# Multifunctions of Bounded Variation

R. B. Vinter R. B. Vinter is with Department of Electrical and Electronic
Engineering, Imperial College London, Exhibition Road, London SW7 2BT, UK,
<EMAIL_ADDRESS>
###### Abstract

Consider control systems described by a differential equation with a control
term or, more generally, by a differential inclusion with velocity set
$F(t,x)$. Certain properties of state trajectories can be derived when it is
assumed that $F(t,x)$ is merely measurable w.r.t. the time variable t. But
sometimes a refined analysis requires the imposition of stronger hypotheses
regarding the time dependence ). Stronger forms of necessary conditions for
minimizing state trajectories can derived, for example, when $F(t,x)$ is
Lipschitz continuous w.r.t. time. It has recently become apparent that
significant addition properties of state trajectories can still be derived,
when the Lipschitz continuity hypothesis is replaced by the weaker requirement
that $F(t,x)$ has bounded variation w.r.t. time. This paper introduces a new
concept of multifunctions $F(t,x)$ that have bounded variation w.r.t. time
near a given state trajectory, of special relevance to control. We provide an
application to sensitivity analysis.

Keywords: Differential Inclusions, Optimal Control, Bounded Variation,
Sensitivity.

AMS classification: 34A60, 26A45, 49J21

##  1 Introduction

A widely used framework for control systems analysis is based on a description
of the dynamic constraint in the form of a differential inclusion

$\dot{x}(t)\in F(t,x(t))\quad\mbox{a.e. }t\in[S,T]\,,$ (1.1)

in which $F(\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\leadsto\mathbb{R}^{n}$ is a
given multifunction. We refer to absolutely continuous functions
$x(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$ satisfying (1.1) as state
trajectories.

It is well known that the assumptions that are made regarding the $t$
dependence of $F(t,x)$ have a critical effect on the qualitative properties of
the set of state trajectories and, if state trajectories minimizing a given
cost function are of primary interest, the assumptions affect the regularity
properties of the value function, the nature of necessary conditions that can
be derived, etc. In previous research on the distinct properties of state
trajectories, depending on the different assumptions that are made about the
regularity of $F(t,x)$ with respect to $t$, the attention has focused on
consequences of hypothesizing:

(a): $t\rightarrow F(t,x)$ is measurable, or

(b): $t\rightarrow F(t,x)$ is Lipschitz continuous.

([13, Chap. 2]) definitions of measurability and Lipschitz continuity of
multifunctions.) Some examples of distinct properties are as follows.

(i): standard necessary conditions of optimality, in state-constrained optimal
control, take a non-degenerate form, under the assumption that $F(\cdot,x)$ is
Lipschitz continuous and other assumptions, but this is no longer in general
the case if $F(\cdot,x)$ is merely measurable [13, Thm. 10.6.1].

(ii): optimal state trajectories have essentially bounded derivatives under
the assumption that $F(\cdot,x)$ is Lipschitz continuous and other
assumptions, but may not be essentially bounded if $F(\cdot,x)$ is merely
measurable [8].

(iii): the Hamitonian evaluated along an optimal state trajectory and co-state
trajectory cannot contain jumps if $F(\cdot,x)$ is Lipschitz continuous, but
may be discontinuous if $F(\cdot,x)$ is merely measurable [6].

Other examples where there are significant differences in the implications of
the two kinds of regularity hypotheses arise in the study of regularity
properties of the value function for state constrained optimal control
problems [3], validity of necessary conditions of optimality for free-time
optimal control problems [6], the interpretation of costate trajectories as
gradients of the value function [2] and in more general sensitivity analysis.

Are there other classes of differential inclusions $F(t,x)$, defined by their
regularity w.r.t. $t$, where interesting, distinct properties are encountered?
It turns out that multifunctions $F(t,x)$ having bounded variation w.r.t. $t$
is an example of such a class. Many properties of the set of state
trajectories that are valid when $F(t,x)$ has Lipschitz $t$-dependence, but
not in general when $F(t,x)$ has measurable $t$-dependence, have analogues
when $F(t,x)$ has bounded variation $t$-dependence.

How should we define ‘$F(t,x)$ has bounded variation $t$-dependence’? An
obvious approach is to require:

$\sup\left\\{\sum_{i=0}^{N-1}\underset{x\in
X}{\sup}\,d_{H}(F(t_{i+1},x),F(t_{i},x))\right\\}\,<\,\infty\,.$ (1.2)

Here, $X$ is some suitably large subset of $\mathbb{R}^{n}$.
$d_{H}(\cdot,\cdot)$ denotes the Hausdorff distance. (See 1.4). The outer
supremum is taken over all possible partitions $\\{t_{0}=S,\ldots,t_{N}=T\\}$
of $[S,T]$. But we follow a more refined approach, for reasons that we now
describe.

In the study of the implications of regularity assumptions regarding the
$t$-dependence of $F(t,x)$, the interest usually focuses on a particular state
trajectory $\bar{x}(.)$ (typically a state trajectory minimizing a given cost
function). We can expect that, in such situations, properties of
$F(\cdot,\cdot)$ only on some neighborhood of the graph of $\bar{x}(.)$ would
be relevant to the ensuing control systems analysis. One way to take account
of the special trajectory $\bar{x}(\cdot)$ would be to let $X$ in (1.2) be a
closed set which contained all possible values of $\bar{x}(.)$, i.e.

$\\{\bar{x}(t)\,|\,t\in[S,T]\\}\;\subset\;X\quad\mbox{for all }t\in[S,T]\,.$

This approach involves making unnecessary assumptions about values of
$F(\cdot,\cdot)$ at points far from the graph of $\bar{x}(\cdot)$. We
therefore adopt a more refined definition of bounded variation multifunctions,
in which the inner suprema in (1.2) are taken, not over $X$, but over smaller
sets (defined by a parameter $\delta>0$) and the outer supremum is taken over
partitions $\\{t_{i}\\}$ of $[S,T]$ with ‘mesh size’, which we write
$\mbox{diam}\,(\\{t_{i}\\})$ (see (2.2) below), not greater than $\epsilon>0$.
Accordingly, we say that $t\rightarrow F(t,\cdot)$ has bounded variation along
$\bar{x}(\cdot)$ if, for some $\delta>0$ and $\epsilon>0$ we have

$\sup\left\\{\sum_{i=0}^{N-1}\underset{x\in\bar{x}(t)+\delta\mathbb{B},t\in[t_{i},t_{i+1}]}{\sup}\,d_{H}(F(t_{i+1},x),F(t_{i},x))\,|\,\mbox{diam}(\\{t_{i}\\})\leq\epsilon\right\\}\,<\,\infty\,.$
(1.3)

We add another refinement; that is to consider multifunctions $F(t,x,a)$,
whose argument includes an additional variable $a$ that ranges over a given
subset $A$ of a finite dimensional linear space. Including the parameter $a$
provides useful flexibility for certain applications [11].

The purpose of this paper is to bring together and prove properties (relevant
to control system analysis) of a multifunction that has bounded variation
along some given state trajectory $\bar{x}(\cdot)$, and of the associated
cummulative variation function. These include one-sided continuity properties
of such multifunctions and the effects on the cummulative variation function
of changes to the multifunction. In the case of a function $m(\cdot,x)$ of
bounded variation along $\bar{x}(\cdot)$ (a function can be regarded as a
special case of a multifunction), it is shown that there is an associated
signed Borel measure. Finally, we show how this theory can be used to obtain
new sensitivity formulae describing how the output of a control system is
affected by a small time delay in the implementation of a control.

The analysis in this paper generalizes some aspects of the classical theory of
functions of a scalar variable having bounded variation, to allow for several
independent variables, when the bounded variation property pertains only to
one of the variables, and when multifunctions replace functions. There is
extensive recent work, treating the properties of bounded variation functions
with several independent variables, for which the monograph [1] is a
comprehensive source of references. The motivation arises from a desire to
investigate regularity properties of minimizers of variational problems in
several independent variables and of solutions to Hamilton Jacobi equations
arising in optimal control (see, for example, [4]). Multi-functions $F(t)$ of
a single variable $t$ (no $x$-dependence) possessing a one-sided bounded
variation property have been investigated by Moreau [10], in connection with
sweeping processes. But the study initiated in this paper of multifunctions
that are $x$-dependent and have bounded variation ‘near’ a given state
trajectory $\bar{x}(.)$ is a new development.

Notation: For vectors $x\in\mathbb{R}^{n}$, $|x|$ denotes the Euclidean
length. $\mathbb{B}$ denotes the closed unit ball in $\mathbb{R}^{n}$. Given a
multifunction $\Gamma(\cdot):\mathbb{R}^{n}\leadsto\mathbb{R}^{k}$, the graph
of $\Gamma(\cdot)$, written Gr$\,\\{\Gamma(\cdot)\\}$, is the set
$\\{(x,v)\in\mathbb{R}^{n}\times\mathbb{R}^{k}\,|\,v\in\Gamma(x)\\}$. Give a
set $A\subset\mathbb{R}^{n}$ and a point $x\in\mathbb{R}^{n}$, we denote by
$d_{A}(x)$ the Euclidean distance of a point $x\in\mathbb{R}^{n}$ from $A$:

$d_{A}(x)\;:=\;\inf\\{|x-y|\,|\,y\in A\\}\;.$

co $A$ denotes the convex hull of a set $A\in\mathbb{R}^{n}$. Given an
interval $I$, we write $\chi_{I}(t)$ for the indicator function of $I$, taking
values $1$ and $0$ when $t\in I$ and $t\notin I$, respectively. For numbers
$a$ an $b$, $a\vee b:=\max\\{a,b\\}$ and $a\wedge b:=\min\\{a,b\\}$. Given two
nonempty sets $A,A\in\mathbb{R}^{k}$, their Hausdorff distance is

$d_{H(A,B)}\,:=\,\inf\\{d_{A}(x)|x\in B\\}\vee\inf\\{d_{B}(x)|x\in A\\}\,.$
(1.4)

A function $r:[S,T]\rightarrow\mathbb{R}$ of bounded variation on the interval
$[S,T]$ has a left limit, written $r(t^{-})$, at every point $t\in(S,T]$ and a
right limit, written $r(t^{+})$, at every point $t\in[S,T)$. We say $r(.)$ is
normalized if it is right continuous on $(S,T)$.

