{array}[]{l}\displaystyle
dx(t)=\tilde{f}(x(t))dt+\tilde{g}(x(t))d\mu(t),\;t\in[0,T]\\\\[4.30554pt]
\displaystyle\mbox{in which }\mu\mbox{ is a non-negative Borel measure on
}[0,T],\mbox{ and }\\\\[4.30554pt]
x(0)=x_{0},\end{array}\right.\qquad\qquad\qquad$

points the way to defining state trajectories that can be interpreted as
limits of non-impulsive state trajectories. For an arbitrary impulsive control
$\mu$, all limits of state trajectories corresponding to absolutely continuous
controls approximating $\mu$ (in a sense specified in the example below) are
the same. If, furthermore, $\mu(\\{\tau\\})=\omega(\tau)$ at a time $\tau$,
then its state trajectory $x$ has a jump $y(1)-x(\tau^{-})$ at $\tau$, where
$y:[0,1]\rightarrow\mathbb{R}^{n}$ is the solution to the differential
equation

$\left\\{\begin{array}[]{l}\dot{y}(s)=\omega(\tau)\tilde{g}(y(s)),\mbox{ a.e.
}s\in[0,1]\\\\[4.30554pt] y(0)=x(\tau^{-})\,.\end{array}\right.$

As is well-documented, the property ‘all sequences of state trajectories
corresponding to some sequence of absolutely controls that approximates the
nominal impulsive control $\mu$ have the same limit’ depends critically on the
assumption that the impulsive control $\mu$ is scalar valued (see [5]). If the
delay-free model is modified to include a $k$-vector valued impulsive control
$\mu=(\mu_{1},\ldots,\mu_{k})$, thus

$\left\\{\begin{array}[]{l}\displaystyle
dx(t)=\tilde{f}(x(t))dt+\sum_{i=1}^{k}\tilde{g}_{i}(x(t))\mu_{i}(dt),\;t\in[0,T]\\\
\mbox{in which $\mu_{i}$, $i=1,\ldots,k$, are non-negative Borel measures on
}[0,T],\mbox{ and }\\\\[4.30554pt] x(0)=x_{0}\,,\end{array}\right.$

the uniqueness property is lost: different approximations of the vector
impulsive controls by absolutely continuous controls can give rise to
different state trajectories in the limit, unless we impose rather stringent
‘commutativity’ hypotheses on the $\tilde{g}_{i}$’s (see [6, 7, 16]). In this
situation, the family of state trajectories associated with a given vector
valued impulsive control $\mu$ can be parameterized by collections of control
functions attached to each point of discontinuity in the distribution of
$\mu$, which determine the evolution of the state during the jump (see [2,
13]).

Now let us return to consideration of impulsive control systems with time
delay, when the impulsive control is scalar valued. We might expect, by
analogy with the delay-free case, that all sequences of state trajectories
associated with some sequence of absolutely continuous controls that
approximates a given impulsive control, would have the same limit and this
could be defined as the state trajectory for the nominal impulsive control.
But instead we encounter a new phenomenon: the uniqueness property is lost
and, to capture the entire possible family of limiting state trajectories, we
need to consider collections of attached controls describing the evolution of
the state during the jump, in a way formerly encountered only for vector
valued impulsive control systems.

######  Example 1.1.

Consider the impulsive control system with scalar state $x$:

$\begin{cases}dx(t)=x(t)x(t-1)d\mu(t)\qquad\text{a.e.
$t\in[0,2]$,}\\\\[4.30554pt] \text{in which $\mu$ is a non-negative Borel
measure on $[0,2]$ and}\\\\[4.30554pt] x(t)=1\quad\forall
t\in[-1,0]\,.\end{cases}$

Take the nominal measure control to be
$\mu=\delta_{\\{1/2\\}}+\delta_{\\{3/2\\}}$, where $\delta_{\\{\bar{t}\\}}$
denotes the unit Borel measure concentrated at $t=\bar{t}$. Take
$\varepsilon_{i}\downarrow 0$ and, for any integer $i\geq 1$, let $u_{i}\,dt$
be the sequence of absolutely continuous measures approximating $\mu$:

$u_{i}(t)=\begin{cases}\varepsilon_{i}^{-1}&\mbox{\hskip 21.68121ptif
}t\in[\frac{1}{2},\frac{1}{2}+\varepsilon_{i}]\cup[\frac{3}{2}-\varepsilon_{i},\frac{3}{2}]\\\
0&\qquad\text{otherwise}.\end{cases}$

(The sequence $(u_{i}\,dt)$ approximates $\mu$ in the sense that
$u_{i}\,dt\overset{*}{\rightharpoonup}\mu$, in the weak∗ $C(0,2)$-topology.)
Write $x_{i}$ for the corresponding state trajectories. A simple calculation
reveals that $x_{i}(t)\rightarrow x(t)$ for all
$t\in[0,2]\setminus\\{\frac{1}{2},\frac{3}{2}\\}$, where $x(t)=1$ if
$t\in[0,1/2[$, $x(t)=e$ if $t\in[1/2,3/2[$ and $x(t)=e^{2}$ if $t\in[3/2,2]$.
Now let $(\tilde{u}_{i}\,dt)$ be a different sequence of absolutely continuous
measures approximating the nominal measure control $\mu$ given by

$\tilde{u}_{i}(t)=\begin{cases}\varepsilon_{i}^{-1}&\mbox{\hskip 21.68121ptif
}t\in[\frac{1}{2}-\varepsilon_{i},\frac{1}{2}]\cup[\frac{3}{2},\frac{3}{2}+\varepsilon_{i}]\\\
0&\mbox{\hskip 21.68121ptotherwise}.\end{cases}$

Write $\tilde{x}_{i}$ for the corresponding state trajectories. We find that
$\tilde{x}_{i}(t)\rightarrow\tilde{x}(t)$ for all
$t\in[0,2]\setminus\\{\frac{1}{2},\frac{3}{2}\\}$, where $\tilde{x}(t)=1$ if
$t\in[0,1/2[$, $\tilde{x}(t)=e$ if $t\in[1/2,3/2[$ and $\tilde{x}(t)=e^{1+e}$
if $t\in[3/2,2]$. Notice that the arcs $x$ and $\tilde{x}$ differ on the
subinterval $[\frac{3}{2},2]$.

In this paper we provide, for the first time, a framework for studying the
multiplicity of state trajectories associated with a given impulsive control,
in the presence of time delays. We pose an optimal control problem where
selection of a state trajectory corresponding to the chosen impulsive control
is part of the optimization procedure. We give conditions for existence of
solutions to this control problem and provide necessary optimality conditions
in the form of a maximum principle.

Let us briefly discuss the methodological challenges of generalizing the
techniques previously employed for studying impulsive control systems, when we
introduce time delays. Take a measure control $\mu$ with associated state
trajectory $x$. The key idea in earlier work concerning delay-free systems,
was to introduce a measure control dependent change of independent variable
$s=\psi(t)$, satisfying $\psi(0)=0$ and $\psi(t)=t+\mu([0,t])$ for
$t\in]0,T]$, and interpret a discontinuous state trajectory $x$ as an
absolutely continuous arc $y$, under the discontinuous change of independent
variable, thus $x(t)=y(\psi(t))$. The discontinuity in $x$ at time $t$ is
captured by the evolution of $y$ on $[\psi(t^{-}),\psi(t^{+})$] (the interval
between left and right limits of $\psi$ at $t$). Each $y$ is governed by a
controlled differential equation. The ‘controls’ in this new system
description generate all possible state trajectories for the original control
system with measure control $\mu$, obtainable as appropriate limits of
absolutely continuous state trajectories. We use this property to study the
set of impulsive state trajectories via a conventional control system. We can,
for example, derive necessary conditions of optimality for impulsive control
problems in this way.

This approach cannot be directly applied, however, when we introduce time
delays, because the control system description that results from the above
change of variable is a controlled delay differential equation, in which the
time delays vary with time and are control dependent. This complicated delay
differential equation is not amenable to analysis. Instead, we use a more
subtle change of independent variable that is, in a sense, uniform over
different time segments of length $h$ and which preserves constancy of the
delays under transformation.

Research is currently in progress into various extensions of this paper. These
include generalizations to allow for end-times that are not integer multiples
of a single time delay, vector-valued measure controls with range a convex
cone, and for non-autonomous control systems with multiple, commensurate time
delays. Conditions under which the infimum cost over classical controls is the
same as that over measure controls, so called ‘gap conditions’, are also under
investigation (see e.g. [17, 8, 9] in the case without delays).

It is worth mentioning that in the literature there are several results on the
stabilizability of delayed impulsive control systems and on the optimization
of some specific related problems, but they all concern the so-called ‘impulse
model’, where impulsive controls essentially reduce to a finite or countable
number of jump instants, with a preassigned jump-function. The notion of
impulsive control system introduced in the present paper, which includes the
use of strategies with a significant freer allocation of the impulses, might
allow the development of new nonlinear models, for instance, of fed-batch
fermentation [24, 12] or of delayed neural networks with impulsive controls
[14]. We also mention [10], in which necessary optimality conditions are
introduced for an impulsive control problem with (possibly non-commensurate)
time delays in the drift term alone of the dynamics.

The paper is structured as follows. In Sec. 2 we give an appropriate
definition of solution to (DIS), in Sec. 3 we introduce an ordinary control
system which can be obtained by (DIS) by means of a suitable change of
independent variable, in Sec. 4 we establish the main properties of solutions
to (DIS), while in Sec. 5 we associate with (DIS) a Mayer cost and an endpoint
constraint and we prove a maximum principle for this optimization problem.

###  1.1. Notation

Given a set $X\subseteq\mathbb{R}^{k}$, we write $W^{1,1}([a,b];X)$,
$L^{1}([a,b];X)$, $L^{\infty}([a,b];X)$, $BV([a,b];X)$, for the set of
absolutely continuous, Lebesgue integrable, essentially bounded, and bounded
variation functions defined on $[a,b]$ and with values in $X$, respectively.
We will not specify domains and codomains when the meaning is clear and we
will use $\|\cdot\|_{L^{\infty}(a,b)}$, or also $\|\cdot\|_{L^{\infty}}$, to
denote the ess-sup norm. As usual, given $T>0$, $S>0$ and an increasing,
surjective map $\sigma:[0,T]\to[0,S]$, $\sigma^{-}(t)$ and $\sigma^{+}(t)$
denote the left and right limits of $\sigma$ at $t$, respectively, when
$t\in(0,T)$, while $\sigma^{-}(0):=0$ and $\sigma^{+}(T):=S$. Note that we
call increasing any monotone non decreasing function. We denote by
$C^{\oplus}(0,T)$ the set of Borel non-negative scalar valued measures on
$[0,T]$ (from now on we will refer to such $\mu$ simply as measures). For any
$\mu\in C^{\oplus}(0,T)$, we use both the notations $\|\mu\|_{TV}$ and
$\int_{[0,T]}d\mu(t)$ for the total variation of $\mu$. In the following,
$\mu$-a.e. means “almost everywhere w.r.t. $\mu$”, and when we do not specify
$\mu$ we implicitly refer to Lebesgue measure. Given a sequence
$(\mu_{i})\subset C^{\oplus}(0,T)$ and $\mu\in C^{\oplus}(0,T)$, as customary
we write $\mu_{i}\overset{*}{\rightharpoonup}\mu$ if
$\lim_{i}\int_{[0,T]}\Phi(t)\mu_{i}(dt)=\int_{[0,T]}\Phi(t)d\mu(t)$ for all
continuous functions $\Phi:[0,T]\to\mathbb{R}$.

The limiting normal cone $N_{\mathcal{T}}(\bar{x})$ to a closed set
$\mathcal{T}\subseteq\mathbb{R}^{k}$ at $\bar{x}\in\mathbb{R}^{k}$ is

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

in which the notation
$x_{i}\stackrel{{\scriptstyle\mathcal{T}}}{{\longrightarrow}}\bar{x}$ means
that $(x_{i})_{i}\subset\mathcal{T}$.

##  2\. Impulsive Controls and Extended Processes

Consider the control system

${\rm(DIS)}\left\\{\begin{array}[]{l}\displaystyle
dx(t)=f(x(t),x(t-h))dt+g(x(t),x(t-h))d\mu(t),\;t\in[0,T]\\\\[4.30554pt] \mbox{
for some }\mu\in C^{\oplus}(0,T)\mbox{ and }x\in
BV([-h,T];\mathbb{R}^{n})\mbox{ satisfying }\\\\[4.30554pt] x(0)=x_{0}\mbox{
and }x(t)=\xi_{0}\mbox{ for }t\in[-h,0[\,,\end{array}\right.$

where $f,g:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$,
$x_{0}$ and $\xi_{0}\in\mathbb{R}^{n}$, and $h>0$, $T>0$. Assume:

(H1): $f$ and $g$ are $C^{1}$ and bounded functions.

(H2): $T=Nh,\mbox{ for some integer }N>0$.