We denote by $NBV^{+}[S,T]$ the space of increasing, real-valued functions
$\mu(.)$ on $[S,T]$ of bounded variation, vanishing at the point $S$ and right
continuous on $(S,T)$. The total variation of a function $\mu(\cdot)\in
NBV^{+}[S,T]$ is written $||\mu(\cdot)||_{\mbox{TV}}$. As is well known, each
point $\mu(\cdot)\in NBV^{+}[S,T]$ defines a unique Borel measure on $[S,T]$.
This associated measure is also denoted $\mu(\cdot)$. The space of continuous
functions $x(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$ with supremum norm is
written $C([S,T];\mathbb{R}^{n})$ and we denote by
$C^{*}([S,T];\mathbb{R}^{n})$ its topological dual space.

A modulus of continuity is a function
$\theta(.):[0,\infty)\rightarrow[0,\infty)$ such that $\lim_{s\downarrow
0}\theta(s)=0$.

Take a lower semicontinuous function
$f(\cdot):\mathbb{R}^{k}\to\mathbb{R}\cup\\{+\infty\\}$ and a point
$\bar{x}\in\mbox{dom}\,f(\cdot)\,:=\,\\{x\in\mathbb{R}^{k}\,|\,f(x)<+\infty\\}$.
The subdifferential of $f(\cdot)$ at $\bar{x}$, denoted $\partial f(\bar{x})$
is the set:

$\displaystyle\partial
f(\bar{x})\;:=\;\Big{\\{}\xi\,|\,\exists\;\xi_{i}\rightarrow\xi\mbox{ and
}x_{i}\stackrel{{\scriptstyle\mbox{{\tiny{\rm
dom}}}\,f(\cdot)}}{{\longrightarrow}}\bar{x}\mbox{ such that}$
$\displaystyle\limsup_{x\rightarrow
x_{i}}\frac{\xi_{i}\cdot(x-x_{i})-\varphi(x)+\varphi(x_{i})}{|x-x_{i}|}\,\leq
0\,\mbox{ for all }i\in\mathbb{N}\Big{\\}}\;.$

Here,
‘$x_{i}\stackrel{{\scriptstyle\mbox{dom}\,f(\cdot)}}{{\rightarrow}}\bar{x}$’
means that all elements in the convergent sequence $\\{x_{i}\\}$ lie in
$\mbox{dom}\,f(\cdot)$. For further information about subdifferentials, and
related constructs in nonsmooth analysis, see [7], [12] and [13].

##  2 Multifunctions of Bounded Variation

Take a bounded interval $[S,T]$, a compact set $A\subset\mathbb{R}^{k},$ a
multifunction $F(.,.,.):[S,T]\times\mathbb{R}^{n}\times
A\leadsto\mathbb{R}^{n}$ and a continuous function
$\bar{x}(.):[S,T]\rightarrow\mathbb{R}^{n}$. Generic elements in the domain of
$F(\cdot,\cdot,\cdot)$ are denoted by $(t,x,a)$.

In this section we define a concept that makes precise the statement
‘$F(t,x,a)$ has bounded variation with respect to the $t$ variable, along
$\bar{x}(.)$, uniformly with respect to $a\in A$’. If $F(t,x,a)$ is
independent of $(x,a)$ and single valued, i.e. $F(t,x,a)=\\{f(t)\\}$ for some
function $f(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$, this concept reduces to
the standard notion ‘$f(\cdot)$ has bounded variation’.

For any $t\in[S,T]$, $\delta>0$ and partition ${\cal
T}\,=\,\\{t_{0}=S,t_{1},\ldots,t_{N-1},t_{N}=t\\}$ of $[S,t]$, define
$I^{\delta}({\cal T})\in\mathbb{R}^{+}\cup\\{+\infty\\}$ to be

$I^{\delta}({\cal
T})\,:=\,\sum_{i=0}^{N-1}\sup\left\\{d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\;|\;x\in\bar{x}([t_{i},t_{i+1}])+\delta
B,a\in A\right\\}\;.$

Here, $\bar{x}([t_{i},t_{i+1}])$ denotes the set
$\\{\bar{x}(t)\,|\,t\in[t_{i},t_{i+1}]\\}$.

Take any $\epsilon>0$. Let
$\eta_{\epsilon}^{\delta}(\cdot):[S,T]\rightarrow\mathbb{R}^{+}\cup\\{+\infty\\}$
be the function defined as follows: $\eta_{\epsilon}^{\delta}(S)=0$ and, for
$t\in(S,T]$,

$\eta_{\epsilon}^{\delta}(t)\;=\;\sup\left\\{I^{\delta}({\cal T})\,|\,{\cal
T}\mbox{ is a partition of }[S,t]\mbox{ s.t. }\mbox{diam}({\cal
T})\leq\epsilon\right\\}\,,$ (2.1)

in which

$\mbox{diam}({\cal T})\,:=\,\sup\\{t_{i+1}-t_{i}\,|\,i=0,\ldots,N-1\\}\,.$
(2.2)

It is clear that, for any $t\in[S,T]$, $\delta>0$, $\delta^{\prime}>0$,
$\epsilon>0$, $\epsilon^{\prime}>0$,

$\delta^{\prime}\leq\delta\mbox{ and }\epsilon^{\prime}\leq\epsilon\implies
0\leq\eta_{\epsilon^{\prime}}^{\delta^{\prime}}(t)\leq\eta_{\epsilon}^{\delta}(t)\;.$
(2.3)

(This relation is valid even when $\eta_{\epsilon}^{\delta}(t)=+\infty$,
according to the rule ‘$+\infty\leq+\infty$’.) We may therefore define the
functions
$\eta^{\delta}(.),\eta(.):[S,T]\rightarrow\mathbb{R}^{+}\cup\\{+\infty\\}$ to
be

$\displaystyle\eta^{\delta}(t)\,:=\,\lim_{\epsilon\downarrow
0}\eta^{\delta}_{\epsilon}(t)\;\mbox{ for }t\in[S,T]\,$ (2.4)
$\displaystyle\eta(t)\,:=\,\lim_{\delta\downarrow 0}\eta^{\delta}(t)\;\mbox{
for }t\in[S,T]\,.$ (2.5)

######  Definition 2.1.

Take a set $A\subset\mathbb{R}^{k}$, a multifunction
$F(\cdot,\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\times A\leadsto\mathbb{R}^{n}$
and a function $\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$. We say that
$F(\cdot,x,a)$ has bounded variation along $\bar{x}(.)$ uniformly over $A$, if
the function $\eta(\cdot)$ given by (2.5) satisfies $\eta(T)\,<\,+\infty\;.$

If $F(\cdot,x,a)$ has bounded variation along $\bar{x}(\cdot)$ uniformly over
$A$, the function $\eta(\cdot)$ defined by (2.5) is called the cummulative
variation function of $F(\cdot,x,a)$ along $\bar{x}(\cdot)$, uniformly over
$A$. We also refer to $\eta^{\delta}_{\epsilon}(\cdot)$ and
$\eta^{\delta}(\cdot)$, defined by (2.1) and (2.4) as the
$(\delta,\epsilon)$-perturbed cummulative variation function and
$\epsilon$-perturbed cummulative variation function respectively.

In what follows we will adhere to the following notational convention: if
$\eta(\cdot)$ is a given cummulative variation function (for some
$F(\cdot,\cdot,\cdot)$ and $\bar{x}(\cdot)$) then, for any $\delta>0$ and
$\epsilon>0$, $\eta^{\delta}(\cdot)$ and $\eta^{\delta}_{\epsilon}(\cdot)$
denotes the $\delta$-perturbed and $\delta,\epsilon$-perturbed cummulative
variation functions associated with $\eta(\cdot)$, according to (2.1) and
(2.4).

If $F(t,x,a)$ does not depend on $a$, we omit mention of the qualifier
‘uniformly over $A$’. A function $L(\cdot,x,a)$ is said to have bounded
variation along $\bar{x}(.)$ uniformly over $A$, if the associated
multifunction $\\{L(\cdot,x,a)\\}$ has this property.

Assume that $F(\cdot,x,a)$ has bounded variation along $\bar{x}(\cdot)$
uniformly over $A$. Then there exist $\bar{\delta}>0$ and $\bar{\epsilon}>0$
for which $\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)<+\infty$. We list the
following elementary properties of the accumulative variation functions
(‘elementary’, in the sense that they are simple consequences of the
definitions): for any $\delta\in(0,\bar{\delta}]$ and
$\epsilon\in(0,\bar{\epsilon}]$,

  * (a):

$t\rightarrow\eta^{\delta}_{\epsilon}(t)$, $t\rightarrow\eta^{\delta}(t)$ and
$t\rightarrow\eta(t)$ are increasing, finite valued functions,

  * (b):

$\eta^{\delta}_{\epsilon}(t)\geq\eta^{\delta}(t)\geq\eta(t)$ for all
$t\in[S,T]$

  * and
  * (c):

given any $[s,t]\subset[S,T]$ such that $t-s\leq\epsilon$,

$d_{H}(F(t,y,a),F(s,y,a))\,\leq\,\eta^{\delta}_{\epsilon}(t)-\eta^{\delta}_{\epsilon}(s),$
(2.6)

$\mbox{ for all }y\in\bar{x}([s,t])+\delta\mathbb{B}\,\mbox{ and }\,a\in A$.

Example. An important potential role of the preceding constructs will be to
derive regularity properties of value functions, minimizing state trajectories
and other functions associated with an optimal control problem, in which the
dynamic constraint is a differential inclusion $\dot{x}\in F(t,x)$, when
$F(t,x)$ has bounded variation with respect to the $t$ variable, ‘near’ a
given state trajectory $\bar{x}(\cdot)$. Regularity properties are typically
related to the cummulative variation function $\eta(.)$ of $F(\cdot,x)$. The
more precise is the information about the cummulative variation the more
informative is the corresponding regularity property that can be derived. This
is the main reason why we have adopted the refined definition, Def. 2.1, for
the formulation of the ‘bounded variation’ hypothesis, in place of a simpler
one based on the condition (1.2), for some closed subset $X$ that strictly
contains the range of $\bar{x}(\cdot)$ in its interior. The purpose of this
example is to show that using the ‘refined’ definition can provide a more
informative cummulative variation function.

Consider the function
$f(\cdot,\cdot):[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ and the function
$\bar{x}(\cdot):[0,1]\rightarrow\mathbb{R}$:

$f(t,x)=t\,x\mbox{ and }\bar{x}(t)=t\quad\mbox{for
$(t,x)\in[0,1]\times\mathbb{R}$.}$

Take $X=\mbox{range}\\{\bar{x}(\cdot)\\}=[0,1]$. The cummulative variation
function $\eta_{\mbox{simple}}(\cdot)$ of $f(\cdot,x)$ related to condition
(1.2) and defined by

$\eta_{\mbox{simple}}(t)\,=\,\sup\left\\{\sum_{i=0}^{N-1}\underset{x\in
X}{\sup}\,d_{H}(F(t_{i+1},x),F(t_{i},x))\right\\}$

in which the outer supremum is taken over all possible partitions of $[0,t]$,
is easily calculated to be:

$\eta_{\mbox{simple}}(t)\,=\,t\,.$

Also, the cummulative variation of $f(\cdot,x)$ along $\bar{x}(.)$ according
to Def. 3.2, is

$\eta(t)\,=\,\frac{1}{2}t^{2}\,.$

Notice that for any (nontrivial) subinterval $[s,t]\subset[0,1]$, $t>s$,

$\displaystyle(\eta_{\mbox{simple}}(t)-\eta_{\mbox{simple}}(s))-(\eta(t)-\eta(s))\;=$
$\displaystyle\hskip
72.26999pt(t-s)-\frac{1}{2}(t^{2}-s^{2})=(t-s)(1-\frac{1}{2}(t+s))\,(\,>\,0),$

from which it can be deduced that the Borel measure induced by $\eta(\cdot)$
strictly minorizes that induced by $\eta_{\mbox{strict}}(\cdot)$ in the sense

$\int_{D}d\eta_{\mbox{simple}}(t)-\int_{D}d\eta(t)>0\,$

for any Borel subset $D\subset[0,1]$ having nonempty interior. This
demonstrates the greater precision that can be achieved in regularity
analysis, by using the more refined definition.

##  3 Continuity Properties

As is well known, an $\mathbb{R}^{n}$-valued function of bounded variation on
a finite interval may be discontinuous, but it has everywhere left and right
limits and it has at most a countable number points of discontinuity. A
multifunction having bounded variation along a given continuous trajectory
uniformly over a given set has similar properties, as described in the
following proposition. We invoke the hypotheses:

  * (C1)

$F(t,x,a)$ is closed and non-empty for each
$(t,x,a)\in[S,T]\times\mathbb{R}^{n}\times A$, $F(\cdot,x,a)$ is measurable
for each $(x,a)\in\mathbb{R}^{n}\times A$ and there exists $c>0$ such that

$F(t,x,a)\,\subset\,c\mathbb{B}\mbox{ for all
$x\in\bar{x}(t)+\bar{\delta}\mathbb{B}$, $t\in[S,T]$, $a\in A$. }$ (3.1)

  * (C2)

There exists a modulus of continuity
$\gamma(.):\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that

$F(t,x,a)\,\subset\,F(t,x^{\prime},a^{\prime})+\gamma(|x-x^{\prime}|+|a-a^{\prime}|)\mathbb{B}$
(3.2)

for all $x,x^{\prime}\in\bar{x}(t)+\bar{\delta}\mathbb{B}$, $t\in[S,T]$ and
$a,a^{\prime}\in A$.

######  Proposition 3.1.

Take a compact set $A\subset\mathbb{R}^{k}$, a continuous function
$\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$ and a multifunction
$F(\cdot,\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\times
A\leadsto\mathbb{R}^{n}$. Suppose that $F(\cdot,x,a)$ has bounded variation
along $\bar{x}(\cdot)$ uniformly over $A$. Take $\bar{\delta}>0$ such that
$\eta^{\bar{\delta}}(T)<+\infty$. Assume (C1) and (C2). Take any
$\delta\in(0,\bar{\delta})$. Then

  * (a):

For any $\bar{s}\in[S,T)$ and $\bar{t}\in(S,T]$ , the one-sided, set-valued
limits

$F(\bar{s}^{+},x,a)\,:=\,\underset{s\downarrow\bar{s}}{\lim}\,F(s,x,a)\,,\quad
F(\bar{t}^{-},y,a)\,:=\,\underset{t\uparrow\bar{t}}{\lim}\,F(t,y,a)$

exist for every $x\in\bar{x}(\bar{s})+\delta\mathbb{B}$,
$y\in\bar{x}(\bar{t})+\delta\mathbb{B}$ and $a\in A$.

  * (b):

For any $\bar{s}\in[S,T)$ and $\bar{t}\in(S,T]$

$\underset{s\downarrow\bar{s}}{\lim}\;\underset{\underset{a\in
A}{x\in\bar{x}(\bar{s})+\delta
B,}}{\sup}\,d_{H}(F(\bar{s}^{+},x,a),F(s,x,a))\,=\,0$

and

$\underset{t\uparrow\bar{t}}{\lim}\;\underset{\underset{a\in
A}{x\in\bar{x}(\bar{t})+\delta
B,}}{\sup}\,d_{H}(F(\bar{t}^{-},x,a),F(t,x,a))\,=\,0$

  * (c):

There exists a countable set ${\cal A}$ such that, for every
$t\in(S,T)\backslash{\cal A}$,

$\underset{t^{\prime}\rightarrow t}{\lim}\;\underset{\underset{a\in
A}{x\in\bar{x}(t)+\bar{\delta}B}}{\sup}\,d_{H}(F(t^{\prime},x,a),F(t,x,a))\,=\,0\,.$

Proof.

(a): We prove only the first assertion. The proof of the second assertion is
similar. Choose any $\bar{s}\in[S,T)$. Take $\epsilon>0$ such that
$\eta^{\bar{\delta}}_{\epsilon}(T)<+\infty$. Fix $\delta\in(0,\bar{\delta})$.
Take any $x\in\bar{x}(\bar{s})+\delta\mathbb{B}$, $a\in A$ and

$v\in\underset{s\downarrow\bar{s}}{\mbox{lim sup}}\,F(s,x,a)\;.$

By definition of ‘lim sup’, there exists $s_{i}\downarrow\bar{s}$ and
$v_{i}\rightarrow v$ such that

$v_{i}\in F(s_{i},x,a)\mbox{ for all }i\;\mbox{ and }\;v_{i}\rightarrow
v\mbox{ as }i\rightarrow\infty\,.$

The assertion (a) will follow if we can show that, also,

$v\in\underset{s\downarrow\bar{s}}{\mbox{lim inf}}\,F(s,x,a)\;,$ (3.3)

i.e. the ‘lim sup’ and ‘lim inf’ coincide, in which case the limit exists. To
show (3.3) we take an arbitrary sequence $t_{j}\downarrow\bar{s}$. Since
$\bar{x}(\cdot)$ is continuous and $x\in\bar{x}(\bar{s})+\delta\mathbb{B}$, we
can arrange, by eliminating elements in the sequence $\\{(s_{i},v_{i})\\}$,
that, for every $j$, $\bar{s}\leq
s_{j}<t_{j},\,t_{j}-\bar{s}\leq\epsilon\mbox{ and
}x\in\bar{x}(t)+\bar{\delta}\mathbb{B}\;\mbox{ for all
}t\in[\bar{s},{t}_{j}]$, $j=1,2,\ldots$ But then, since
$t_{j}-s_{j}\leq\epsilon$ and by property (2.6) of the
$(\delta,\epsilon)$-perturbed cummulative variation function,

$d_{H}(F(t_{j},x,a),F(s_{j},x,a))\,\leq\,\eta_{\epsilon}^{\bar{\delta}}(t_{j})-\eta_{\epsilon}^{\bar{\delta}}(s_{j})\;.$

This means that, for each $j$, there exists $w_{j}\in F(t_{j},x,a)$ and

$|v_{j}-w_{j}|\,\leq\,\eta_{\epsilon}^{\bar{\delta}}(t_{j})-\eta_{\epsilon}^{\bar{\delta}}(s_{j})\;.$

We know however that, since $\eta_{\epsilon}^{\bar{\delta}}(.)$ is a finite
valued, monotone function, it has a right limit
$\eta_{\epsilon}^{\bar{\delta}}(\bar{s}^{+})$ at $\bar{s}$. Hence

$\lim_{j\rightarrow\infty}|v_{j}-w_{j}|\,\leq\,\lim_{j\rightarrow\infty}\left(\eta_{\epsilon}^{\bar{\delta}}(t_{j})-\eta_{\epsilon}^{\bar{\delta}}(s_{j})\right)\,\leq\,\eta_{\epsilon}^{\bar{\delta}}(\bar{s}^{+})-\eta_{\epsilon}^{\bar{\delta}}(\bar{s}^{+})=0\;.$

It follows that $v_{j}-w_{j}\rightarrow 0$. But then
$v=\lim_{j}v_{j}=\lim_{j}w_{j}$. Since $t_{j}\downarrow\bar{s}$ was an
arbitrary sequence, we conclude (3.3). We have confirmed (a).

(b) These assertions may be deduced from (a), together with the compactness of
the set $A$ and of closed $\delta$ balls about $\bar{x}(\bar{s})$ and
$\bar{x}(\bar{t})$, and with the assumed continuity properties of
$(x,a)\rightarrow F(t,x,a)$.