######  Definition 2.1 (Impulsive and Strict Sense Controls).

An impulsive control $(\mu,\\{w^{r}\\}_{r\in[0,h]})$ comprises a function
$\mu\in C^{\oplus}(0,T)$ and a family of measurable functions
$w^{r}=(w^{r}_{1},\ldots,w^{r}_{N}):[0,1]\rightarrow[0,+\infty[^{N}$,
parameterized by $r\in[0,h]$, with the following properties:

  * (i):

For any $r\in[0,h]$,
$\displaystyle\sum_{i=1}^{N}w^{r}_{i}(s)=\sum_{i=1}^{N}\int_{0}^{1}w^{r}_{i}(s)ds,\
\mbox{ a.e. }s\in[0,1];$

  * (ii):

For any $r\in]0,h[$,
$\displaystyle\int_{0}^{1}w^{r}_{i}(s)ds=\mu(\\{r+(i-1)h\\}),\ \ \mbox{for
}i=1,\dots,N.$

  * (iii):

$\displaystyle\int_{0}^{1}[w^{h}_{i}(s)+w^{0}_{i+1}(s)]ds=\mu(\\{ih\\}),\ \
\mbox{for }i=0,\dots,N$, where $w^{h}_{0}\equiv 0$ and $w^{0}_{N+1}\equiv 0$.

A strict sense control is an impulsive control
$(\mu,\\{w^{r}\\}_{r\in[0,h]})$, in which $d\mu(t)=u(t)dt$ for some
nonnegative function $u\in L^{1}(0,T)$. With a slight abuse of notation, we
also call $u$ strict sense control. (Notice that, for a strict sense control,
for any $r\in[0,h]$ and $i=1,\dots,N$, we have $w^{r}_{i}(s)=0$ a.e. since
$d\mu(t)=u(t)dt$ has no atoms.)

######  Definition 2.2 (Extended and Strict Sense Processes).

We say that a triple $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ is an extended process
(with extended state trajectory $x$) if $x\in BV([-h,T];\mathbb{R}^{n})$ and
$(\mu,\\{w^{r}\\}_{r\in[0,h]})$ is an impulsive control such that

(i): $x(t)=\xi_{0}$, for $t\in[-h,0[$;

(ii): for each $i=1,\ldots,N$, $x(t)=\begin{cases}x_{0}\qquad\mbox{if }i=1\\\
\zeta_{i}^{0}(1)\qquad\mbox{if }i>1\end{cases}$ for $t=(i-1)h,$

$\begin{split}x(t)=\zeta^{0}_{i}(1)&+\int_{(i-1)h}^{t}f(x(t^{\prime}),x(t^{\prime}-h))dt^{\prime}+\int_{[(i-1)h,t]}g(x(t^{\prime}),x(t^{\prime}-h))d\mu^{c}(t^{\prime})\\\
&+\underset{r\in]0,\,t-(i-1)h]}{\sum}(\zeta^{r}_{i}(1)-x((r+(i-1)h)^{-})\qquad\mbox{
for }t\in](i-1)h,ih[\end{split}$

and $x(T)=\zeta^{h}_{N}(1)$. Here, for $r\in[0,h]$, the functions
$\zeta^{r}_{1},\ldots,\zeta^{r}_{N}:[0,1]\rightarrow\mathbb{R}^{n}$ satisfy
the coupled differential equations

$\frac{d}{ds}\zeta^{r}_{i}(s)\,=\,w_{i}^{r}(s)\,g(\zeta^{r}_{i}(s),\zeta^{r}_{i-1}(s)),\mbox{
a.e. }s\in[0,1],\quad\mbox{ for }i=1,\ldots,N\,,$ (2.1)

with boundary conditions

$\zeta^{r}_{i}(0)=\left\\{\begin{array}[]{ll}x^{-}(r+(i-1)h)&\mbox{if
}r\in]0,h]\\\ \zeta_{i-1}^{h}(1)&\mbox{if }r=0\,.\end{array}\right.$ (2.2)

In these relations,
$\zeta^{r}_{0}(s):=\left\\{\begin{array}[]{ll}\xi_{0}&\mbox{ if $s\in[0,1[$ or
$r\in[0,h[$}\\\ x_{0}&\mbox{if $s=1$ and $r=h$}\end{array}\right.$ and
$\mu^{c}$ denotes the continuous component of the measure $\mu$. (Extended
state trajectories have bounded variation and are right continuous on
$]0,T[$.)

An extended state process $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ corresponding to a
strict sense control $u$, also written simply $(x,u)$, is called a strict
sense process. In these circumstances $x$, referred to as a strict sense state
trajectory, is an absolutely continuous function that satisfies the delayed
differential equation

$\dot{x}(t)=f(x(t),x(t-h))+g(x(t),x(t-h))u(t),\mbox{ a.e. }t\in[0,T]\,.$

Note that if, for some $i=1,\dots,N$, $t\in](i-1)h,ih[$ is not an atom of
$\mu$, then $w^{t-(i-1)h}_{i}=0$ a.e and the trajectory $x$ is continuous at
$t$, as the function $\zeta^{t-(i-1)h}_{i}$ describing the jump of the state
at the instant $t$, turns out to be constant on $[0,1]$. On the other hand,
when $t=ih$ for $i=1,\dots,N-1$, the possible atom of the measure $\mu$ at $t$
(hence, the corresponding jump of the trajectory $x$) is determined by the
combined effect of both the attached controls $w^{h}_{i}$ and $w^{0}_{i+1}$,
due to the presence of the time delay $h$.

######  Remark 2.3.

In the earlier Example 1.1 (in which $h=1$, $N=2$, and $T=Nh=2$), we showed
how different approximations of a measure control $\mu$ by absolutely
continuous controls can give rise to different state trajectories. We show how
these observations are consistent with the preceding definition of state
trajectories corresponding to impulsive controls. Choose once again
$\mu=\delta_{\\{1/2\\}}+\delta_{\\{3/2\\}}\,.$ Now consider the following
impulsive control $(\mu,\\{w^{r}\\}_{r\in[0,1]})$, in which $w^{r}\equiv 0$ if
$r\not=\frac{1}{2}$ and

$w^{r=\frac{1}{2}}_{1}(t)=\begin{cases}0&\mbox{if }0\leq t\leq\frac{1}{2}\\\
2&\mbox{if }\frac{1}{2}<t\leq 1\end{cases}\qquad\mbox{ and }\qquad
w^{r=\frac{1}{2}}_{2}(t)=\begin{cases}2&\mbox{if }0\leq t\leq\frac{1}{2}\\\
0&\mbox{if }\frac{1}{2}<t\leq 1.\end{cases}$

Furthermore, consider $(\mu,\\{\tilde{w}^{r}\\}_{r\in[0,1]})$ in which
$\tilde{w}^{r}\equiv 0$, if $r\not=\frac{1}{2}$ and

$\tilde{w}^{r=\frac{1}{2}}_{1}(t)=\begin{cases}2&\mbox{if }0\leq
t\leq\frac{1}{2}\\\ 0&\mbox{if }\frac{1}{2}<t\leq 1\end{cases}\qquad\mbox{ and
}\qquad\tilde{w}^{r=\frac{1}{2}}_{2}(t)=\begin{cases}0&\mbox{if }0\leq
t\leq\frac{1}{2}\\\ 2&\mbox{if }\frac{1}{2}<t\leq 1.\end{cases}$

Based on the constructions implicit in the preceding definition, the state
trajectories $x$ and $\tilde{x}$, corresponding to the impulsive controls
$(\mu,\\{w^{r}\\}_{r\in[0,1]})$ and $(\mu,\\{\tilde{w}^{r}\\}_{r\in[0,1]})$,
respectively, are

$x(t)=\begin{cases}1&\mbox{if }t\in[0,\frac{1}{2}[\\\ e&\mbox{if
}t\in[\frac{1}{2},\frac{3}{2}[\\\ e^{2}&\mbox{if
}t\in[\frac{3}{2},2]\end{cases}\quad\text{and}\quad\tilde{x}(t)=\begin{cases}1&\mbox{if
}t\in[0,\frac{1}{2}[\\\ e&\mbox{if }t\in[\frac{1}{2},\frac{3}{2}[\\\
e^{1+e}&\mbox{if }t\in[\frac{3}{2},2].\end{cases}$

These formulae reproduce the two distinct limits of state trajectories
corresponding to absolutely continuous controls, identified in the earlier
analysis of this example.

##  3\. Reparameterization

Processes for (DIS) will be analysed with the help of a change of independent
variable that replaces possibly discontinuous state trajectories by continuous
functions. The new set of processes that arise in this way are called
reparameterized processes for (DIS).

######  Definition 3.1.

A reparameterized process for (DIS) is a collection of elements
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$, comprising a number $S>0$, controls
$\alpha_{i}\in L^{\infty}(0,S)$, $i=1,\ldots,N$, and functions $y_{i}\in
W^{1,1}([0,S];\mathbb{R}^{n})$, $i=0,\ldots,N$, that satisfy

$\
\left\\{\begin{array}[]{l}\displaystyle\dot{y}_{i}(s)=\Big{(}1-\sum_{i=1}^{N}\alpha_{i}(s)\Big{)}f(y_{i}(s),y_{i-1}(s))+\alpha_{i}(s)g(y_{i}(s),y_{i-1}(s)),\\\
\hskip 180.67499pt\mbox{ a.e. $s\in[0,S]$, for $i=1,\ldots,N$},\\\
\displaystyle\alpha_{i}(s)\geq 0\mbox{ for each }i=1,\dots,N\mbox{ and
}\sum_{i=1}^{N}\alpha_{i}(s)\leq 1,\mbox{ a.e. }s\in[0,S],\\\\[4.30554pt]
y_{0}(s)=\xi_{0}\ \text{for $s\in[0,S[$},\ y_{0}(S)=x_{0},\ \
y_{i+1}(0)=y_{i}(S),\mbox{ for }i=0,\ldots N-1\,,\\\\[4.73611pt]
\displaystyle\int_{0}^{S}\Big{(}1-\sum_{i=1}^{N}\alpha_{i}(s)\Big{)}ds=h.\end{array}\right.$

######  Remark 3.2.

Because $f$ and $g$ are $C^{1}$, bounded functions, for any $S>0$ and all
controls $\alpha_{1},\dots,\alpha_{N}\in L^{1}(0,S)$ such that
$\alpha_{i}(s)\geq 0$ for each $i$, $\sum_{i=1}^{N}\alpha_{i}(s)\leq 1$ a.e.,
and $\int_{0}^{S}(1-\sum_{i=1}^{N}\alpha_{i}(s))ds=h$, there exists a unique
set of functions $y_{1},\dots,y_{N}\in W^{1,1}([0,S];\mathbb{R}^{n})$ such
that $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ is a reparameterized process for
(DIS).

######  Theorem 3.3.

The following statements hold true:

  * (A):

Let $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ be a reparameterized process for (DIS).
Then, there exists an extended process $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for
(DIS) such that $x(T)=y_{N}(S)$ and

$\|\mu\|_{\mbox{TV}}={\sum}_{i=1}^{N}\;\int_{0}^{S}\alpha_{i}(s)ds.$ (3.1)

  * (B):

Let $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ be an extended process for (DIS). Then
there exists a reparameterized process $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ for
(DIS) such that $y_{N}(S)=x(T)$ and (3.1) is satisfied.

Proof of this theorem requires the following lemma.

######  Lemma 3.4.

Take an increasing function $A:[0,a]\to[0,b]$ such that $A(0)=0$, $A(a)=b$,
and $A$ is right continuous on $]0,a[$. Define its right inverse as the
function $B:[0,b]\to[0,a]$ such that

$B(0):=0,\qquad B(r):=\inf\\{s\in[0,a]:\ \ A(s)>r\\}\ \text{for
}r\in]0,b[,\qquad B(b):=a.$

Then, $B$ is increasing, right continuous on $]0,b[$, and it is continuous iff
$A$ is strictly increasing. For any $r\in[0,b]$ one has $A(B(r))\geq r$ and,
when $A$ is continuous at $B(r)$, $A(B(r))=r$. Moreover, $A$ is the right
inverse of $B$, namely

$A(s)=\inf\\{r\in[0,b]:\ \ B(r)>s\\}\ \text{for }s\in]0,a[.$

Furthermore, if $F:[0,a]\to\mathbb{R}^{n}$ is a Borel function, then

$\int_{[B(r_{1}),B(r_{2})]}F(s)dA(s)=\int_{[r_{1},r_{2}]}F(B(r))dr\quad\text{for
every $0\leq r_{1}<r_{2}\leq b$,}$

where $dA(s)$ is the measure associated with the increasing function $A$.