(c) Let ${\cal A}$ be the empty or countable subset of $(S,T)$ comprising
points at which the finite-valued, monotone function
$\eta_{\epsilon}^{\bar{\delta}}(\cdot)$ is discontinuous. Fix a point
$t\in(S,T)\backslash{\cal A}$. Take any $\rho>0$. Since
$\eta_{\epsilon}^{\bar{\delta}}(\mathbb{R}^{n}\times A)$ is continuous at $t$,
we may choose $\sigma>0$ such that

$\eta_{\epsilon}^{\bar{\delta}}(t+\sigma)-\eta_{\epsilon}^{\bar{\delta}}(t-\sigma)\,\leq\,\rho\;.$

So by (2.6), for any $t^{\prime}\in[S,T]$ such that
$|t^{\prime}-t|\leq\sigma\wedge\epsilon$ we have

$\displaystyle\sup\\{d_{H}(F(t^{\prime},x,a),F(t,x,a))\,|\,x\in\bar{x}(t)+\bar{\delta}\mathbb{B},\,a\in
A\\}$ $\displaystyle\leq$
$\displaystyle\eta_{\epsilon}^{\bar{\delta}}(t^{\prime}\vee
t)-\eta_{\epsilon}^{\bar{\delta}}(t^{\prime}\wedge t)$ $\displaystyle\leq$
$\displaystyle\eta_{\epsilon}^{\bar{\delta}}(t+\sigma)-\eta_{\epsilon}^{\bar{\delta}}(t-\sigma)\leq\rho\;.$

The continuity properties of $F(.,x,a)$ at $t$ have been confirmed. $\square$

The following proposition provides information about how the cummulative
variation function of a multifunction, and its $\delta$-perturbation, are
affected by changes of $\delta$ and the parameter space for $a$.

######  Proposition 3.2.

Take compact sets $A_{1},A\subset\mathbb{R}^{k}$ such that $A_{1}\subset A$, a
continuous function $\bar{x}(.)$ and a multifunction
$F(\cdot,\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\times\mathbb{R}^{k}\leadsto\mathbb{R}^{n}$.
Suppose that $F(\cdot,x,a)$ has bounded variation along $\bar{x}(.)$ uniformly
over $A$. Write the cummulative variation function $\eta_{A}(\cdot)$. Let
$\bar{\delta}>0$ be such that $\eta^{\bar{\delta}}_{A}(T)<\infty$. Suppose
that hypotheses (C1) and (C2) of Prop. 3.1 are satisfied.

Then $F(\cdot,x,a)$ has bounded variation along $\bar{x}(.)$ uniformly over
$A_{1}$. (Write the cummulative variation function $\eta_{A_{1}}(\cdot)$.)
Furthermore, for any $[s,t]\subset[S,T]$ and $\delta>0$ and
$\delta^{\prime}>0$ such that $0<\delta^{\prime}\leq\delta\leq\bar{\delta}$,

$\eta_{A_{1}}(t)-\eta_{A_{1}}(s)\;\leq\;\eta^{\delta^{\prime}}_{A_{1}}(t)-\eta^{\delta^{\prime}}_{A_{1}}(s)\;\leq\;\eta^{\delta}_{A}(t)-\eta^{\delta}_{A}(s)\;.$
(3.4)

Proof. The facts that $F(\cdot,x,a)$ has bounded variation along
$\bar{x}(\cdot)$ uniformly over $A_{1}$ and
$\eta^{\bar{\delta}}_{A_{1}}(T)<\infty$, follow immediately from the
definitions, since $A_{1}\subset A$.

Take any $[s,t]\subset[S,T]$ and
$0<\delta^{\prime}\leq\delta\leq\bar{\delta}$, such that
$\eta^{\bar{\delta}}_{A}(T)<\infty$. Choose $\epsilon>0$ such that
$\eta_{A\,\epsilon}^{\bar{\delta}}(T)<\infty$.

Concerning assertion (3.4), we need prove only the right-side inequality
$\eta^{\delta^{\prime}}_{A_{1}}(t)-\eta^{\delta^{\prime}}_{A_{1}}(s)\;\leq\;\eta^{\delta}_{A}(t)-\eta^{\delta}_{A}(s)$,
since the left-side inequality of (3.4) follows immediately by passing to the
limit as $\delta^{\prime}\downarrow 0$.

Presently, we shall make use of the following fact: take
$[\sigma_{1},\sigma_{2}]\subset[S,T]$ such that
$|\sigma_{1}-\sigma_{2}|\leq\epsilon$. Then, for any
$\sigma\in[\sigma_{1},\sigma_{2}]$ and
$x^{\prime}\in\bar{x}([\sigma_{1},\sigma_{2}])+\delta^{\prime}\mathbb{B}$,

$\displaystyle\sup\\{d_{H}(F(\sigma_{1},x^{\prime},a),F(\sigma_{2},x^{\prime},a))|a\in
A\\}$
$\displaystyle\,\leq\sup\\{d_{H}(F(\sigma_{1},x,a),F(\sigma,x,a))|x\in\bar{x}([\sigma_{1},\sigma])+\delta^{\prime}\mathbb{B},a\in
A\\}$
$\displaystyle\;+\sup\\{d_{H}(F(\sigma,x,a),F(\sigma_{2},x,a))|x\in\bar{x}([\sigma,\sigma_{2}])+\delta^{\prime}\mathbb{B},a\in
A\\}+2\gamma(c\epsilon)$ (3.5)

in which $c$ and $\gamma(\cdot)$ are as in hypotheses (C1) and (C2). Define

$G^{\delta^{\prime}}_{A_{1}\epsilon}(\sigma_{1},\sigma_{2})\,:=\,\sup\,\sum_{i=0}^{N}\sup\\{d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))|x\in\bar{x}([t_{i},t_{i+1}])+\delta^{\prime}\mathbb{B},a\in
A_{1}\\}\,.$

(The outer supremum is taken over partitions $\\{t_{i}\\}$ of
$[\sigma_{1},\sigma_{2}]$ with diam $(\\{t_{i}\\})\leq\epsilon$.)

In view of (3.5) and by consideration of arbitrary partitions of $[S,t]$
having diameter at most $\epsilon$, and their modifications to include an
extra grid point $s$, we can show

$\eta^{\delta^{\prime}}_{A_{1}\,\epsilon}(t)\,\leq\,\eta^{\delta^{\prime}}_{A_{1}\,\epsilon}(s)+2\,\gamma(c\,\epsilon)+G^{\delta^{\prime}}_{A_{1}\epsilon}(s,t)\,,$
(3.6)

From the definition of $\eta^{\delta}_{A\,\epsilon}(.)$

$\eta^{\delta}_{A\,\epsilon}(t)\,\geq\,\eta^{\delta}_{A\,\epsilon}(s)+G^{\delta}_{A\epsilon}(s,t)\;.$
(3.7)

Since $A_{1}\subset A$ and $\delta^{\prime}\leq\delta$,
$G^{\delta^{\prime}}_{A_{1}\epsilon}(s,t)\leq G^{\delta}_{A\epsilon}(s,t)$.
Using this relation, combining (3.6) and (3.7) and passing to the limit as
$\epsilon\downarrow 0$, yields

$\eta^{\delta^{\prime}}_{A_{1}}(t)-\eta^{\delta^{\prime}}_{A_{1}}(s)\,\leq\,\eta^{\delta}_{A}(t)-\eta^{\delta}_{A}(s)\,.\qquad\square$

The next proposition relates the cummulative variation function of the
multifunction $F(\cdot,x,a)$ to that of the derived multifunction
$\tilde{F}(\cdot,x,a)$, obtained by replacing the end-point values by left and
right limits.

######  Proposition 3.3.

Take a compact set $A\subset\mathbb{R}^{k}$, a continuous function
$\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$ and a multifunction
$F(\cdot,\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\times
A\rightarrow\mathbb{R}^{n}$. Suppose that $F(\cdot,x,a)$ has bounded variation
along $\bar{x}(\cdot)$ uniformly over $A$. Denote by $\eta(\cdot)$ the
cummulative variation function. Take $\bar{\delta}>0$ such that
$\eta^{\bar{\delta}}(T)<\infty$. Assume that hypotheses (C1) and (C2). Let
$\tilde{F}(\cdot,\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\times
A\rightarrow\mathbb{R}^{n}$ be a multifunction such that, for
$(t,x,a)\in[S,T]\times\mathbb{R}^{n}\times A$,

$\tilde{F}(t,x,a)\,=\,\left\\{\begin{array}[]{ll}F(S^{+},x,a)&\mbox{if
}t=S\mbox{ and }|x-\bar{x}(S)|<\bar{\delta}\\\ F(T^{-},x,a)&\mbox{if
}t=T\mbox{ and }|x-\bar{x}(T)|<\bar{\delta}\\\ F(t,x,a)&\mbox{otherwise .
}\end{array}\right.$ (3.8)

(The limit sets $F(S^{+},x,a)$ and $F(T^{-},x,a))$ exist, by the preceding
proposition.)

Then $\tilde{F}(\cdot,\cdot,\cdot)$ has bounded variation along
$\bar{x}(\cdot)$ uniformly over $A$. Write $\tilde{\eta}(\cdot)$ for its
cummulative variation function.

Take any $\delta\in(0,\bar{\delta})$. Then $\tilde{\eta}^{\delta}(.)$ is right
continuous at $S$ and left continuous at $T$, i.e.

$\tilde{\eta}^{\delta}(S)=\lim_{s\downarrow
S}\tilde{\eta}^{\delta}(s)\quad\mbox{and}\quad\tilde{\eta}^{\delta}(T)=\lim_{t\uparrow
T}\tilde{\eta}^{\delta}(t)\,.$ (3.9)

Furthermore, the $\delta$-perturbed cummulative variation functions of
$F(.,.,.)$ and $\tilde{F}(.,.,.)$ are related as follows:

$\tilde{\eta}^{\delta}(t)\,=\,\left\\{\begin{array}[]{ll}\eta^{\delta}(t)-\underset{x\in\bar{x}(S)+\delta\mathbb{B},\,a\in
A}{\sup}\,d_{H}(F(S,x,a),F(S^{+},x,a))&\mbox{for }t\in(S,T)\\\
\eta^{\delta}(T)-\underset{x\in\bar{x}(S)+\delta\mathbb{B},\,a\in
A}{\sup}\,d_{H}(F(S,x,a),F(S^{+},x,a))&\\\ \hskip
57.81621pt-\underset{x\in\bar{x}(T)+\delta\mathbb{B},\,a\in
A}{\sup}\,d_{H}(F(T,x,a),F(T^{-},x,a))&\mbox{for }t=T\,.\end{array}\right.$
(3.10)

Relation (3.10) implies, in particular, that

$\tilde{\eta}^{\delta}(t)-\tilde{\eta}^{\delta}(s)=\eta^{\delta}(t)-\eta^{\delta}(s)\,\;\mbox{for
any }[s,t]\subset(S,T)\,.$ (3.11)

The relations (3.9)-(3.11) remain valid when $\delta=0$, under the
interpretation ‘ $\eta^{0}(\cdot)=\eta(\cdot)$ and
$\tilde{\eta}^{0}(\cdot)=\tilde{\eta}(\cdot)$’.