###### Proof.

If we consider $A$ as the restriction to $[0,a]$ of an increasing and right
continuous function, continuous outside $[0,a]$, the properties of $B$ follow
from [20, Lemma 4.8, Ch. 0]. For each $\ell=1,\dots,n$, let $F^{+}_{\ell}$ and
$F^{-}_{\ell}$ be the positive and the negative part of the component
$F_{\ell}$ of $F$, so that $F^{+}_{\ell}$, $F^{-}_{\ell}\geq 0$, and
$F_{\ell}(s)=F^{+}_{\ell}(s)-F^{-}_{\ell}(s)$ for all $s\in[0,a]$. Fix $0\leq
r_{1}<r_{2}\leq b$. Then, if we extend $F^{+}_{\ell}$, $F^{-}_{\ell}$ as
nonnegative Borel functions on $[0,+\infty[$, [20, Prop. 4.9, Ch. 0] implies
that

$\int_{[0,+\infty[}F^{\pm}_{\ell}(s)\chi_{[B(r_{1}),B(r_{2})]}(s)dA(s)=\int_{[0,+\infty[}F^{\pm}_{\ell}(B(r))\chi_{[r_{1},r_{2}]}(r)dr,$

from which we immediately derive
$\int_{[B(r_{1}),B(r_{2})]}F_{\ell}(s)dA(s)=\int_{[r_{1},r_{2}]}F_{\ell}(B(r))dr,$
for each value of the index $\ell$. ∎

######  Proof of Thm. 3.3.

(A): Take a reparameterized process $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ for
(DIS). Define the mapping $\psi:[0,S]\rightarrow\mathbb{R}$ to be

$\psi(s):=\int_{0}^{s}\Big{(}1-\sum_{i=1}^{N}\alpha_{i}(s^{\prime})\Big{)}ds^{\prime},\;\mbox{for
}s\in[0,S]\,.$ (3.2)

Directly from the definition of reparameterized processes for (DIS), we know
that $\psi(S)=h$. Because $\psi$ is a continuous, increasing function, by
Lemma 3.4 its right inverse function $\sigma:[0,h]\to[0,S]$, given by

$\sigma(0):=0,\qquad\sigma(r):=\inf\\{s\in[0,S]:\ \ \psi(s)>r\\}\ \text{for
}r\in]0,h[,\qquad\sigma(h):=S,$

is strictly increasing, right continuous on $]0,h[$, and satisfies

$\sigma(r)\geq r,\quad\psi(\sigma(r))=r,\quad\text{for any $r\in[0,h]$.}$

Moreover, it has at most a countable set
$(r_{k})_{k\in\mathbb{N}}\subset[0,h]$ of discontinuity points, where
$\sigma^{-}(r_{k})<\sigma^{+}(r_{k})$, which correspond to the constancy
intervals of $\psi$, namely
$\psi^{-1}(\\{r_{k}\\})=[\sigma^{-}(r_{k}),\sigma^{+}(r_{k})]$ for all
$k\in\mathbb{N}$. Notice that, by the definition of $\psi$, for any
$r\in[0,h]$,

$\sum_{j=1}^{N}\alpha_{j}(s)=1,\mbox{ for a.e.
}s\in[\sigma^{-}(r),\sigma^{+}(r)].$ (3.3)

(If $r\notin(r_{k})_{k}$, we trivially have
$[\sigma^{-}(r),\sigma^{+}(r)]=\\{\sigma(r)\\}$.) Integrating across this
relation over the subinterval $[\sigma^{-}(r),\sigma^{+}(r)]$, yields

$\sigma^{+}(r)-\sigma^{-}(r)=\sum_{j=1}^{N}\int_{\sigma^{-}(r)}^{\sigma^{+}(r)}\alpha_{j}(s)\,ds,\quad\text{for
any $r\in[0,h]$.}$ (3.4)

Let us also introduce the continuous, increasing function
$\tilde{\psi}:[0,NS]\to\mathbb{R}$, obtained by an $N$-fold time advance and
concatenation of $\psi$, namely

$\tilde{\psi}(s):=(i-1)h+\psi(s-(i-1)S),\quad\text{for
$s\in[(i-1)S,iS]$,}\quad i=1,\dots,N,$

and its right inverse $\tilde{\sigma}:[0,T]\to[0,NS]$. Note that

$\tilde{\sigma}^{-}(r)=\sigma^{-}(r-(i-1)h)+(i-1)S,\quad\tilde{\sigma}^{+}(r)=\sigma^{+}(r-(i-1)h)+(i-1)S,$
(3.5)

for any $i=1,\dots,N$ and $r\in](i-1)h,ih[$, while

$\begin{array}[]{l}\tilde{\sigma}^{-}(0)=\sigma^{-}(0)=0,\quad\tilde{\sigma}^{+}(0)=\sigma^{+}(0),\\\\[6.45831pt]
\tilde{\sigma}^{-}(ih)=\sigma^{-}(h)+(i-1)S,\quad\tilde{\sigma}^{+}(ih)=\sigma^{+}(0)+iS,\quad
i=1,\dots,N-1,\\\\[6.45831pt]
\tilde{\sigma}^{-}(Nh)=\sigma^{-}(h)+(N-1)S,\quad\tilde{\sigma}^{+}(Nh)=\sigma^{+}(h)+(N-1)S=NS.\end{array}$
(3.6)

We now construct an extended process $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for
(DIS) with the desired properties. Consider the functions $\tilde{y}\in
W^{1,1}([0,NS];\mathbb{R}^{n})$ and $\tilde{\alpha}\in L^{1}(0,NS)$, given by

$\begin{array}[]{l}\tilde{y}(s)=y_{i}(s-(i-1)S),\quad\text{for
$s\in[(i-1)S,iS]$, \ $i=1,\dots,N$}\\\\[6.45831pt]
\tilde{\alpha}(s)=\alpha_{i}(s-(i-1)S),\quad\text{a.e. $s\in[(i-1)S,iS[$, \
$i=1,\dots,N$.}\end{array}$ (3.7)

Define $\mu\in C^{\oplus}(0,T)$ and $x:[0,T]\to\mathbb{R}^{n}$ according to
the following relations:

$\mu([0,t]):=\int_{0}^{\tilde{\sigma}^{+}(t)}\tilde{\alpha}(s)\,ds\quad\text{for
any $t\in[0,T]$,}$ (3.8)

and $x(t):=\tilde{y}(\tilde{\sigma}(t))$ for any $t\in[0,T]$, $x(t)=\xi_{0}$
for $t<0$ (notice that $\tilde{\sigma}^{+}(t)=\tilde{\sigma}(t)$ for all
$t\in]0,T]$). Finally, take any $r\in[0,h]$. For all $i=1,\dots,N$, define
$w_{i}^{r}:[0,1]\rightarrow[0,+\infty[$ and
$\zeta^{r}_{i}:[0,1]\rightarrow\mathbb{R}^{n}$ as follows:

$\begin{split}&w_{i}^{r}(s):=(\sigma^{+}(r)-\sigma^{-}(r))\alpha_{i}(\sigma^{-}(r)+s\,(\sigma^{+}(r)-\sigma^{-}(r))),\;\mbox{
a.e. }s\in[0,1],\\\
&\zeta_{i}^{r}(s):=y_{i}(\sigma^{-}(r)+s\,(\sigma^{+}(r)-\sigma^{-}(r))),\;\mbox{
for all }s\in[0,1].\end{split}$

Moreover, set $\zeta_{0}^{r}(s):=\xi_{0}$ if $s\in[0,1[$ or $r\in[0,h[$ and
$\zeta_{0}^{h}(1):=x_{0}$. From (3.3) it follows that
$\sum_{i=1}^{N}w^{r}_{i}(s)=(\sigma^{+}(r)-\sigma^{-}(r))=\sum_{i=1}^{N}\int_{\sigma^{-}(r)}^{\sigma^{+}(r)}\alpha_{i}(s)ds$
for a.e. $s\in[0,1]$.

By changing the independent variable in the integral, we see that

$\int_{0}^{1}w^{r}_{i}(s)ds=\int_{\sigma^{-}(r)}^{\sigma^{+}(r)}\alpha_{i}(s)ds\quad\text{for
$i=1,\dots,N$}.$ (3.9)

Therefore, condition (i) in Definition 2.1 of impulsive control is satisfied.
It can be deduced from (3.8) and (3.5) that, for $r\in]0,h[$ and
$i=1,\dots,N$,

$\mu(\\{r+(i-1)h\\})=\int_{\sigma^{-}(r)+(i-1)S}^{\sigma^{+}(r)+(i-1)S}\alpha_{i}(s-(i-1)S)\,ds=\int_{\sigma^{-}(r)}^{\sigma^{+}(r)}\alpha_{i}(s)\,ds.$

Together with (3.9), this relation implies condition (ii) in Definition 2.1.
Finally, condition (iii) follows from (3.8), (3.6), and (3.9). Indeed
$\mu(\\{0\\})=\int_{\sigma^{-}(0)}^{\sigma^{+}(0)}\alpha_{1}(s)\,ds=\int_{0}^{1}w^{0}_{1}(s)ds$,
$\mu(\\{Nh\\})=\int_{\sigma^{-}(h)}^{\sigma^{+}(h)}\alpha_{N}(s)\,ds=\int_{0}^{1}w^{h}_{N}(s)ds$,
and, for $i=1,\dots,N-1$,
$\mu(\\{ih\\})=\int_{\tilde{\sigma}^{-}(ih)}^{\tilde{\sigma}^{+}(ih)}\tilde{\alpha}(s)\,ds$,
so that:

$\begin{split}\mu(\\{ih\\})&=\int_{\sigma^{-}(h)+(i-1)S}^{S+(i-1)S}\alpha_{i}(s-(i-1)S)\,ds+\int_{iS}^{\sigma^{+}(0)+iS}\alpha_{i+1}(s-iS)\,ds\\\
&=\int_{\sigma^{-}(h)}^{\sigma^{+}(h)}\alpha_{i}(s)\,ds+\int_{\sigma^{-}(0)}^{\sigma^{+}(0)}\alpha_{i+1}(s)\,ds=\int_{0}^{1}w^{h}_{i}(s)ds+\int_{0}^{1}w^{0}_{i+1}(s)ds.\end{split}$

Thus, $(\mu,\\{w^{r}\\}_{r\in[0,h]})$ is an impulsive control. Notice that,
from (3.8),

$||\mu||_{TV}=\sum_{i=1}^{N}\,\int_{(i-1)S}^{iS}\alpha_{i}(s-(i-1)S)\,ds=\sum_{i=1}^{N}\,\int_{0}^{S}\alpha_{i}(s)\,ds.$

Take $r\in[0,h]$. Observe that, for $i=1,\dots,N$ the $\zeta^{r}_{i}$’s are
absolutely continuous functions such that
$\frac{d}{ds}\zeta^{r}_{i}(s)\,=\,w_{i}^{r}(s)\,g(\zeta^{r}_{i}(s),\zeta^{r}_{i-1}(s))$
a.e. $s\in[0,1]$ and $\zeta^{r}_{i}(0)=y_{i}(\sigma^{-}(r)).$ Furthermore,

$y_{i}(\sigma^{+}(r))-y_{i}(\sigma^{-}(r))=\zeta_{i}^{r}(1)-\zeta^{r}_{i}(0)\,,\quad\text{for
every $i=1,\ldots,N$.}$ (3.10)

Consider now $t\in](i-1)h,ih[$ for some $i=1,\ldots,N$. Then
$\tilde{\sigma}^{+}(t)\in](i-1)S,iS[$ and from the previous calculations it
follows that

$\begin{split}\tilde{y}(\tilde{\sigma}^{+}(t))-\tilde{y}(\tilde{\sigma}^{-}(t))&=y_{i}(\sigma^{+}(t-(i-1)h))-y_{i}(\sigma^{-}(t-(i-1)h))\\\
&=\zeta_{i}^{t-(i-1)h}(1)-\zeta_{i}^{t-(i-1)h}(0)\,.\end{split}$ (3.11)

If instead $r=ih$ for some $i=1,\ldots,N$, (3.7) (together with the fact that
$y_{i+1}(0)=y_{i}(S)$) yield

$\begin{split}&\tilde{y}(\tilde{\sigma}^{+}(ih))-\tilde{y}(\tilde{\sigma}^{-}(ih))=\tilde{y}(\sigma^{+}(0)+iS)-\tilde{y}(\sigma^{-}(h)+(i-1)S)\\\
&\qquad=[y_{i+1}(\sigma^{+}(0))-y_{i+1}(\sigma^{-}(0))]+[y_{i}(\sigma^{+}(h))-y_{i}(\sigma^{-}(h))]\\\
&\qquad=[\zeta_{i+1}^{0}(1)-\zeta_{i+1}^{0}(0)]+[\zeta_{i}^{h}(1)-\zeta_{i}^{h}(0)]\,,\quad\text{for
$i=1,\dots,N-1$,}\\\
&\tilde{y}(\tilde{\sigma}^{+}(Nh))-\tilde{y}(\tilde{\sigma}^{-}(Nh))=y_{N}(\sigma^{+}(h))-y_{N}(\sigma^{-}(h))=\zeta_{N}^{h}(1)-\zeta^{h}_{N}(0)\,.\end{split}$
(3.12)

For each $t\in](i-1)h,ih[$ and $i=1,\dots,N$, by (3.7) and (3.5), we have

$\begin{split}x(t)=\tilde{y}(\tilde{\sigma}(t))=\tilde{y}(\tilde{\sigma}^{+}((i-1)h))&+\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}f(\tilde{y}(s),\tilde{y}(s-S))\frac{d\tilde{\psi}}{ds}(s)\,ds\\\
&+\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}g(\tilde{y}(s),\tilde{y}(s-S))\tilde{\alpha}(s)ds\,,\end{split}$

where $\tilde{y}(\tilde{\sigma}^{+}((i-1)h))=\zeta_{i}^{0}(1)$. We deduce from
Lemma 3.4 and (3.5) that

$\begin{split}\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}f(\tilde{y}(s),\tilde{y}(s-S))\frac{d\tilde{\psi}}{ds}(s)\,ds&=\int_{(i-1)h}^{t}f(\tilde{y}(\tilde{\sigma}^{+}(t^{\prime})),\tilde{y}(\tilde{\sigma}^{+}(t^{\prime})-S))\,dt^{\prime}\\\
&=\int_{(i-1)h}^{t}f(x(t^{\prime}),x(t^{\prime}-h))dt^{\prime}\,.\end{split}$

Define the set of nondegenerate intervals corresponding to the discontinuity
points $(t_{k})_{k\in\mathbb{N}}$ of $\tilde{\sigma}$ on $[0,T]$, namely
$D:=\cup_{k\in\mathbb{N}}[\tilde{\sigma}^{-}(t_{k}),\tilde{\sigma}^{+}(t_{k})]\subset[0,NS]$.