The proof of Prop. 3.3 is given in the Appendix.

##  4 The Partial Variation Measure of a Function of a Scalar and a Vector
Variable

In this section we examine in more detail the properties of a function
$m(\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{r}$ that has
bounded variation with respect to the first variable, along a given trajectory
$\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$. We restrict attention to a
special case of the multifunctions earlier considered, in which the
multifunction is point valued (i.e. $m(\cdot,\cdot)$ is a function), and no
longer depends on the parameter $a$.

The motivation for these investigation is a desire to make sense of integrals
arising in sensitivity analysis, of the form

$\int_{S}^{T}p^{T}(t)\frac{\partial m}{\partial t}\,(t,\bar{x}(t))dt\,,$ (4.1)

in circumstances when $m(t,x)$ has bounded variation with respect to the first
variable, but fails to be continuously differentiable with respect to this
variable. Here, $p(\cdot)$ is a given continuous function. Notice that, if
$m(\cdot,\cdot)$ is a continuously differentiable function, the integral can
be written as

$\int_{S}^{T}p^{T}(t)d\mu(t)\,,$ (4.2)

where $\mu(\cdot)$ is the (signed) Borel measure on $[S,T]$ defined by
$d\mu(t)\,=\,\alpha(t)dt\,,$ in which $\alpha(t)$ is the integrable function

$\alpha(t)=\frac{\partial m}{\partial t}\,(t,\bar{x}(t))\,.$

For functions $m(t,x)$’s that are merely of bounded variation with respect to
$t$, the idea is to define the integral according to (4.2), but now taking
$\mu(\cdot)$ to be some measure constructed from limits of finite difference
approximations of the function $m(\cdot,\cdot)$. The first step is to define
the measure $\mu(\cdot)$ to replace $\frac{\partial m}{\partial
t}\,(t,\bar{x}(t))$. The construction of the measure is based on the following
lemma, which invokes the hypotheses:

  * (BV1):

$\bar{x}(\cdot)$ is continuous and there exists $\delta^{\prime}>0$ such that
$m(t,.)$ is continuously differentiable on the interior of
$\bar{x}(t)+\delta^{\prime}\,\mathbb{B}$ for all $t\in[S,T]$.

  * (BV2):

    * (i):

$m(\cdot,x)$ has bounded variation along $\bar{x}(\cdot)$

    * (i):

$\nabla_{x}m(\cdot,x)$ has bounded variation along $\bar{x}(\cdot)$.

######  Proposition 4.1.

Take functions
$m(\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{r}$ and
$\bar{x}(.):[S,T]\rightarrow\mathbb{R}^{n}$. Assume (BV1) and (BV2) are
satisfied. Take any sequence of partitions $\\{t_{i}^{j}\\}_{i=0}^{N_{j}}$,
$j=1,2,\ldots$ of $[S,T]$ such that diam$(\\{t^{j}_{i}\\})\rightarrow 0$ as
$j\rightarrow\infty$ and any sequence $\rho^{j}\downarrow 0$. Take also any
sequence of collections of $n$-vectors $\\{\xi^{j}_{i}\\}_{i=0}^{N_{j}}$ such
that

$\xi^{j}_{i}\,\in\,\bar{x}(t)+\rho^{j}\mathbb{B}\quad\mbox{for some
}t\in[t^{j}_{i},t^{j}_{i+1}]$

for each $i$ and $j$. Define the sequence of discrete measures
$\mu^{j}(\cdot)$, $j=1,2,\ldots$ to be

$\mu^{j}(t)\,=\,\sum_{i=0}^{N_{j}-1}\left[m(t^{j}_{i+1},\xi^{j}_{i})-m(t^{j}_{i},\xi^{j}_{i})\right]\delta(t-t^{j}_{i})$

(‘pseudo-density’ notation), in which $\delta(\cdot)$ denotes the Dirac delta
function.

Then there exists a (signed) Borel measure $\mu(\cdot)$ on $[S,T]$ such that

$\mu^{j}(\cdot)\rightarrow\mu(\cdot)\quad\mbox{with respect to the
weak${}^{*}$ topology on $C^{*}([S,T];\mathbb{R}^{r})$, i.e.}$
$\int_{[S,T]}g^{T}(t)\,d\mu^{j}(t)\rightarrow\int_{[S,T]}g^{T}(t)\,d\mu(t)\mbox{
for every $g(.)\in C([S,T];\mathbb{R}^{r})$}\,.$

Furthermore, the limit measure $\mu(\cdot)$ does not depend on the choice of
sequences of partitions $\\{t^{j}\\}_{i=0}^{N_{j}}$, the sequence
$\\{\rho_{j}\\}$ or the sequence of collections of vectors
$\\{\xi^{j}\\}_{i=0}^{N_{j}}$ satisfying the stated conditions.

Proof. Denote by $\eta(\cdot)$ and $\tilde{\eta}(\cdot)$ cummulative variation
functions of $m(\cdot,x)$ and $\nabla_{x}m(\cdot,x)$, respectively, along
$\bar{x}(\cdot)$. We can choose $\bar{\epsilon}>0$ and $\bar{\delta}>0$ such
that $\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)<\infty$,
$\tilde{\eta}^{\bar{\delta}}_{\bar{\epsilon}}(T)<\infty$ and
$\theta_{\bar{x}}(\bar{\epsilon})\,\leq\,\bar{\delta}\,,$ where
$\theta_{\bar{x}}(\cdot)$ is a continuity modulus for $\bar{x}(\cdot)$.

Take any sequence of partitions $\\{t^{j}_{i}\\}$ of $[S,T]$, $j=1,2,\ldots$,
any sequence of real numbers $\rho^{j}\downarrow 0$ and any sequence
$\\{\xi^{j}_{i}\\}$, $j=1,2,\ldots$ of collections of $n$-vectors with the
properties listed in the statement of the lemma. Write

$\epsilon_{j}:=\sup_{j^{\prime}\geq
j}\,\mbox{diam}(\\{t^{j^{\prime}}_{i}\\})\,.$

By assumption $\epsilon_{j}\downarrow 0$, as $j\rightarrow\infty$. Fix $j$ and
$j^{\prime}(>j)$. Consider the case

  * (T):

$\\{t^{j}_{i}\\}\subset\\{t^{j^{\prime}}_{i}\\}$ , i.e.
$\\{t^{j^{\prime}}_{i}\\}$ is a sub-partition of $\\{t^{j}_{i}\\}$ .

We relabel the sequence $\\{t^{j}_{i}\\}$ as $\\{s_{0}\ldots s_{M}\\}$. Then,
since $\\{t^{j^{\prime}}_{i}\\}$ is a sub-partition of $\\{t^{j}_{i}\\}$,
$\\{t^{j^{\prime}}_{i}\\}$ can be written

$\\{t^{j^{\prime}}_{i}\\}=\\{s_{0\ell}\\}_{\ell=0}^{\ell_{0}}\cup\ldots\cup\\{s_{(M-1)\ell}\\}_{\ell=0}^{\ell_{M-1}}\,.$

Here $s_{00}=s_{0}$, and, for $i=1,\ldots,M-1$, $s_{i0}=s_{i}$ and
$s_{i\ell_{i}}=s_{i+1}(=s_{(i+1)0})$. Relabel the $n$-vectors associated with
these two partitions as $\xi_{0},\ldots\xi_{M-1}$ and as $\xi_{i\ell}$,
$\ell=0,\ell_{i}-1$, $i=1,\dots,M-1$. For any continuous function
$g(.):[S,T]\rightarrow\mathbb{R}^{r}$

$\displaystyle\langle\mu^{j}-\mu^{j^{\prime}},g(\cdot)\rangle$
$\displaystyle:=$
$\displaystyle\int_{[S,T]}g(t)(d\mu^{j}(t)-d\mu^{j^{\prime}}(t))$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{M-1}\left[g^{T}(s_{i})\left(m(s_{i+1},\xi_{i})-m(s_{i},\xi_{i})\right)\right.$
$\displaystyle\hskip
14.45377pt-\sum_{\ell=0}^{\ell_{i}-1}\left.g^{T}(s_{i\ell})\left(m(s_{i(l+1)},\xi_{i\ell})-m(s_{i\ell},\xi_{i\ell})\right)\right]\,.$

Using the fact that
$|m(s_{i(\ell+1)},\xi_{i\ell})-m(s_{i\ell},\xi_{i\ell})|\,\leq\,\eta^{\bar{\delta}}_{\bar{\epsilon}}(s_{i(\ell+1)})-\eta^{\bar{\delta}}_{\bar{\epsilon}}(s_{i\ell})\,,\mbox{etc.,}$

we can write

$\langle\mu^{j}-\mu^{j^{\prime}},g(\cdot)\rangle\,=\,a+e_{1}\,,$ (4.3)

where

$\displaystyle
a=\left.\sum_{i=0}^{M-1}g^{T}(s_{i})\right[\left(m(s_{i+1},\xi_{i})-m(s_{i},\xi_{i})\right)$
$\displaystyle\hskip
93.95122pt-\left.\sum_{\ell=0}^{\ell_{i}-1}\left(m(s_{i(l+1)},\xi_{i\ell})-m(s_{i\ell},\xi_{i\ell})\right)\right]\,.$
(4.4)

and $e_{1}$ is an ‘error term’ that satisfies

$\displaystyle|e_{1}|\,\leq\,\theta_{g}(\epsilon_{j})\times\sum_{i,\ell}\left(\eta^{\bar{\delta}}_{\bar{\epsilon}}(s_{i(\ell+1)})-\eta^{\bar{\delta}}_{\bar{\epsilon}}(s_{i\ell})\right)\,=\,\theta_{g}(\epsilon_{j})\times\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)\,.$