Write $\tilde{\alpha}^{0}(s)=\left\\{\begin{array}[]{ll}0&\mbox{if }s\in D\\\
\tilde{\alpha}(s)&\mbox{if }s\notin D\end{array}\right.$. The continuous
component $\mu^{c}$ of $\mu$ is the measure with distribution function
$\mu^{c}([0,t])=\int_{0}^{\tilde{\sigma}^{+}(t)}\tilde{\alpha}^{0}(s^{\prime})ds^{\prime}$
for $t\in[0,T]$. Notice that, since
$\tilde{\sigma}(\tilde{\psi}(s^{\prime}))=s^{\prime}$ for all
$s^{\prime}\notin D$, from Lemma 3.4 (applied to $A=\tilde{\sigma}$,
$B=\tilde{\psi}$, and
$F(t^{\prime})=\tilde{\alpha}^{0}(\tilde{\sigma}(t^{\prime}))$) for
$t\in[0,T]$ it follows that

$\int_{[0,t]=[\tilde{\psi}(0),\tilde{\psi}(\tilde{\sigma}(t))]}\tilde{\alpha}^{0}(\tilde{\sigma}(t^{\prime}))\,d\tilde{\sigma}(t^{\prime})=\int_{0}^{\tilde{\sigma}(t)}\tilde{\alpha}^{0}(s^{\prime})ds^{\prime}=\mu^{c}([0,t]).$
(3.13)

For each $t\in](i-1)h,ih[$ and $i=1,\dots,N$, we have

$\begin{split}&\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}g(\tilde{y}(s),\tilde{y}(s-S))\tilde{\alpha}(s)ds\\\
&\quad\qquad=\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}g(\tilde{y}(s),\tilde{y}(s-S))\tilde{\alpha}^{0}(s)ds+\underset{t^{\prime}\in](i-1)h,t]}{\sum}(\tilde{y}(\tilde{\sigma}^{+}(t^{\prime}))-\tilde{y}(\tilde{\sigma}^{-}(t^{\prime}))).\end{split}$

In view of (3.13), from Lemma 3.4 (applied to
$F(\cdot)=g(\tilde{y}(\tilde{\sigma}(\cdot)),\tilde{y}(\tilde{\sigma}(\cdot)-S))\tilde{\alpha}^{0}(\tilde{\sigma}(\cdot)$,
$A=\tilde{\sigma}$ and $B=\tilde{\psi}$) it follows that

$\begin{array}[]{l}\displaystyle\int_{[(i-1)h,t]}g(x(t^{\prime}),x(t^{\prime}-h))d\mu^{c}(t^{\prime})\\\
\qquad\qquad\displaystyle=\int_{[\tilde{\psi}(\tilde{\sigma}((i-1)h)),\tilde{\psi}(\tilde{\sigma}(t))]}g(\tilde{y}(\tilde{\sigma}(t^{\prime})),\tilde{y}(\tilde{\sigma}(t^{\prime})-S))\tilde{\alpha}^{0}(\tilde{\sigma}(t^{\prime}))\,d\tilde{\sigma}(t^{\prime})\\\
\qquad\qquad\displaystyle=\int_{\tilde{\sigma}^{+}((i-1)h)}^{\tilde{\sigma}^{+}(t)}g(\tilde{y}(s),\tilde{y}(s-S))\tilde{\alpha}^{0}(s)ds.\end{array}$

Furthermore, (3.11), (3.12) imply that

$\underset{t^{\prime}\in](i-1)h,t]}{\sum}(\tilde{y}(\tilde{\sigma}^{+}(t^{\prime}))-\tilde{y}(\tilde{\sigma}^{-}(t^{\prime})))=\underset{r\
\in]0,t-(i-1)h]}{\sum}(\zeta^{r}_{i}(1)-\zeta^{r}_{i}(0))$

if $i=1,\dots,N-1$ and $t\in](i-1)h,ih[$, or if $i=N$ and $t\in](N-1)h,Nh]$.

Notice that the function $x$ is right continuous on $]0,T[$, so that the
boundary conditions on the $\zeta^{r}_{i}$’s can be expressed in terms of the
function $x$:

$\zeta^{r}_{i}(0)=\left\\{\begin{array}[]{ll}x^{-}(r+(i-1)h)&\mbox{if
}r\in]0,h]\\\ \zeta_{i-1}^{h}(1)&\mbox{if }r=0\end{array}\right.\quad
i=1,\ldots,N.$

Thus these $\zeta^{r}_{i}$’s are consistent with those appearing in Def. 2.2.
Reviewing the above relations, we see that $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$
is a process for (DIS).

(B): Take an extended process $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for (DIS). For
any $i=1,\dots,N$, let $\mu_{i}\in C^{\oplus}(0,h)$ be the Borel measure with
distribution function

$\mu_{i}([0,r]):=\begin{cases}\displaystyle\int_{0}^{1}w_{i}^{0}(s)\,ds+\mu(](i-1)h,r+(i-1)h])\quad\text{if
$r\in[0,h[$,}\\\
\displaystyle\int_{0}^{1}w_{i}^{0}(s)\,ds+\mu(](i-1)h,h[)+\int_{0}^{1}w_{i}^{h}(s)\,ds\quad\text{if
$r=h$.}\end{cases}$

Notice that $\mu_{i}(\\{r\\})=\mu(\\{r+(i-1)h\\})$ if $r\in]0,h[$, while
$\mu_{i}(\\{0\\})=\int_{0}^{1}w_{i}^{0}(s)\,ds$ and
$\mu_{i}(\\{h\\})=\int_{0}^{1}w_{i}^{h}(s)\,ds$, so that Def. 2.1 (ii) ,(iii)
imply the relations

$\begin{array}[]{l}\displaystyle\int_{0}^{1}w_{i}^{r}(s)\,ds=\mu_{i}(\\{r\\})\quad\text{for
$r\in[0,h]$, \ $i=1,\dots,N$,}\\\\[6.45831pt]
\displaystyle\mu_{i}(\\{h\\})+\mu_{i+1}(\\{0\\})=\mu(\\{ih\\}),\quad\mbox{for
}i=1,\dots,N-1.\end{array}$ (3.14)

Define the strictly increasing function,

$\phi(0):=0,\quad\phi(r):=r+\sum_{i=1}^{N}\mu_{i}([0,r])\quad\mbox{for
}r\in]0,h]\,,$ (3.15)

and let $S:=\phi(h)$. The function $\phi$ is right continuous on $]0,h[$. Take
$\eta:[0,S]\rightarrow[0,h]$ be the right inverse of $\phi$, as defined in
Lemma 3.4. The function $\eta$ is increasing, 1-Lipschitz continuous, such
that $\eta(0)=0$ and $\eta(S)=h$. Write $\\{r_{j}\\}$ for the countable set of
discontinuity points of $\phi$ and define
$D:=\underset{j}{\cup}[\phi^{-}(r_{j}),\phi^{+}(r_{j})]\,.$ For each
$r\in[0,h]$, $\eta^{-1}(r)=[\phi^{-}(r),\phi^{+}(r)]$ and $\eta$ is constant
exactly on the intervals in $D$. Then, the Lebesgue measure $\mu_{L}$ and the
measures $\mu_{i}^{c}$, $i=1,\ldots,N$, are absolutely continuous w.r.t. the
measure $d\phi$. Let $m_{L}$ and $m_{i}^{c}$ be the Radon-Nicodym derivatives
of $\mu_{L}$ and $\mu_{i}^{c}$, $i=1,\ldots,N$ w.r.t. $d\phi$, respectively.
Hence, $0\leq m_{L}(r)\leq 1$ and $0\leq m_{i}^{c}\leq 1$, $i=1,\ldots,N$,
$d\phi$-a.e. Furthermore,

$m_{L}(r)+\sum_{i=1}^{N}\,m_{i}^{c}(r)=1,\quad\mbox{$d\phi$-a.e.
}r\in[0,h]\backslash\underset{j}{\cup}\\{r_{j}\\}.$ (3.16)

Now $0=\mu_{L}(\\{r_{j}\\})=m_{L}(r_{j})[\phi^{+}(r_{j})-\phi^{+}(r_{j})]$ and
$0=\mu_{i}^{c}(\\{r_{j}\\})=m_{i}^{c}(r_{j})[\phi^{+}(r_{j})-\phi^{+}(r_{j})]$,
so that $m_{L}(r_{j})=0$, and $m_{i}^{c}(r_{j})=0$ for all $j\in\mathbb{N}$.

We define the Borel measurable functions
$\alpha_{0}:[0,S]\rightarrow\mathbb{R}$ and
$\alpha_{i}:[0,S]\rightarrow\mathbb{R}$, $i=1,\ldots,N$ to be

$\alpha_{0}(s):=m_{L}(\eta(s)),\quad\text{for $s\in[0,S]$}$ (3.17)

(notice that $\alpha_{0}(s)=0$ on $D$) and

$\alpha_{i}(s):=\left\\{\begin{array}[]{ll}m_{i}^{c}(\eta(s))&\mbox{if
}s\notin D\\\
\frac{1}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\,w^{r_{j}}_{i}\left(\frac{s-\phi^{-}(r_{j})}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\right)&\mbox{if
}s\in[\phi^{-}(r_{j}),\phi^{+}(r_{j})]\mbox{ for some }j\,.\end{array}\right.$
(3.18)

We show that

$\alpha_{0}(s)+\sum_{i=1}^{N}\alpha_{i}(s)=1,\quad\mbox{$\mu_{L}$-a.e.
}s\in[0,S].$ (3.19)

This is certainly true for $\mu_{L}$-a.e. $s\notin D$ by (3.16). On the other
hand, for any $j$ and $\mu_{L}$-a.e. $s\in[\phi^{-}(r_{j}),\phi^{+}(r_{j})]$,
we know that $\alpha_{0}(s)=0$ and

$\begin{split}\sum_{i=1}^{N}\alpha_{i}(s)&=\frac{1}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\,\sum_{i=1}^{N}w^{r_{j}}_{i}\left(\frac{s-\phi^{-}(r_{j})}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\right)=\frac{\sum_{i=1}^{N}\mu_{i}(r_{j})}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\,,\end{split}$

by Def. 2.1 (i), and (3.14). Since
$\phi^{+}(r_{j})-\phi^{-}(r_{j})=\sum_{i=1}^{N}\mu_{i}(r_{j})$, we obtain that
(3.19) is true for $\mu_{L}$-a.e. $s\in[0,S]$. Finally, using Lemma 3.4 for
$B=\eta$, $A=\phi$, and $F=m_{L}$, for any $s\in[0,S]$, we get

$\begin{split}\int_{0}^{s}\left(1-\sum_{i=1}^{N}\alpha_{i}(s^{\prime})\right)\,ds^{\prime}&=\int_{0}^{s}\alpha_{0}(s^{\prime})\,ds^{\prime}=\int_{0}^{\phi(\eta(s))}m_{L}(\eta(s))\,ds\\\
&=\int_{[0,\eta(s)]}m_{L}(r)d\phi(r)=\eta(s).\end{split}$ (3.20)

Therefore, the functions $\eta$ and $\phi$ coincide with the function $\psi$
defined as in (3.2) and its right inverse $\sigma$, respectively. Notice that
the controls $\alpha_{i}\in L^{1}(0,S)$ are nonnegative,
$\sum_{i=1}^{N}\alpha_{i}(s)\leq 1$ a.e. $s\in[0,S]$, and
$\int_{0}^{S}(1-\sum_{i=1}^{N}\alpha_{i}(s))ds=\eta(S)=h$. We next construct
functions $y_{i}:[0,S]\rightarrow\mathbb{R}^{n}$ from the process
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ and the corresponding functions
$\zeta^{r}_{i}:[0,1]\rightarrow\mathbb{R}^{n}$ appearing in Def. 2.2. Then, we
show that $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ is the (unique) reparameterized
process associated with $S>0$ and $\\{\alpha_{i}\\}$. Set $y_{0}(s):=\xi_{0}$
if $s\in[0,S[$ and $y_{0}(S):=x_{0}$. For $i=1,\ldots,N$, define

$y_{i}(0):=\begin{cases}x_{0}&\quad\text{if $i=1$}\\\
\zeta^{h}_{i-1}(1)&\quad\text{if $i>1$}\end{cases},\qquad
y_{i}(S):=\zeta^{h}_{i}(1)$
$y_{i}(s):=\begin{cases}x((i-1)h+\eta(s))&\quad\text{if}\quad
s\in]0,S[\setminus D\\\
\zeta^{r_{j}}_{i}\left(\frac{s-\phi^{-}(r_{j})}{\phi^{+}(r_{j})-\phi^{-}(r_{j})}\right)&\quad\mbox{if
}s\in]0,S[\cap[\phi^{-}(r_{j}),\phi^{+}(r_{j})]\mbox{ for some
$j$.}\end{cases}$

We deduce from the differential equations and boundary conditions for the
$\zeta^{r}_{i}$’s that, for any index value $j$ of the discontinuity points
$\\{r_{j}\\}$ of $\phi$, the $y_{i}$’s satisfy

$\frac{dy_{i}}{ds}(s)\,=\,\alpha_{i}(s)g(y_{i}(s),y_{i-1}(s)),\mbox{ a.e.
}s\in[\phi^{-}(r_{j}),\phi^{+}(r_{j})]\mbox{ for }i=1,\ldots,N\,,$ (3.21)

with boundary conditions

$y_{i}(\phi^{-}(r_{j}))=\zeta^{r}_{i}(0)\qquad\text{($\zeta^{r}_{i}(0)$ as in
\eqref{jump_ode2}).}$ (3.22)

For $i=1,\ldots,N$, take any $s\in]0,S[$. Let $t=(i-1)h+\eta(s)$. Then,

$\begin{split}x(t)=\zeta^{0}_{i}(1)&+\int_{(i-1)h}^{t}f(x(t^{\prime}),x(t^{\prime}-h))dt^{\prime}+\int_{[(i-1)h,t]}g(x(t^{\prime}),x(t^{\prime}-h))d\mu^{c}(t^{\prime})\\\
&+\underset{\\{j\,:\,r_{j}\in]0,t-(i-1)h]\\}}{\sum}y_{i}(\phi^{+}(r_{j}))-y_{i}(\phi^{-}(r_{j})).\end{split}$

(We have made use of properties of $y_{i}$ on $D$ to obtain the last term on
the right.) But, in consequence of Lemma 3.4,

$\begin{split}&\int_{(i-1)h}^{t}f(x(t^{\prime}),x(t^{\prime}-h))dt^{\prime}\\\
&\ \ \
=\int_{0}^{t-(i-1)h}f(x(r^{\prime}+(i-1)h),x(r^{\prime}+(i-1)h-h))dr^{\prime}\\\
&\ \ \
=\int_{[0,t-(i-1)h]}f(x(r^{\prime}+(i-1)h),x(r^{\prime}+(i-1)h-h))m_{L}(r^{\prime})d\phi(r^{\prime})\\\
&\ \ \
=\int_{[0,\phi^{+}(t-(i-1)h)]}f(x(\eta(s^{\prime})+(i-1)h),x(\eta(s^{\prime})+(i-1)h-h))m_{L}(\eta(s^{\prime}))\,ds^{\prime}\\\
&\ \ \
=\int_{[0,s]}f(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{0}(s^{\prime})\,ds^{\prime}\,.\end{split}$

(We have used the facts that $y_{i}(s^{\prime})=x(\eta(s^{\prime})+(i-1)h)$
for $m_{L}$-a.e. $s^{\prime}\notin D$, and $\alpha_{0}(s)=0$, a.e. on
$[\phi^{-}(t-(i-1)h),\phi^{+}(t-(i-1)h)]$.)

Furthermore (since $m_{i}^{c}(\eta(s))=0$ a.e. on $[\phi^{-}(r),\phi^{+}(r)]$,
so that $s^{\prime}=\phi(\eta(s^{\prime}))$ if $s^{\prime}\notin D$), applying
Lemma 3.4 we get

$\begin{split}&\int_{[(i-1)h,t]}g(x(t^{\prime}),x(t^{\prime}-h))d\mu^{c}(t^{\prime})\\\
&\ \ \
=\int_{[0,t-(i-1)h]}g(x(r^{\prime}+(i-1)h),x(r^{\prime}+(i-1)h-h))d\mu_{i}^{c}(r^{\prime})\\\
&\ \ \
=\int_{[0,t-(i-1)h]}g(x(r^{\prime}+(i-1)h),x(r^{\prime}+(i-1)h-h))m_{i}^{c}(r^{\prime})d\phi(r^{\prime})\\\
&\ \ \
=\int_{[0,\phi^{+}(t-(i-1)h)]}g(x(\eta(s^{\prime})+(i-1)h),x(\eta(s^{\prime})+(i-1)h-h))m_{i}^{c}(\eta(s^{\prime}))\,ds^{\prime}\\\
&\ \ \
=\int_{[0,s]}g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})\chi_{[0,S]\backslash
D}(s^{\prime})ds^{\prime}\,.\end{split}$

If $s\in]0,S[\setminus D$, then we conclude from the above relations that

$\begin{split}y_{i}(s)=\zeta^{0}_{i}(1)&+\int_{[0,s]}f(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{0}(s^{\prime})\,ds^{\prime}\\\
&+\int_{[0,s]}g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})\chi_{[0,S]\backslash
D}(s^{\prime})ds^{\prime}\\\
&+\underset{\\{j\,:\,r_{j}\in]0,t-(i-1)h]\\}}{\sum}y_{i}(\phi^{+}(r_{j}))-y_{i}(\phi^{-}(r_{j})).\end{split}$

But $\zeta^{0}_{i}(1)=y_{i}(0)+(y_{i}(\phi^{+}(0))-y_{i}(\phi^{-}(0)))$, so

$\begin{split}\zeta^{0}_{i}(1)&+\underset{\\{j\,:\,r_{j}\in]0,r-(i-1)h]\\}}{\sum}y_{i}(\phi^{+}(r_{j}))-y_{i}(\phi^{-}(r_{j}))\\\
&=y_{i}(0)+\underset{\\{j\,:\,r_{j}\in[0,t-(i-1)h]\\}}{\sum}y_{i}(\phi^{+}(r_{j}))-y_{i}(\phi^{-}(r_{j}))\\\
&=y_{i}(0)+\int_{[0,s]}g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})\chi_{D}(s^{\prime})ds^{\prime}\,.\end{split}$

It follows that, for each $i$,

$y_{i}(s)=y_{i}(0)+\int_{[0,s]}f(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{0}(s^{\prime})ds^{\prime}+\int_{[0,s]}g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})ds^{\prime}.$
(3.23)

Now assume that $s\in]0,S[\cap[\phi^{-}(r_{j}),\phi^{+}(r_{j})]$ for some
index value $j$. Since $\phi^{-}(r_{j})$ is a density point of
$[0,S]\backslash D$ and the $y_{i}$’s are continuous, we can deduce from
(3.23) that, for each $i$,

$\begin{split}y_{i}(\phi^{-}(r_{j}))=y_{i}(0)&+\int_{[0,\phi^{-}(r_{j})]}f(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})ds^{\prime}\\\
&+\int_{[0,\phi^{-}(r_{j})]}g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))\alpha_{i}(s^{\prime})\,ds^{\prime}.\end{split}$

Exploiting the facts that

$y_{i}(s)-y_{i}(\phi^{-}(r_{j}))=\int_{\phi^{-}(r_{j})}^{s}\alpha_{i}(s^{\prime})g(y_{i}(s^{\prime}),y_{i-1}(s^{\prime}))ds^{\prime}$
(3.24)

and $\alpha_{0}(s)=0$ for $\mu_{L}$-a.e.
$s\in[\phi^{-}(r_{j}),\phi^{+}(r_{j})]$, we see that (3.23) is valid for
arbitrary $s\in]0,S[$ . By definition, $y_{i}(S)=\zeta_{i}^{h}(1)$. Hence, if
$S\notin D$ or $i=N$, then $y_{i}$ satisfies (3.23) on $]0,S]$ by continuity.
If, on the other hand, $i\in\\{1,\dots,N-1\\}$ and
$S\in[\phi^{-}(r_{j}),\phi^{+}(r_{j})]$ for some index value $j$, then
$r_{j}=h$ and we derive that (3.23) still holds on $]0,S]$ by (3.24).
Furthermore, notice that the $y_{i}$’s satisfy the boundary conditions in Def.
3.1. In particular, $y_{1}(0)=x_{0}$ and
$y_{i+1}(0)=\zeta_{i}^{h}(1)=y_{i}(S).$ Reviewing the preceding relations, we
see that assertion (B) has been confirmed. ∎

The mapping, constructed in the proof of Part (B) of the theorem, between
extended and reparameterized processes, is actually invertible.

######  Corollary 3.5.

Let ${\mathcal{I}}$ be the map which associates with any extended process
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ the reparameterized process
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ for (DIS) constructed in the proof of Thm.
3.3, (B). Then, the function ${\mathcal{I}}$ is invertible and the extended
process
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})={\mathcal{I}}^{-1}(S,\\{y_{i}\\},\\{\alpha_{i}\\})$
coincides with the extended process associated with
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ constructed in the proof of Thm. 3.3, (A).

###### Proof.

This assertion is an easy consequence of relation (3.20), which implies that
the change of variable $r\mapsto\phi(r)$ (see (3.15)) utilized in the proof of
Thm. 3.3, statement (B), to obtain $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ from
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ is the right inverse of the change of
variable $s\mapsto\psi(s)$ (see (3.2)) employed in the proof of part (A) to
construct an extended process from $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$. (See
also the arguments in the proof of Prop. 4.1 below). ∎

##  4\. Existence, Density and Compactness Properties of Extended Trajectories

We establish some fundamental properties of extended processes.

######  Proposition 4.1.

Given an impulsive control $(\mu,\\{w^{r}\\}_{r\in[0,h]})$, there exists an
extended trajectory $x\in BV([-h,T];\mathbb{R}^{n})$ to (DIS) associated with
it and it is unique.

###### Proof.

Take an impulsive control $(\mu,\\{w^{r}\\}_{r\in[0,h]})$. Define $S>0$ and
the controls $\\{\alpha_{i}\\}$ as in the proof of assertion (B) in Thm. 3.3,
and let $\\{y_{i}\\}$ be the corresponding (unique) solution to the
reparameterized system. Then, with the reparameterized process
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ we can associate an extend process
$(\hat{x},\hat{\mu},\\{\hat{w}^{r}\\}_{r\in[0,h]})$ as in the proof of
assertion (A). Since $\sigma\equiv\phi$ (we use the notation of the proof), it
is not difficult to deduce that the impulsive control
$(\hat{\mu},\\{\hat{w}^{r}\\}_{r\in[0,h]})$ actually coincides with
$(\mu,\\{w^{r}\\}_{r\in[0,h]})$. Therefore, $\hat{x}$ is an extended
trajectory associated with the given impulsive control. Suppose now that $x$
is another $BV$ function such that $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ is an
extended process. To this process there corresponds the same reparameterized
process $(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ as above, since, given $S>0$ and
$\\{\alpha_{i}\\}$, the solution $\\{y_{i}\\}$ is unique. The equality
$x=\hat{x}$ now follows from the properties of the $y_{i}$’s constructed in
the proof of (B). ∎

For any $K>0$, we define the $K$-extended reachable set for (DIS), as

${\mathcal{R}}^{{\rm
e}}_{K}(T):=\\{x(T):\,(x,\mu,\\{w^{r}\\}_{r\in[0,h]})\mbox{ is extended}\mbox{
process for }{\rm(DIS)}\mbox{ s.t. }||\mu||_{\mbox{TV}}\leq K\\}$

and the $K$-strict sense reachable set for (DIS), as

${\mathcal{R}}^{{\rm s}}_{K}(T):=\\{x(T):\,(x,u)\mbox{ is a strict sense }\\\
\mbox{process for }{\rm(DIS)}\mbox{ s.t. }\int_{0}^{T}u(t)\,dt\leq K\\}.$

Clearly, ${\mathcal{R}}^{{\rm s}}_{K}(T)$ is nonempty and ${\mathcal{R}}^{{\rm
s}}_{K}(T)\subseteq{\mathcal{R}}^{{\rm e}}_{K}(T)$. In control theory an
extension of a family of processes is said to be a ‘relaxation’ of that
family, if the reachable set of the original family is dense in the reachable
set of the extension and if the extended reachable is closed. The role of the
extension is then to be ‘close’ to the original family of processes, but to
include extra elements ensuring closure properties. Relaxation procedures have
had an important role in optimal control and system theory, both in the
investigation of conditions that guarantee existence of solutions to optimal
control problems and in the derivation of necessary conditions of optimality.
Proposition 4.2 below justifies interpreting extended sense processes for
(DIS) as a relaxation of the family of strict sense processes.

######  Proposition 4.2.

For any $K>0$, ${\mathcal{R}}^{{\rm e}}_{K}(T)$ is a non-empty compact set and
${\mathcal{R}}^{{\rm e}}_{K}(T)=\overline{{\mathcal{R}}^{{\rm s}}_{K}(T)}$.