Here, $\theta_{g}(\cdot)$ is a continuity modulus for $g(\cdot)$. Observe next
that
$\xi_{i\ell},\,\xi_{i}\in\bar{x}(s_{i})+\theta_{\bar{x}}(\epsilon_{j})\mathbb{B}$
for each $i$ and $\ell=0,\ldots,l_{i}$. It follows from (BV1) that, for $j$
sufficiently large, the functions $m(s_{i(\ell+1)},\cdot)$ and
$m(s_{i\ell},\cdot)$ in (4.4) are continuously differentiable on a ball
containing $\xi_{i\ell}$ and $\xi_{i}$. Employing an exact first order Taylor
expansion of these functions about $\xi_{i}$, we can write terms in the inner
summation on the right of (4.4)

$\displaystyle
g^{T}(s_{i})\left(m(s_{i(\ell+1)},\xi_{i\ell})-m(s_{i\ell},\xi_{i\ell})\right)$
$\displaystyle\hskip
72.26999pt=\,g^{T}(s_{i})\left(m(s_{i(\ell+1)},\xi_{i})-m(s_{i\ell},\xi_{i})\right)$
$\displaystyle\hskip
72.26999pt\,+\,g^{T}(s_{i})\left(\nabla_{x}m(s_{i(\ell+1)},\tilde{\xi}_{i\ell})-\nabla_{x}m(s_{i\ell},\tilde{\xi}_{i\ell})\right)\cdot\left(\xi_{il}-\xi_{i}\right)$
(4.5)

for $n$-vectors $\tilde{\xi}_{i\ell}\in\mbox{co}\,\\{\xi_{i},\xi_{i\ell}\\}$.
We can show that, for each $i$ and $\ell=0,\dots,l_{i-1}$, and for all values
of $j$ sufficently large,

$|\tilde{\xi}_{i\ell}-\bar{x}(s_{i\ell})|\leq
3\times(\theta_{\bar{x}}(\epsilon_{j})+\rho_{j})\leq\bar{\delta}\,.$

Here, $\theta_{\bar{x}}(.)$ is a continuity modulus for $\bar{x}(.)$. We note
also that

$|\xi_{il}-\xi_{i}|\leq 2\times(\theta_{\bar{x}}(\epsilon_{j})+\rho_{j})\,.$

Substituting (4.5) into (4.4), noting cancellation of terms and, finally,
using the fact that $\nabla_{x}m(\cdot,x)$ has bounded variation along
$\bar{x}(\cdot)$, we arrive at

$a\,=\,0+\ldots 0+e_{2}\,,$

where $e_{2}$ is an error term that satisfies
$|e_{2}|\,\leq\,2\,(\theta_{\bar{x}}(\epsilon_{j})+\rho^{j})\,||g(.)||_{C}\,\tilde{\eta}^{\bar{\delta}}_{\bar{\epsilon}}(T)\,.$

It now follows from (4.3) and (4.4) that

$\langle\mu^{j}-\mu^{j^{\prime}},g(.)\rangle\,\leq\,\theta_{g}(\epsilon_{j})\,\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)+2\,(\theta_{\bar{x}}(\epsilon_{j})+\rho_{j})\,||g(.)||_{C}\,\tilde{\eta}^{\bar{\delta}}_{\bar{\epsilon}}(T)\,.$
(4.6)

Recall that (4.6) has been proved in the case (T). Suppose that (T) is not
satisfied, i.e. $\\{t^{j^{\prime}}_{i}\\}$ is not a sub-partition of
$\\{t^{j}_{i}\\}$. We shall show that a similar estimate is valid. The key
observation here is that, given the two partitions, we can construct a new
partition $\\{\tilde{t}_{i}\\}$ of $[S,T]$, simply by combining all the
discretization times of the two partitions. Write $\tilde{\mu}(\cdot)$ for the
measure

$\tilde{\mu}(t)\,=\,\sum_{i}\left[m(\tilde{t}_{i+1},\bar{x}(\tilde{t}_{i}))-m(\tilde{t}_{i},\bar{x}(\tilde{t}_{i}))\right]\delta(t-\tilde{t}_{i})\;.$

Applying the preceding analysis, first to $\mu^{j}(\cdot)$ and
$\tilde{\mu}(\cdot)$ and then to $\mu^{j^{\prime}}$ and $\tilde{\mu}$, and
noting the triangle inequality, we arrive at:

$\displaystyle|\langle\mu^{j}-\mu^{j^{\prime}},g(\cdot)\rangle|\,\leq\,|\langle\mu^{j}-\tilde{\mu},g(\cdot)\rangle|+|\langle\mu^{j^{\prime}}-\tilde{\mu},g(\cdot)\rangle|$
$\displaystyle\hskip 36.135pt\leq
2\left(\theta_{g}(\epsilon_{j})\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)+2\,(\theta_{\bar{x}}(\epsilon_{j})+\rho_{j})\,||g(.)||_{C}\,\tilde{\eta}^{\bar{\delta}}_{\bar{\epsilon}}(T)\right)\,.$

This relation implies

$\lim_{j\rightarrow\infty}\sup_{j^{\prime}\geq
j}\langle\mu^{j}-\mu^{j^{\prime}},g(\cdot)\rangle\,=\,0\,.$

We have shown that, for arbitrary continuous $g(\cdot)$,
$\\{\langle\mu^{j},g(\cdot)\rangle\\}$ is a Cauchy sequence in $\mathbb{R}$.
The sequence therefore has a limit.

In consequence of property (2.6) of functions having bounded variation, the
measures $\\{\mu^{j}(\cdot)\\}$ are bounded by
$\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)$, for $j$ sufficiently large. Since
closed balls in $C^{*}([S,T],\mathbb{R}^{r})$ are weak∗ compact there exists a
Borel measure $\mu(\cdot)$ on $[S,T]$ and a subsequence
$\\{\mu^{j_{k}}(\cdot)\\}$ of $\\{\mu^{j}\\}$ such that

$\mu^{j_{k}}(\cdot)\rightarrow\mu(\cdot)\quad\mbox{with respect to the
weak${}^{*}$ topology},$

as $k\rightarrow\infty$. But then, by the preceding analysis,

$\lim_{j\rightarrow\infty}\,\langle\mu^{j},g(\cdot)\rangle=\lim_{k\rightarrow\infty}\,\langle\mu^{j_{k}},g(\cdot)\rangle=\langle\mu,g(\cdot)\rangle\,,$

for any $g(.)\in C([S,T];\mathbb{R}^{r})$. We have demonstrated that there
exists a Borel measure $\mu(\cdot)$ such that $\mu^{j}(\cdot)$ converges to
$\mu(\cdot)$ in the manner claimed (weak∗ convergence in the dual space).

We now prove the final assertion of the lemma (‘uniqueness of the limit’). If
it were not true, there would exist two sequences of Borel measures
$\\{\mu^{j}(\cdot)\\}$ and $\\{\tilde{\mu}^{j}(\cdot)\\}$ on $[S,T]$ that
converge to different limits $\mu(\cdot)$ and $\tilde{\mu}(\cdot)$
(respectively), with respect to the weak∗ topology. The fact that the limits
are distinct means that there exists some $g^{*}(.)\in
C([S,T];\mathbb{R}^{r})$ such that

$\langle\mu,g^{*}(\cdot)\rangle\not=\langle\tilde{\mu},g^{*}(\cdot)\rangle\,.$
(4.7)

Now construct a new sequence $\\{\tilde{\tilde{\mu}}^{j}(\cdot)\\}$ by
alternating elements in the two sequences. By the preceding analysis, there
exists a Borel measure $\tilde{\tilde{\mu}}$ such that
$\tilde{\tilde{\mu^{j}}}\rightarrow\tilde{\tilde{\mu}}(\cdot)$ in the weak∗
topology, as $j\rightarrow\infty$. So

$\lim_{j\rightarrow\infty}\langle\tilde{\tilde{\mu}}^{j},g^{*}(\cdot)\rangle=\langle\tilde{\tilde{\mu}}(\cdot),g^{*}(\cdot)\rangle\,.$

But the sequence $\\{\langle\tilde{\tilde{\mu}}^{j},g^{*}(\cdot)\rangle\\}$
cannot converge, because there exist two subsequences, one with limit
$\langle\mu,g^{*}(\cdot)\rangle$ and the other with limit
$\langle\tilde{\mu},g^{*}(\cdot)\rangle$, which are distinct by (4.7). This
contradiction completes the proof. $\square$

######  Definition 4.2.

Take functions
$m(\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{r}$ and
$\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$. Assume hypothesis (BV1) and
(BV2) are satisfied. Then the partial variation measure of $m(\cdot,x)$ along
$\bar{x}(\cdot)$, written

$B\rightarrow\int_{B}d_{t}m(t,\bar{x}(t))\,,$

is the Borel measure on $[S,T]$:

$\mu(\cdot)\,=\,\lim_{j}\mu^{j}(\cdot)\,,$

in which the limit is taken with respect to the weak∗ topology on
$C^{*}([S,T];\mathbb{R}^{r})$. Here, $\\{\mu^{j}(.)\\}$ is any sequence of
discrete Borel measures, each of the form

$\mu^{j}\,=\,\sum_{i=0}^{N_{j}-1}\left[m(t^{j}_{i+1},\xi^{j}_{i})-m(t^{j}_{i},\xi^{j}_{i})\right]\delta(t-t^{j}_{i})\,,$

in which $\\{t^{j}_{i}\\}_{0=1}^{N_{j}}$, $j=1,2,\ldots$, is a sequence of
partitions of $[S,T]$ such that diam$(\\{t^{j}_{i}\\})\rightarrow 0$ as
$j\rightarrow\infty$. $\\{\xi^{j}_{i}\\}_{i=0}^{N_{j}-1}$ is a collection of
$n$-vectors such that $\xi^{j}_{i}=\bar{x}(t)$ for some
$t\in[t^{j}_{i},t^{j}_{i+1}]$. (The definition of
$B\rightarrow\int_{B}d_{t}m(t,\bar{x}(t))$ is unambiguous since, according the
preceding analysis, the limiting measure $\mu(\cdot)$ is the same for all
choices of sequences $\\{\mu^{j}(\cdot)\\}$.)