###### Proof.

Let us first show that ${\mathcal{R}}^{{\rm
e}}_{K}(T)\subseteq\overline{{\mathcal{R}}^{{\rm s}}_{K}(T)}$. Take
$z\in{\mathcal{R}}^{{\rm e}}_{K}(T)$ and let us consider an extended process
$(x,\mu,\\{w_{i}^{r}\\}_{r\in[0,h]})$ with $||\mu||_{\mbox{TV}}\leq K$ and
such that $x(T)=z$. Then, by Thm. 3.3, there exists
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ which is a reparameterized process for
(DIS) such that $x(T)=y_{N}(S)$ and
$\sum_{i=1}^{N}\int_{0}^{S}\alpha_{i}(s)\,ds=||\mu||_{\mbox{TV}}$. Set
$\alpha_{0}(s):=1-\sum_{i=1}^{N}\alpha_{i}(s)$ for $s\in[0,S]$, and
$\nu:=\sum_{i=1}^{N}\int_{0}^{S}\alpha_{i}(s)\,ds$. For each index value
$j\in\mathbb{N}$, $j\geq 1$, and a.e. $s\in[0,S]$, define
$\alpha_{0}^{j}(s):=\frac{h}{h+\frac{\nu}{j}}\left[\alpha_{0}(s)+\frac{1}{j}\sum_{i=1}^{N}\alpha_{i}(s)\right],$
and

$\alpha^{j}_{i}(s):=\left\\{\begin{array}[]{ll}\displaystyle\left(1-\alpha_{0}^{j}(s)\right)\frac{\alpha_{j}(s)}{\sum_{i=1}^{N}\alpha_{i}(s)}&\mbox{if
}\sum_{i=1}^{N}\alpha_{i}(s)\neq 0\\\
\displaystyle\frac{\left(1-\alpha_{0}^{j}(s)\right)}{N}&\mbox{if
}\sum_{i=1}^{N}\alpha_{i}(s)=0\end{array}\right.,\qquad i=1,\ldots,N.$

Notice that $\alpha_{0}^{j}(s)+\sum_{i=1}^{N}\alpha^{j}_{i}(s)=1$ and
$0<\frac{1}{j}\leq\alpha_{0}^{j}(s)\leq 1$ a.e. Moreover, since $S=h+\nu$, one
has
$\int_{0}^{S}\left(1-\sum_{i=1}^{N}\alpha^{j}_{i}(s)\right)\,ds=\int_{0}^{S}\alpha_{0}^{j}(s)\,ds=h$
and

$\int_{0}^{S}\sum_{i=1}^{N}\alpha^{j}_{i}(s)\,ds=\int_{0}^{S}\left(1-\alpha_{0}^{j}(s)\right)\,ds=S-h=\nu\leq
K.$

Let $(y_{1}^{j},\dots,y_{N}^{j})$ be the unique solution to the
reparameterized system associated with $S$ and the $\alpha_{i}^{j}$’s. Then,
for any $j\geq 1$, the element $(S,\\{y_{i}^{j}\\},\\{\alpha_{i}^{j}\\})$ is a
reparameterized process for (DIS) to which by Thm. 3.3 it corresponds a strict
sense process $(x^{j},u^{j})$ such that $x^{j}(T)=y^{j}_{N}(S)$ and
$\int_{0}^{T}u^{j}(s)\,ds=\nu\leq K$. We see that, for any $i=1,\dots,N$, the
controls $\alpha_{i}^{j}$ converge to $\alpha_{i}$ in the $L^{\infty}$-norm,
so that, by the continuity of the input-output map, one has
$|x^{j}(T)-x(T)|=|y^{j}_{N}(S)-y_{N}(S)|\to 0$. Therefore, we have shown that
all points in ${\mathcal{R}}^{{\rm e}}_{K}(T)$ lie in the closure of
${\mathcal{R}}^{{\rm s}}_{K}(T)$.

Let us now prove that ${\mathcal{R}}^{{\rm e}}_{K}(T)$ is compact. Take a
sequence $(z_{j})_{j}\subset{\mathcal{R}}^{{\rm e}}_{K}(T)$. For each $j$
then, there exists an extended process
$(x^{j},\mu^{j},\\{w^{j,r}\\}_{r\in[0,h]})$ such that $x^{j}(T)=z_{j}$ and
$||\mu_{j}||_{TV}\leq K$. By Thm. 3.3 there exists a reparameterized process
$(S^{j},\\{y^{j}_{i}\\},\\{\alpha^{j}_{i}\\})$ for (DIS) such that
$y^{j}_{N}(S^{j})=z_{j}$. Furthermore, it holds
$||\mu^{j}||_{TV}=\underset{i}{\sum}\;\int_{0}^{S^{j}}\alpha^{j}_{i}(s)ds$.
But then one obtains
$h=\int_{[0,S^{j}]}(1-\sum_{i}\alpha^{j}_{i}(s))ds=S^{j}-||\mu^{j}||_{TV},$ so
that $S^{j}\leq h+K$. Since the sequence $(S^{j})_{j}$ is bounded, possibly
passing to a subsequence (we do not relabel) we have that $S^{j}$ converges to
some $S\leq h+K$. Let us now add to the reparameterized system the equations
of time and total variation, denoted by $\tau$ and $v$, respectively, that is
we consider the system

$\
\left\\{\begin{array}[]{l}\displaystyle\dot{\tau}(s)=\Big{(}1-\sum_{i=1}^{N}\alpha_{i}(s)\Big{)}\\\
\displaystyle\dot{v}(s)=\sum_{i=1}^{N}\alpha_{i}(s)\\\
\displaystyle\dot{y}_{i}(s)=\Big{(}1-\sum_{i=1}^{N}\alpha_{i}(s)\Big{)}f(y_{i}(s),y_{i-1}(s))+\alpha_{i}(s)g(y_{i}(s),y_{i-1}(s)),\
\ \mbox{$i=1,\ldots,N$},\end{array}\right.$ (4.1)

with the previous boundary conditions on $\\{y_{i}\\}$ and $\tau(0)=v(0)=0$.
For each $j\geq 1$, let $(\tau^{j},v^{j},\\{y_{i}^{j}\\})$ denote the solution
to (4.1) on $[0,S^{j}]$, associated with the controls $\\{\alpha_{i}^{j}\\}$
and satisfying these endpoint conditions. Notice that $\tau^{j}(S^{j})=h$ and
$v^{j}(S^{j})\leq K$. We may apply a standard convergence analysis, based on
the Compactness of Trajectories Theorem [23, Thm. 2.5.3] and the fact that the
velocity sets associated with (4.1) are convex, to show that there exists a
solution $(\tau,v,\\{y_{i}\\})$ to (4.1) on $[0,S]$, corresponding to some
control $\\{\alpha_{i}\\}$, such that, along a subsequence (we do not re-
label), $(\tau^{j},v^{j},\\{y_{i}^{j}\\})$ converges uniformly to
$(\tau,v,\\{y_{i}\\})$ on $[0,S]$ (possibly extending by a constant,
continuous extrapolation all functions to $[0,h+K]$). In particular,
$(S,\\{y_{i}\\},\\{\alpha_{i}\\})$ is a reparameterized process for (DIS) with
$\sum_{i=1}^{N}\int_{0}^{S}\alpha_{i}(s)\,ds\leq K$, since
$\tau(S)=\lim_{j}\tau^{j}(S^{j})=h$ and $v(S)=\lim_{j}v^{j}(S^{j})\leq K$.
Moreover, $\lim_{j}z^{j}=\lim_{j}y^{j}_{N}(S^{j})=y_{N}(S)$. Again, appealing
to Thm. 3.3, we deduce that there exists an extended process
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ such that $x(T)=y_{N}(S)$ and
$\|\mu\|_{TV}\leq K$. This proves that ${\mathcal{R}}^{{\rm e}}_{K}(T)$ is
compact and ${\mathcal{R}}^{{\rm
e}}_{K}(T)\subseteq\overline{{\mathcal{R}}^{{\rm s}}_{K}(T)}$. Since
${\mathcal{R}}^{{\rm s}}_{K}(T)\subseteq{\mathcal{R}}^{{\rm e}}_{K}(T)$, the
compactness of ${\mathcal{R}}^{{\rm e}}_{K}(T)$ yields ${\mathcal{R}}^{{\rm
e}}_{K}(T)=\overline{{\mathcal{R}}^{{\rm s}}_{K}(T)}$. ∎

##  5\. Existence of Optimal Extended Processes and a Maximum Principle

We have defined extended processes for the delayed impulsive control system
(DIS). We now consider a related optimal control problem over extended
processes for (DIS) with terminal cost, involving constraints on the total
variation of the measure control and the location of the terminal state.

${\rm(P)}\left\\{\begin{array}[]{l}\mbox{Minimize }\Psi(x(T))\\\ \mbox{over
extended processes $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for {\rm(DIS)},
satisfying}\\\\[4.30554pt] \|\mu\|_{TV}\leq K,\\\\[6.45831pt]
x(T)\in\mathcal{T}\,.\end{array}\right.$

The additional data for this problem comprise a function
$\Psi:\mathbb{R}^{n}\rightarrow\mathbb{R}$, a nonempty set
$\mathcal{T}\subset\mathbb{R}^{n}$, and $K>0$. We shall say that an extended
process $(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for (DIS) is admissible if
$x(T)\in\mathcal{T}$ and $\|\mu\|_{TV}\leq K$. An admissible extended process
$(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$ for (DIS) that minimizes
$\Psi(x(T))$ over all admissible extended processes
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ for (DIS) is said to be an optimal extended
process.

######  Theorem 5.1 (Existence of Optimal Controls).

Consider the optimal control problem (P). Assume that $\mathcal{T}$ is closed
and $\Psi:\mathbb{R}^{n}\to\mathbb{R}$ is lower semicontinuous and that there
exists an admissible extended process. Then, there exists an optimal extended
process.

###### Proof.

Under the stated hypotheses, by Prop. 4.2 the set ${\mathcal{R}}^{{\rm
e}}_{K}(T)\cap\mathcal{T}$ is non-empty and compact. Since the function $\Psi$
is lower semicontinuous, there exists $\bar{z}\in{\mathcal{R}}^{{\rm
e}}_{K}(T)\cap\mathcal{T}$ that minimizes $\Psi$ over this set. Let
$(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$ be an admissible extended
process such that $\bar{x}(T)=\bar{z}$. Take any admissible extended process
$(x,\mu,\\{w^{r}\\}_{r\in[0,h]})$ and write $z=x(T)$. Then, by the minimizing
property of $\bar{z}$,

$\Psi(\bar{x}(T))=\Psi(\bar{z})\leq\Psi(z)=\Psi(x(T)).$

In other words $(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$ is a
minimizer for (P). ∎

The maximum principle of this section is derived by transforming the impulsive
optimal control problem into a conventional, free end-time optimal control
problem, to which we apply standard necessary conditions of optimality. In the
following, given a reference extended process
$(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$, for any function
$h=h(x_{1},x_{2})$, $(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$, we
set $\bar{h}(t):=h(\bar{x}(t),\bar{x}(t-h))$ and write $\bar{h}_{x_{1}}(t)$,
$\bar{h}_{x_{2}}(t)$ for the Jacobian of $h$ at $(\bar{x}(t),\bar{x}(t-h))$
w.r.t. $x_{1}$ and $x_{2}$, respectively. For instance,
$\bar{f}(t)=f(\bar{x}(t),\bar{x}(t-h))$ and $\bar{f}_{x_{1}}(t)=\frac{\partial
f}{\partial x_{1}}(\bar{x}(t),\bar{x}(t-h))$.

######  Theorem 5.2 (Maximum Principle).

Let $(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$ be a minimizing
process for (P). For $i=1,\ldots,N$ and $r\in[0,h]$, let
$\bar{\zeta}^{r}_{i}:[0,1]\rightarrow\mathbb{R}^{n}$ be the corresponding
functions describing the instantaneous evolution of the state of this process,
defined by (2.1) and (2.2). Assume that $\mathcal{T}\subseteq\mathbb{R}^{n}$
is closed and $\Psi:\mathbb{R}^{n}\to\mathbb{R}$ is a $C^{1}$ function. Then,
there exist $\lambda\geq 0$, $c\in\mathbb{R}$, $d\geq 0$ and a function $p\in
BV([0,+\infty[;\mathbb{R}^{n})$, right continuous on $]0,T[$, with the
following properties: $d=0$ if $\|\bar{\mu}\|_{TV}<K$ and

(A): $\lambda+\|p\|_{L^{\infty}}\neq 0$,

(B): for each $i=1,\ldots,N$ and $t\in[(i-1)h,ih]$,

$\begin{split}&p(t)=\eta_{i}^{0}(1),\mbox{ when $i=2,\dots,N$ and
$t=(i-1)h$,}\\\
&p(t)=\eta_{i}^{0}(1)-\int_{(i-1)h}^{t}p(t^{\prime})\cdot\bar{f}_{x_{1}}(t^{\prime})dt^{\prime}-\int_{[(i-1)h,t]}p(t^{\prime})\cdot\bar{g}_{x_{1}}(t^{\prime})\bar{d}\mu^{c}(t^{\prime})\\\
&\ \
-\int_{(i-1)h}^{t}p(t^{\prime}+h)\cdot\bar{f}_{x_{2}}(t^{\prime}+h)dt^{\prime}-\int_{[(i-1)h,t]}p(t^{\prime}+h)\cdot\bar{g}_{x_{2}}(t^{\prime}+h))\bar{d}\mu^{c}(t^{\prime}+h)\\\
&\ \
-\underset{r\in]0,t-(i-1)h]}{\sum}(\eta^{r}_{i}(0)-\eta^{r}_{i}(1)),\mbox{
when $t\in](i-1)h,ih[$,}\end{split}$

Furthermore,

$p(T)=\eta^{h}_{N}(1)\mbox{ and }p(t)=0\mbox{ for }t>T.$

Here, for each $r\in[0,h]$,
$(\eta^{r}_{1},\ldots,\eta^{r}_{N}):[0,1]\rightarrow(\mathbb{R}^{n})^{N}$
satisfies the differential equations

$\displaystyle\frac{d}{ds}\eta^{r}_{i}(s)\,=\,-\eta^{r}_{i}(s)\cdot
g_{x_{1}}(\bar{\zeta}_{i}^{r}(s),\bar{\zeta}_{i-1}^{r}(s))\bar{w}_{i}^{r}(s)\,$
(5.1) $\displaystyle\qquad-\;\eta^{r}_{i+1}(s)\cdot
g_{x_{2}}(\bar{\zeta}_{i+1}^{r}(s),\bar{\zeta}_{i}^{r}(s))\bar{w}_{i+1}^{r}(s)\qquad\mbox{
a.e. }s\in[0,1],\ \ i=1,\ldots,N\,,$

with boundary conditions

$\eta^{r}_{i}(0)=\left\\{\begin{array}[]{ll}p^{-}(r+(i-1)h)&\mbox{if
}r\in]0,h]\\\ \eta_{i-1}^{h}(1)&\mbox{if }r=0\,.\end{array}\right.$ (5.2)

In these relations, we interpret $\eta^{h}_{0}(1):=p(0)$ and
$\eta^{r}_{N+1}(s):=0$ for all $r\in[0,h]$ and $s\in[0,1]$.

(C): $-p(T)\in\lambda\,\nabla\Psi(\bar{x}(T))+N_{\mathcal{T}}(\bar{x}(T))$,

(D):

  * (i):

$\underset{j=1}{\overset{N}{\sum}}\,p(r+(j-1)h)\cdot\bar{f}(r+(j-1)h)-c=0$ for
all $r\in]0,h[$,

and also at $r=0$ if
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{0}_{j}(s)\,ds=0$, and
at $r=h$ if
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{0}_{j}(s)\,ds=0$ and
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{h}_{j}(s)\,ds=0$,

  * (ii):

$p(t)\cdot\bar{g}(t)-d\leq 0$ for all $t\in[0,T]$,

  * (iii):

$\mbox{supp}\,\\{{\bar{\mu}}\\}\subset\left\\{t\in[0,T]:\ \
p(t)\cdot\bar{g}(t)-d=0\right\\}$,

  * (iv):

for any $r\in[0,h]$ such that
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}_{j}^{r}(s)\,ds>0$,

    * (a):

$\eta^{r}_{i}(s)\cdot
g(\bar{\zeta}^{r}_{i}(s),\bar{\zeta}^{r}_{i-1}(s))=\underset{j}{\max}\;\eta^{r}_{j}(s)\cdot
g(\bar{\zeta}^{r}_{j}(s),\bar{\zeta}^{r}_{j-1}(s))$ ,

for $\mathcal{L}$-a.e. $s\in[0,1]$ such that $\bar{w}^{r}_{i}(s)>0$.

    * (b):

$\underset{j=1}{\overset{N}{\sum}}\eta^{r}_{j}(s)\cdot
f(\bar{\zeta}_{j}^{r}(s),\bar{\zeta}_{j-1}^{r}(s)))-c\leq\max_{j}\eta_{j}^{r}(s)\cdot
g(\bar{\zeta}_{j}^{r}(s),\bar{\zeta}_{j-1}^{r}(s))-d=0$

for all $s\in[0,1]$.

Comments.

(1): Condition (B) is the costate equation in integral form for $p$. Its
corresponding infinitesimal form, obtained by differentiating across the
relevant equations at points of differentiability is a generalized ‘advance’
differential equation (equivalently, a ‘delay’ differential equation in
reverse time), consistent with standard first order necessary optimality
conditions for impulsive-free optimal control problems. Notice that the
‘attached’ controls $s\mapsto\bar{w}^{r}_{i}(s)$ affect the jumps in the
costate function $p$. In particular, let us observe that $p$ may jump at some
$t\in[(i-1)h,ih]$ where $\bar{\mu}$ (and $\bar{x}$) does not, as the ODE (5.1)
involves both the attached controls $\bar{w}_{i}^{t-(i-1)h}$ and
$\bar{w}_{i+1}^{t-(i-1)h}$.

(2): Condition (D), (i) and (D), (iii) are a version of the ‘constancy of the
Hamiltonian condition’ appropriate for the impulsive control problem.
Specifically, the equality in condition (D), (i) is satisfied also at $r=0$ if
$t=0$ is not an atom for $\bar{\mu}$ and at $r=h$ if all points $t=ih$,
$h=1,\dots,N$ are not atoms for $\bar{\mu}$.

(3): Condition (D), (iv), which is related to the Weierstrass condition of
classical optimal control, combines with (D), (iii) to give information about
the location of the support of the measure $\bar{\mu}$. In particular,
condition (D), (iv) describes properties of the attached controls determining
the instantaneous evolution of the state at atoms of $\bar{\mu}$.

(4): In view of the nontriviality relation (A), the transversality condition
(C) and the linearity of the adjoint equations in (B), if the final point
$x(T)$ is unconstrained we can choose $\lambda=1$.

###### Proof of Thm. 5.2.

Let $(\bar{S},\\{\bar{y}_{i}\\},\\{\bar{\alpha}_{i}\\})$ be the
reparameterized process associated with the optimal extended process
$(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$, constructed as in the
proof of Thm. 3.3. Define
$\bar{\tau}(s):=\int_{0}^{s}\Big{(}1-\underset{j=1}{\overset{N}{\sum}}\bar{\alpha}_{j}(s^{\prime})\Big{)}\,ds^{\prime}$
and
$\bar{v}(s):=\int_{0}^{s}\underset{j=1}{\overset{N}{\sum}}\bar{\alpha}_{j}(s^{\prime}))\,ds^{\prime}$
for $s\in[0,\bar{S}]$, and

$\Lambda^{(N)}:=\Big{\\{}(a_{1},\dots,a_{N})\in\mathbb{R}^{N}:\quad a_{j}\geq
0\text{ for each $j=1,\dots,N$, and }\sum_{j=1}^{N}a_{j}\leq 1\Big{\\}}.$

It can be deduced from Thm. 3.3 that
$(\bar{S},\bar{\tau},\bar{v},\\{\bar{y}_{i}\\},\\{\bar{\alpha}_{i}\\})$ is a
minimizer for the following optimal control problem:

$\left\\{\begin{array}[]{l}\mbox{Minimize }\Psi(y_{N}(S))\\\ \mbox{over
controls $S>0$, $\alpha_{i}\in L^{\infty}(0,S)$, and functions }\\\
(\tau_{i},v_{i},y_{i})\in W^{1,1}([0,S];\mathbb{R}^{1+1+n}),\text{ for
$i=1,\ldots,N$, that satisfy}\\\
\displaystyle\dot{\tau}(s)=1-\sum_{j=1}^{N}\alpha_{j}(s)\qquad\text{a.e.
$s\in[0,S]$,}\\\ \dot{v}(s)=\sum_{j=1}^{N}\alpha_{j}(s)\qquad\text{a.e.
$s\in[0,S]$,}\\\
\dot{y}_{i}(s)=(1-\sum_{j=1}^{N}\alpha_{j}(s))f(y_{i}(s),y_{i-1}(s))+\alpha_{i}(s)g(y_{i}(s),y_{i-1}(s))\\\
\hskip 180.67499pt\mbox{a.e. $s\in[0,S]$, for $i=1,\ldots,N$},\\\
(\alpha_{1}(s),\dots,\alpha_{N}(s))\in\Lambda^{(N)}\quad\mbox{a.e.
}s\in[0,S],\\\ y_{1}(0)=x_{0}\mbox{ and }y_{i+1}(0)=y_{i}(S),\mbox{ for
}i=1,\ldots,N-1,\quad y_{0}\equiv\xi_{0},\\\ \tau(0)=0,\quad
v(0)=0,\quad\tau(S)=h,\quad v(S)\leq K,\quad
y_{N}(S)\in\mathcal{T}.\end{array}\right.$

This is a standard ‘free end-time’ optimal control problem for which necessary
conditions are well known (see, e.g. [23, Thm. 8.7.1]). We deduce existence of
$\lambda\geq 0$, $c\in\mathbb{R}$, $d\geq 0$ ($-c$ and $-d$ is the Lagrange
multiplier associated with $\bar{\tau}$ and $\bar{v}$, respectively) and
absolutely continuous functions $q_{i}:[0,\bar{S}]\rightarrow\mathbb{R}^{n}$
such that (a): $|c|+d+\lambda+\|(q_{1},\dots,q_{N})\|_{L^{\infty}}\not=0$,

(b): for each $i=1,\ldots,N$ and a.e. $s\in[0,\bar{S}]$

$-\dot{q}_{i}(s)=q_{i}(s)\cdot
f_{x_{1}}(\bar{y}_{i}(s),\bar{y}_{i-1}(s))(1-\underset{j=1}{\overset{N}{\sum}}\,\bar{\alpha}_{j}(s))$

$+q_{i}(s)\cdot
g_{x_{1}}(\bar{y}_{i}(s),\bar{y}_{i-1}(s))\bar{\alpha}_{i}(s)+q_{i+1}(s)\cdot
g_{x_{2}}(\bar{y}_{i+1}(s),\bar{y}_{i}(s))\bar{\alpha}_{i+1}(s)$,

$+\;q_{i+1}(s)\cdot
f_{x_{2}}(\bar{y}_{i+1}(s),\bar{y}_{i}(s))(1-\underset{j=1}{\overset{N}{\sum}}\,\bar{\alpha}_{j}(s))$

where $q_{N+1}\equiv 0$.

(c):
$-q_{N}(\bar{S})\in\lambda\nabla\Psi(\bar{y}_{N}(\bar{S}))+N_{\mathcal{T}}(\bar{y}_{N}(\bar{S}))$
and, for $i=1,\ldots,N-1$,

$q_{i}(\bar{S})=q_{i+1}(0)$,

(d): for a.e. $s\in[0,\bar{S}]$,111Actually, the second equality involving the
maximized Hamiltonian is satisfied for all $s\in[0,\bar{S}]$, by the
continuity of the functions $s\mapsto q_{i}(s)\cdot
g(\bar{y}_{i}(s),\bar{y}_{i-1}(s))$ and $s\mapsto q_{i}(s)\cdot
f(\bar{y}_{i}(s),\bar{y}_{i-1}(s))$.

$H(\\{\bar{y}_{i}(s)\\},c,d,\\{q_{i}(s)\\},\\{\bar{\alpha}_{i}(s)\\})=\underset{\\{a_{i}\\}\in\Lambda^{(N)}}{\max}H(\\{\bar{y}_{i}(s)\\},c,d,\\{q_{i}(s)\\},\\{a_{i}\\})=0,$

where

$\begin{array}[]{l}H(\\{y_{i}\\},c,d,\\{q_{i}\\},\\{a_{i}\\})=H(y_{0},y_{1},\dots,y_{N},c,d,q_{1},\dots,q_{N},a_{1},\dots,a_{N})\\\
\qquad:=-c(1-\sum_{j=1}^{N}a_{j})-d\sum_{j=1}^{N}a_{j}\\\
\qquad\qquad+\sum_{j=1}^{N}\,q_{j}\cdot\Big{(}f(y_{j},y_{j-1})(1-\sum_{j=1}^{N}a_{j})+g(y_{j},y_{j-1}(s))a_{j}\Big{)}.\end{array}$

Notice that the non-triviality condition (a) can be replaced by (a)′:
$\lambda+\|(q_{1},\dots,q_{N})\|_{L^{\infty}}\not=0$. To prove (a)′, suppose
to the contrary that $(q,\lambda)=(0,0)$ but $|c|+d\not=0$. Then, integrating
condition (d) we obtain that
$-c\,\bar{\tau}(\bar{S})-d\,\bar{v}(\bar{S})=-c\,h-d\,\bar{v}(\bar{S})=0$, so
$c=-\frac{\bar{v}(\bar{S})}{h}\,d\leq 0$. However, for $a_{1}=\dots=a_{N}=0$
the maximality condition in (d) implies that $c\geq 0$. Hence $c=d=0$ and we
get a contradiction (we remember that, if $\bar{v}(\bar{S})<K$, then $d=0$).

As we shall see, the assertions of Theorem 5.2 are consequences of relations
(a)′–(d) concerning the re-parameterized process
$(\bar{S},\\{\bar{y}_{i}\\},\\{\bar{\alpha}_{i}\\})$, re-expressed in terms of
the original minimizing extended process
$(\bar{x},\bar{\mu},\\{\bar{w}^{r}\\}_{r\in[0,h]})$.