The next proposition relates the value of the partial variation measure on a
subinterval $[a,b]\subset[S,T]$ and the difference in values of $m(\cdot,x)$
at $a$ and $b$.

######  Proposition 4.3.

Take functions
$m(\cdot,\cdot):[S,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{r}$ and
$\bar{x}(\cdot):[S,T]\rightarrow\mathbb{R}^{n}$. Assume hypothesis (BV1) and
(BV2) are satisfied, for some $\delta^{\prime}>0$. Denote by $\eta(\cdot)$ and
$\tilde{\eta}(\cdot)$ the cummulative variation functions of $m(\cdot,x)$ and
$\nabla_{x}m(\cdot,x)$, respectively.

Take a closed subinterval $[a,b]\subset[S,T]$, $\delta\in(0,\delta^{\prime})$
such that $\eta^{\delta}(T)<\infty$ and $\tilde{\eta}^{\delta}(T)<\infty$ and
$\xi\in\mathbb{R}^{n}$ such that $\xi=\bar{x}(t)$ for some $t\in[a,b]$. Assume
that

$\theta_{\bar{x}}(|b-a|)\leq\delta\,,$ (4.8)

where $\theta_{\bar{x}}(.)$ is a continuity modulus for $\bar{x}(.)$. Then

$\displaystyle\left|\int_{[a,b]}d_{t}m(t,\bar{x}(t))-\left(m(b,\xi)-m(a,\xi)\right)\right|\,\leq\,$
$\displaystyle\hskip
28.90755pt\theta_{\bar{x}}(|b-a|)\times\left(\tilde{\eta}^{\delta}(b)-\tilde{\eta}^{\delta}(a)\right)+\left(\eta(a)-\lim_{a^{\prime}\uparrow
a}\eta(a^{\prime})\right)+\left(\lim_{b^{\prime}\downarrow
b}\eta(b^{\prime})-\eta(b)\right)\,.$

(The second term on the right is interpreted as $0$ if $a=S$, and the third as
$0$ if $b=T$.)

Proof. Fix $[a,b]\subset[S,T]$ and $\delta>0$ such that
$\eta^{\delta}(T)<\infty$ and $\tilde{\eta}^{\delta}(T)<\infty$ and (4.8) is
satisfied.

Let $\\{G_{k}(.):[S,T]\rightarrow\mathbb{R}^{r\times r}\\}$ be a sequence
continuous functions such that

$\displaystyle G_{k}(t)=I_{r\times r}\mbox{ for }t\in[a,b]$
$\displaystyle|G_{k}(t)|=0\mbox{ if }t\leq a-k^{-1}\mbox{ or }b+k^{-1}\leq t$
$\displaystyle|G_{k}(t)|\leq 1\mbox{ if }a-k^{-1}\leq t\leq a\mbox{ or }b\leq
t\leq b+k^{-1}\,.$

Take any index value $k$ and $\epsilon>0$ sufficiently small that
$\eta^{\delta}_{\epsilon}(T)<\infty$ and
$\tilde{\eta}^{\delta}_{\epsilon}(T)<\infty$. Let
$\\{t^{j}_{i}\\}_{i=0}^{N_{j}}$ be a sequence of partitions of $[S,T]$ such
that diam$(\\{t^{j}_{i}\\})\rightarrow 0$ as $j\rightarrow 0$ and such that
$\\{t^{j}_{i}\\}$ contains $a$ and $b$ for each $j$. Now define

$\mu^{j}(t)\,=\,\sum_{i=0}^{N_{j}-1}\left[m(t^{j}_{i+1},\bar{x}(t^{j}_{i}))-m(t^{j}_{i},\bar{x}(t^{j}_{i}))\right]\delta(t-t^{j}_{i})\;.$

By Lemma 4.1, applied component-wise,

$\int_{[S,T]}G_{k}(t)\,d\mu^{j}(t)\rightarrow\int_{[S,T]}G_{k}(t)\,d_{t}m(t,\bar{x}(t))\mbox{
as $j\rightarrow\infty$ }\,.$ (4.9)

For each $j$, let $m^{j}_{1}$ and $m^{j}_{2}$ be the values of the index $i$
defined by $t^{j}_{m^{j}_{1}}=a$ and $t^{m^{j}_{2}}_{i}=b$. Then, for each
$j$,

$\int_{[S,T]}G_{k}(t)\,d\mu^{j}(t)-(m(b,\xi)-m(a,\xi))\,=\,\Sigma_{1}+\Sigma_{2}+\Sigma_{3}\,,$
(4.10)

in which

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{m^{j}_{1}-1}G_{k}(t_{i})\left[m(t^{j}_{i+1},\bar{x}(t^{j}_{i}))-m(t^{j}_{i},\bar{x}(t^{j}_{i}))\right]\,,$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=m^{j}_{1}}^{m^{j}_{2}-1}\left[m(t^{j}_{i+1},\bar{x}(t^{j}_{i}))-m(t^{j}_{i},\bar{x}(t^{j}_{i}))\right]-\left[m(b,\xi)-m(a,\xi)\right]\,,$
$\displaystyle\Sigma_{3}$ $\displaystyle=$
$\displaystyle\sum_{i=m^{j}_{2}}^{N_{j}-1}G_{k}(t_{i})\left[m(t^{j}_{i+1},\bar{x}(t^{j}_{i}))-m(t^{j}_{i},\bar{x}(t^{j}_{i}))\right]\,.$

Consider the term $\Sigma_{2}$. Take any $\nu\in R^{r}$. Then, using the exact
first order Taylor expansion formula, we can show, for $j$ sufficiently large,

$\displaystyle\nu^{T}\sum_{i=m^{j}_{1}}^{m^{j}_{2}-1}\left[m(t^{j}_{i+1},\bar{x}(t^{j}_{i}))-m(t^{j}_{i},\bar{x}(t^{j}_{i}))\right]$
$\displaystyle\qquad\,=\,\nu^{T}\sum_{i=m^{j}_{1}}^{m^{j}_{2}-1}\left[m(t^{j}_{i+1},\xi)-m(t^{j}_{i},\xi)+\nabla_{x}m(t^{j}_{i+1},\xi^{\nu
j}_{i})-\nabla_{x}m(t^{j}_{i},\xi^{\nu
j}_{i})\right]\cdot\left(\bar{x}(t^{j}_{i})-\xi\right)$
$\displaystyle\qquad\,\leq\,\nu^{T}\left(\left[m(b,\xi)+0\ldots
0-m(a,\xi)\right]\right.$ $\displaystyle\hskip
65.04256pt+\;\theta_{\bar{x}}(|b-a|)\times\sum_{i=m^{j}_{1}}^{m^{j}_{2}-1}\left[\nabla_{x}m(t^{j}_{i+1},\xi^{\nu_{j}}_{i})-\nabla_{x}m(t^{j}_{i},\xi^{\nu
j}_{i})\right])\,,$

in which $\xi^{\nu j}_{i}\in\bar{x}(a)+\delta\mathbb{B}$, for each $i$ and all
$j$ sufficiently large. (We have used the fact that $G_{k}(t)\equiv I_{r\times
r}$ for $t\in[a,b]$.) But $\nabla_{x}m(\cdot,x)$ has bounded variation along
$\bar{x}(\cdot)$. We can therefore conclude that, for $j$ sufficiently large,

$|\nu^{T}\Sigma_{2}|\,\leq\,|\nu|\left(\tilde{\eta}^{\delta}_{\epsilon}(b)-\tilde{\eta}^{\delta}_{\epsilon}(a)\right)\times\left(\theta_{\bar{x}}(|b-a|)\right)\,.$

Since $\nu$ is an arbitrary $r$-vector,

$|\Sigma_{2}|\,\leq\,(\tilde{\eta}^{\delta}_{\epsilon}(b)-\tilde{\eta}^{\delta}_{\epsilon}(a))\;\theta_{\bar{x}}(|b-a|)\,.$
(4.11)

Now take any $\delta^{\prime}\in(0,\delta)$. Since $G_{k}(.)$ satisfies
$|G_{k}(.)|\leq 1$ on $[S,T]\backslash[a,b]$ and vanishes on
$[S,T]\backslash[a-k^{-1},b+k^{-1}]$, we deduce from property (2.6) of
cummulative variation functions that, for sufficiently large $j$,

$|\Sigma_{1}|\leq\eta^{\delta^{\prime}}_{\epsilon}(a)-\eta^{\delta^{\prime}}_{\epsilon}(S\vee(a-k^{-1}))\mbox{
and
}|\Sigma_{3}|\leq\eta^{\delta^{\prime}}_{\epsilon}(T\vee(b+k^{-1}))-\eta^{\delta^{\prime}}_{\epsilon}(b)\,.$
(4.12)

Noting (4.9), (4.10), (4.11) and (4.12) and passing to the limit as
$j\rightarrow\infty$ gives

$\displaystyle|\int_{[S,T]}G_{k}(t)\,d\mu(t)-(m(b,\xi)-m(a,\xi))|\,\leq\,\left(\tilde{\eta}_{\epsilon}^{\delta}(b)-\tilde{\eta}_{\epsilon}^{\delta}(a)\right)\times\theta_{\bar{x}}(|b-a|)$
$\displaystyle\hskip
28.90755pt+(\eta^{\delta^{\prime}}_{\epsilon}(a)-\eta^{\delta^{\prime}}_{\epsilon}(S\vee(a-1/k))+(\eta^{\delta^{\prime}}_{\epsilon}(T\vee(b+1/k))-\eta^{\delta^{\prime}}_{\epsilon}(b))$
(4.13)

But $\delta^{\prime}>0$ and $\epsilon>0$ are arbitrary, sufficiently small
numbers. We may therefore pass to the limit as first $\epsilon\downarrow 0$
and second as $\delta^{\prime}\downarrow 0$, to deduce the validity of the
preceding relation when $\eta^{\delta}_{\epsilon}(\cdot)$ and
$\eta^{\delta^{\prime}}_{\epsilon}(\cdot)$ are replaced by
$\eta^{\delta}(\cdot)$ and $\eta(\cdot)$, respectively.