Let the continuous mapping $\tilde{\psi}:[0,N\bar{S}]\rightarrow\mathbb{R}$
and its right inverse $\tilde{\sigma}:[0,T]\to[0,N\bar{S}]$ be as in the proof
of Thm. 3.3, when $\\{\alpha_{i}\\}=\\{\bar{\alpha}_{i}\\}$. Precisely, let
$\tilde{\psi}(s)=(i-1)h+\psi(s-(i-1)\bar{S})$ for any
$s\in[(i-1)\bar{S},i\bar{S}]$ and $i=1,\dots,N$, where
$\psi(s)=\int_{0}^{s}(1-\sum_{j=1}^{N}\bar{\alpha}_{j}(s^{\prime}))ds^{\prime}$
for $s\in[0,\bar{S}]$. Observe that, in view of Cor. 3.5 one has
$\bar{x}(t)=\tilde{y}(\tilde{\sigma}(t))$ for any $t\in[0,T]$, where

$\tilde{y}(s)=\begin{cases}\bar{y}_{i}(s-(i-1)\bar{S})\qquad&\text{if
$s\in[(i-1)\bar{S},i\bar{S}]$ for some $i=1,\dots,N$,}\\\
\xi_{0}\qquad&\text{if $s<0$.}\end{cases}$

Note also that the measure $\bar{\mu}$ has distribution

$\bar{\mu}([0,t])=\int_{0}^{\tilde{\sigma}^{+}(t)}\tilde{\alpha}(s)ds\qquad\text{for
$t\in[0,T]$},$ (5.3)

where $\tilde{\alpha}(s)=\bar{\alpha}_{i}(s-(i-1)\bar{S})$ for any
$i=1,\dots,N$ and $s\in[(i-1)\bar{S},i\bar{S}[$. Moreover, for each
$i\in\\{1,\dots,N\\}$ and $r\in[0,h]$, one has

$\left\\{\begin{array}[]{l}\bar{w}_{i}^{r}(s)=(\sigma^{+}(r)-\sigma^{-}(r))\,\bar{\alpha}_{i}(\sigma^{-}(r)+s(\sigma^{+}(r)-\sigma^{-}(r)))\,\mbox{
a.e. }s\in[0,1],\\\\[6.45831pt]
\bar{\zeta}_{i}^{r}(s)=\bar{y}_{i}(\sigma^{-}(r)+s(\sigma^{+}(r)-\sigma^{-}(r))),\;\mbox{
all }s\in[0,1]\,,\end{array}\right.$ (5.4)

where $\sigma$ denotes the right inverse of $\psi$ above. Here, the
$\bar{\zeta}^{r}_{i}$’s are as in the theorem’s statement. For each
$i=1,\dots,N$ and $r\in[0,h]$, we set
$\eta_{i}^{r}(s)=q_{i}(\sigma^{-}(r)+s(\sigma^{+}(r)-\sigma^{-}(r)))$ for all
$s\in[0,1]$. The $\eta_{i}^{r}$’s are the functions used in the theorem
statement to calculate the jumps in the costate. Moreover, we introduce the
continuous function $\tilde{q}:[0,N\bar{S}]\to\mathbb{R}^{n}$, given by
$\tilde{q}(s):=q_{i}(s-(i-1)\bar{S})$ for any $s\in[(i-1)\bar{S},i\bar{S}]$
and $i=1,\dots,N$, and define $p:[0,+\infty[\to\mathbb{R}^{n}$ by
$p(t):=\tilde{q}(\tilde{\sigma}(t))$ for $t\in[0,T]$ and $p(t)=0$ for $t>T$.
Notice that $p$ is right continuous on $]0,T[$. Since
$\tilde{\sigma}(T)=N\bar{S}$, our construction together with relation (c)
yields condition (C) in the theorem statement.

By the above definitions, it immediately follows that $p(T)=\eta_{N}^{h}(1)$,
$p((i-1)h)=\eta_{i}^{0}(1)$ for any $i=2,\dots,N$ (since
$\tilde{\sigma}((i-1)h)=i\bar{S}+\sigma^{+}(0)$), and $p(0)=\eta_{1}^{0}(0)$.
Moreover, it is straightforward to check that the functions $\eta_{i}^{r}$
satisfy (5.1)-(5.2). In particular, for the boundary condition (5.2) we use
that $q_{i}(h)=q_{i+1}(0)$ for $i=1,\dots,N-1$.

Condition (A) of the theorem’s statement follows by relation (a)′ above.
Indeed, by (a)′ and the very definition of $\tilde{q}$ we get
$\lambda+\|\tilde{q}\|_{L^{\infty}}\neq 0$. Hence, if it were
$\lambda+\|p\|_{L^{\infty}}=0$, then $\lambda=0$ and $\tilde{q}\equiv 0$ on
$[0,N\bar{S}]\setminus D$, as for any $s\in[0,N\bar{S}]\setminus D$ there
exists some $s\in[0,T]$ such that $s=\tilde{\sigma}(r)$. In the previous
relations, $D=\cup_{k}[\tilde{\sigma}^{-}(r_{k}),\tilde{\sigma}^{+}(r_{k})]$,
where $\\{r_{k}\\}$ is the countable set of jumps of $\bar{\mu}$. By
continuity, $\tilde{q}\equiv 0$ on the boundary of $[0,N\bar{S}]\setminus D$,
so that $\eta_{i}^{r}(0)=\eta_{i}^{r}(1)=0$ for any $i=1,\dots,N$ and each
$r\in[0,h]$. Accordingly, the linearity of the adjoint equations yields that
$\eta_{i}^{r}\equiv 0$ for any $i=1,\dots,N$ and any $r\in[0,h]$, which
implies $\tilde{q}\equiv 0$ on $D$. A contradiction.

Let $t\in](i-1)h,ih[$ for some $i=1,\dots,N$, then condition (b) implies

$\begin{split}&p(t)=\tilde{q}(\tilde{\sigma}(t))=\tilde{q}(\tilde{\sigma}((i-1)h))+\int_{\tilde{\sigma}((i-1)h)}^{\tilde{\sigma}(t)}\frac{d\tilde{q}}{ds}(s)\,ds=\eta_{i}^{0}(1)\\\
&\qquad-\int_{\tilde{\sigma}((i-1)h)}^{\tilde{\sigma}(t)}[\tilde{q}(s)\cdot
f_{x_{1}}(\tilde{y}(s),\tilde{y}(s-\bar{S}))+\tilde{q}(s+\bar{S})\cdot
f_{x_{2}}(\tilde{y}(s+\bar{S}),\tilde{y}(s))]\frac{d\tilde{\psi}}{ds}(s)\,ds\\\
&\qquad-\int_{\tilde{\sigma}((i-1)h)}^{\tilde{\sigma}(t)}[\tilde{q}(s)\cdot
g_{x_{1}}(\tilde{y}(s),\tilde{y}(s-\bar{S}))\tilde{\alpha}(s)\\\
&\qquad\qquad\qquad+\tilde{q}(s+\bar{S})\cdot
g_{x_{2}}(\tilde{y}(s+\bar{S}),\tilde{y}(s))\tilde{\alpha}(s+\bar{S})]ds,\end{split}$

in which $\tilde{q}\equiv 0$ for $s>N\bar{S}$. Employing analogous analysis to
that earlier used in the proof of Thm. 3.3 part (A) (based on Lemma 3.4), we
deduce the ‘advance’ differential equation in integral form in condition (B).

The second equality in condition (d) tells us: for all $s\in[0,\bar{S}]$ and
$(a_{1},\dots,a_{N})\in\Gamma^{(N)}$,

$\begin{split}&\Big{(}-c+\underset{j=1}{\overset{N}{\sum}}q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))\Big{)}\Big{(}1-\underset{j=1}{\overset{N}{\sum}}a_{j}\Big{)}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\underset{j=1}{\overset{N}{\sum}}\Big{(}-d+q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))\Big{)}\,a_{j}\leq 0.\end{split}$

Choosing $(a_{1},\dots,a_{N})=e_{j}$ for some element $e_{j}$ of the canonical
basis of $\mathbb{R}^{N}$, we get

$-d+q_{j}(s)\cdot g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))\leq 0\quad\forall
s\in[0,\bar{S}],\ \ \forall j=1,\dots,N,$ (5.5)

while the choice $(a_{1},\dots,a_{N})=0$ yields
$-c+\underset{j=1}{\overset{N}{\sum}}q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))\leq 0$ for all $s\in[0,\bar{S}]$. Take
$s\in[0,\bar{S}]$ and let $(\bar{a}_{1},\dots,\bar{a}_{N})\in\Lambda^{(N)}$ be
a point where the maximum in condition (d) is obtained. Then,
$\max_{j=1,\dots,N}\left\\{-d+q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))\right\\}=0$ if
$(\bar{a}_{1},\dots,\bar{a}_{N})\neq 0$ (actually, $-d+q_{i}(s)\cdot
g(\bar{y}_{i}(s),\bar{y}_{i-1}(s))=0$ for each $i$ such that $\bar{a}_{i}>0$),
$-c+\underset{j=1}{\overset{N}{\sum}}q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))=0$ if
$\underset{j=1}{\overset{N}{\sum}}\bar{a}_{j}<1$, and, for any
$s\in[0,\bar{S}]$,

$\max\Big{\\{}\underset{j=1}{\overset{N}{\sum}}q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-c\,,\,\max_{j=1,\dots,N}q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-d\Big{\\}}=0.$ (5.6)

It follows from the above relations and the very definition of
$\\{\bar{\alpha}_{i}\\}$ that there exists a measurable set $\mathcal{N}$ of
zero Lebesgue measure, such that, for all
$s\in[0,\bar{S}]\setminus\mathcal{N}$:

$\displaystyle\underset{j=1}{\overset{N}{\sum}}\bar{\alpha}_{j}(s)>0\
\implies\ \max_{j=1,\dots,N}q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-d=0,$ (5.7)
$\displaystyle\bar{\alpha}_{i}(s)>0\ \text{for some $i\in\\{1,\dots,N\\}$}\
\implies\ q_{i}(s)\cdot g(\bar{y}_{i}(s),\bar{y}_{i-1}(s))-d=0,$ (5.8)
$\displaystyle\underset{j=1}{\overset{N}{\sum}}\bar{\alpha}_{j}(s)<1\
\implies\ \underset{j=1}{\overset{N}{\sum}}q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-c=0.$ (5.9)

Expressing (5.5) and (5.8) in terms of $\tilde{q}$ and $\tilde{y}$, we obtain

$\tilde{q}(s)\cdot g(\tilde{y}(s),\tilde{y}(s-\bar{S}))-d\leq 0\quad\text{for
all }s\in[0,N\bar{S}],$ (5.10)

and

$\tilde{q}(s)\cdot g(\tilde{y}(s),\tilde{y}(s-\bar{S}))-d=0\qquad\text{for
a.e. }s\in[0,N\bar{S}]\ \text{such that}\ \tilde{\alpha}(s)>0,$ (5.11)

respectively. Inserting into these relations $s=\tilde{\sigma}(t)$ for any
$t\in[0,T]$, from (5.10) we derive immediately condition (D), (ii). As for
(D), (iii), take any $t\in\mbox{supp}\,(\bar{\mu})$. Set
$s:=\tilde{\sigma}(t)$. From (5.3) it follows that the set
$\\{s^{\prime}\in[0,N\bar{S}]\,:\,\tilde{\alpha}(s^{\prime})>0\mbox{ and
}|s^{\prime}-s|\leq\epsilon\\}$ has positive Lebesgue measure for every
$\epsilon>0$. So, by (5.11) and ‘continuity’ we get that $\tilde{q}(s)\cdot
g(\tilde{y}(s),\tilde{y}(s-\bar{S}))-d=0$. Since $s:=\tilde{\sigma}(t)$, this
yields condition (D), (iii).

Let $r\in]0,h[$ and write $s=\sigma^{+}(r)$. Then $s<\bar{S}$ and consequently
the set
$\\{s^{\prime}\in[0,\bar{S}]\,:\,\sum_{i}\bar{\alpha}_{i}(s^{\prime})<1\mbox{
and }|s^{\prime}-s|\leq\epsilon\\}$ has positive Lebesgue measure for all
$\epsilon>0$. Using (5.9), we can show that, for some point arbitrarily close
to $s$ and therefore, by continuity, at $s$ itself,

$\underset{j=1}{\overset{N}{\sum}}\tilde{q}(s+(j-1)\bar{S})\cdot
f(\tilde{y}(s+(j-1)\bar{S}),\tilde{y}(s+(j-2)\bar{S}))-c=0\,.$ (5.12)

Since $s=\sigma^{+}(r)$ and
$\tilde{\sigma}^{+}(r+(j-1)h)=\sigma^{+}(r)+(j-1)\bar{S}$, this implies
condition (D), (i). Notice that if
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{0}_{j}(s)\,ds=\sigma^{+}(0)-\sigma^{-}(0)=\sigma^{+}(0)=0$,
then the equality in condition (D), (i) is valid also at $r=0$. Similarly, it
is satisfied also at $r=h$ whenever both
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{0}_{j}(s)\,ds=0$ and
$\underset{j=1}{\overset{N}{\sum}}\int_{0}^{1}\bar{w}^{h}_{j}(s)\,ds=\sigma^{+}(h)-\sigma^{-}(h)=\bar{S}-\sigma^{-}(h)=0$.
In this case indeed, (5.12) is satisfied for $s=\bar{S}=\sigma^{-}(h)$ and
$\tilde{\sigma}(ih)=i\bar{S}$ by (3.6).

Now take any $r\in[0,h]$ such that
$\sum_{i=1}^{N}\int_{0}^{1}\bar{w}^{r}_{i}(s)\,ds>0$. Then
$\sigma^{+}(r)>\sigma^{-}(r)$ and $\sum_{j}\bar{\alpha}_{j}(s^{\prime})=1$,
for a.e. $s^{\prime}\in[\sigma^{-}(r),\sigma^{+}(r)]$. It follows from
(5.6)–(5.9) and ‘continuity’ that $\sum_{j}\,q_{j}(s)\cdot
f(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-c\leq\max_{j}\,q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s))-d=0$ for all
$s\in[\sigma^{-}(r),\sigma^{+}(r)]$ and, for any $i$ such that
$\bar{\alpha}_{i}(s)>0$, $\,q_{i}(s)\cdot
g(\bar{y}_{i}(s),\bar{y}_{i-1}(s))=\max_{j}\,q_{j}(s)\cdot
g(\bar{y}_{j}(s),\bar{y}_{j-1}(s)$ for a.e.
$s\in[\sigma^{-}(r),\sigma^{+}(r)]$. It follows now from (5.4) and the very
definition of $\eta_{i}^{r}$ that condition (D), (iv) is satisfied. ∎

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