So far $k$ has been fixed. Finally, we pass to the limit as
$k\rightarrow\infty$. Since $G_{k}(t)\rightarrow I_{r\times
r}\times\chi_{[a,b]}$ everywhere and the monotone function $\eta(.)$ has
everywhere one-sided limits, we deduce with the help of the Dominated
Convergence Theorem that

$\displaystyle|\int_{[a,b]}d_{t}m(t,\bar{x}(t))-(m(b,\xi)-m(a,\xi))|\,\leq\,$
$\displaystyle\hskip
21.68121pt\theta_{\bar{x}}(|b-a|)\times\left(\tilde{\eta}^{\delta}(b)-\tilde{\eta}^{\delta}(a)\right)+\left(\eta(a)-\lim_{a^{\prime}\uparrow
a}\eta(a^{\prime})\right)+\left(\lim_{b^{\prime}\downarrow
b^{\prime}}\eta(b)-\eta(b)\right)\,$

in which $\lim_{a^{\prime}\uparrow a}\eta(a^{\prime}):=\eta(S)$ if $a=S$ and
$\lim_{b^{\prime}\downarrow b}\eta(b^{\prime}):=\eta(T)$ if $b=T$. The proof
is complete. $\square$

##  5 An Application

Consider a control system relating the control function $u(\cdot)$ to an
output function $y(t)$ according to

$(S)\quad\left\\{\begin{array}[]{l}\dot{x}(t)=f(x(t),u(t))\mbox{ a.e
}t\in[S,T]\,,\\\ u(t)\in\Omega\mbox{ a.e }t\in[S,T]\,,\\\ x(0)=x_{0}\,,\\\
y(t)=g(x(t))\mbox{ for }t\in[S,T]\,,\end{array}\right.$

the data for which is: functions
$f(\cdot,\cdot):\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
and $g(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$, a set
$\Omega\subset\mathbb{R}^{m}$ and an $n$-vector $x_{0}$.

Let $\bar{u}(\cdot)$ be a control function that has been chosen to give a
desired value to the output at time $T$, which we write

$J(u(\cdot))\,:=\,g(x(T;u(\cdot),x_{0}))$ (5.1)

where $t\rightarrow x(t;u(\cdot),x_{0})$ denotes the solution to the
differential equation in the control system description, for a given control
function $u(\cdot)$ and initial condition $x_{0}$. (Hypotheses will be imposed
ensuring the existence and uniqueness solutions.) Write
$\bar{x}(t)=x(t;\bar{u}(\cdot),x_{0})$.

In this section we focus our attention on the following phenomenon: in control
engineering it is often the case that a feedback control cannot be implemented
perfectly, but only with a time delay. This is especially evident in process
control, where controlled chemical reactors are routinely modelled with a pure
delay at the input, to take account of the finite rate of flow of fluids
between reactors, etc. (See, for example, the widely studied Tennessee Eastman
challenge controller design problem, in which the system equations take the
form of a matrix of first order lags with pure time delay [9]). The presence
of a time delay complicates the controller design and so, if it is small, it
is often ignored. To justify the use of idealized ‘delay-free’ models, it then
becomes necessary to carry out a sensitivity analysis, to quantify the errors
in the output $J(u(\cdot))$ when small delays are introduced into the
controller implementation. We need then to look at consequences of applying
the control

$u^{h}(t):=\left\\{\begin{array}[]{ll}\bar{u}(S)&\mbox{if }t-h<S\\\
\bar{u}(t-h)&\mbox{if }S\leq t-h\leq T\\\ \bar{u}(T)&\mbox{if
}T<t-h\,.\end{array}\right.$ (5.2)

Notice we allow $h$ to be both positive (a delay) or negative (an advance).
The effect of introducing the delay on the output at time $T$ is quantified by

$J(u^{h}(.)):=g(x(T;u^{h}(\cdot),x_{0}))\,.$

Suppose that $f(\cdot,\cdot)$ is continously differentiable and globally
Lipschitz continuous. If the control $\bar{u}(\cdot)$ is an absolutely
continuous function, a routine analysis yields the information that
$h\rightarrow J(u^{h}(\cdot))$ is differentiable at the origin with gradient

$\frac{d}{dh}J(u^{h}(\cdot))|_{h=0}\,=\,\int_{[S,T]}p^{T}(t)\nabla_{u}f(\bar{x}(t),\bar{u}(t))\,\frac{d\bar{u}}{dt}(t)dt\,,$
(5.3)

in which $p(.):[S,T]\rightarrow\mathbb{R}^{n}$ is the solution to the costate
equation:

$\left\\{\begin{array}[]{l}-\dot{p}(t)=\nabla_{x}f^{T}(\bar{x}(t),\bar{u}(t))\,p(t)\\\
p(T)=\nabla_{x}g^{T}(\bar{x}(T))\,.\end{array}\right.$ (5.4)

It is sometimes required to consider controls $\bar{u}(\cdot)$ that are not
absolutely continuous (‘bang-bang’ controls arising from the solution to
minimum time problems, for example). Is it possible to establish regularity
properties of $h\rightarrow J(u^{h}(\cdot))$ and to derive a formula akin to
(5.3) for a larger class of controls $\bar{u}(\cdot)$, and when $f(x,u)$ is no
longer assumed to be differentiable w.r.t. the $u$ variable? The Prop. 5.1
below provides a positive answer, when $\bar{u}(\cdot)$ is a function of
bounded variation.

We shall invoke the following hypotheses: there exists $k_{1}>0$, $\delta>0$
and a modulus of continuity $\theta(.)$ such that

  * (S1):

$g(\cdot)$ is a $C^{1}$ function,

  * (S2):

$f(\cdot,\cdot)$ is continuous, $f(\cdot,u)$ is a $C^{1}$ function for each
$u\in\Omega$ and

    * (i):

$|f(x,u)|\,\leq\,c[1+|x|]$ for all $x\in\mathbb{R}^{n}$, $u\in\Omega$

    * (ii):

$|\nabla_{x}f(x,u)|\leq K$ for all $x\in\mathbb{R}^{n}$ and $u\in\Omega$,

    * (iii):

$|f(x,u)-f(x^{\prime},u)-\nabla_{x}f(x^{\prime},u)|\,\leq\,\theta(|x-x^{\prime}|)\times|x-x^{\prime}|$

for all $x,x^{\prime}\in\bar{x}(t)+\delta\mathbb{B}$ and $u\in\Omega$,

  * (S3):

$|f(x,u)-f(x,u^{\prime})|+|\nabla_{x}f(x,u)-\nabla_{x}f(x,u^{\prime})|\leq
k_{1}|u-u^{\prime}|$

for all $x\in\bar{x}(t)+\delta\mathbb{B}$, $u,u^{\prime}\in\Omega$ and
$t\in[S,T]\,.$

  * (BV):

$\bar{u}(\cdot)$ has bounded variation.

######  Proposition 5.1.

Consider the control system (S) and a control function $\bar{u}(\cdot)$.
Assume that hypotheses (S1)-(S3) and (BV) are satisfied.

For any number $h\in\mathbb{R}$ define
$u^{h}(\cdot):[S,T]\rightarrow\mathbb{R}^{m}$ according to (5.2). Write
$x^{h}(\cdot)$ for the solution on $[S,T]$ of $\dot{x}(t)=f(x(t),u^{h}(t))$,
$x(S)=x_{0}$ and also

$m^{h}(t,x):=f(x,u^{h}(t))\mbox{ and
}\nabla_{x}m^{h}(t,x):=\nabla_{x}f(x,u^{h}(t))\,.$ (5.5)

Then, for all $h$ in some neighborhood of $0$:

  * (a):

$m^{h}(\cdot,x)$ and $\nabla_{x}m^{h}(\cdot,x)$ have bounded variation along
$\bar{x}(\cdot)$,

  * (b):

$h^{\prime}\rightarrow J(u^{h^{\prime}}(\cdot))$ (given by (5.1)) has one
sided derivatives (from left and right) at $h^{\prime}=h$:

$\lim_{h^{\prime}\downarrow
h}\,\frac{J(u^{h^{\prime}}(.))-J(u^{h}(.))}{{h^{\prime}-h}}=-\int_{[S,T)}p_{h}^{T}(t)\,d_{t}m^{h}(t,\bar{x}(t))$
(5.6)

and

$\lim_{h^{\prime}\uparrow
h}\,\frac{J(u^{h}(.))-J(u^{h}(.))}{{h-h^{\prime}}}=-\int_{(S,T]}p^{T}_{h}(t)\,d_{t}m^{h}(t,\bar{x}(t))\,.$
(5.7)

(In these relations, $p_{h}(\cdot)$ is the solution to (5.4) when
$u^{h}(\cdot)$ and $x^{h}(\cdot)$ replace $\bar{u}(\cdot)$ and
$\bar{x}(\cdot.)$, and $B\rightarrow\int_{B}d_{t}m^{h}(t,\bar{x}(t))$ is the
partial variation measure associated with $m^{h}(\cdot,\cdot)$.)

  * (c):

If $\bar{u}(\cdot)$ is continuous at both endpoints $S$ and $T$, the mapping
$h^{\prime}\rightarrow J(u^{h^{\prime}}(\cdot))$ is differentiable at $h$ and
its derivative is

$\lim_{h^{\prime}\rightarrow
h}\,\frac{J(u^{h^{\prime}}(\cdot))-J(u^{h}(\cdot))}{{h^{\prime}-h}}=-\int_{[S,T]}p_{h}^{T}(t)\,d_{t}m^{h}(t,\bar{x}(t))\,.$
(5.8)

Discussion: The property that the sensitivity function $h\rightarrow J(u^{h})$
is differentiable when $\bar{u}(\cdot)$ has bounded variation and continuous
at the two end-times (part (c) of the proposition) is highly non-trivial,
since $f(x,u)$ is not assumed to be differentiable w.r.t. $u$. To convey the
nature of this property in its simplest terms, let us consider the case of
control system (S) when $f(x,u)$ is independent of $x$ (write the function
$f(u)$). Assume that

  * (a):

$f(\cdot)$ is Lipschitz continuous,

  * (b):

$\bar{u}(\cdot)$ is continuously differentiable.

It is straightforward to show that, under these hypotheses, the sensitivity
function $V(h):=J(u^{h}(\cdot))$ is Lipschitz continuous. A standard analysis
based on perturbing $h$ and using the properties of Clarke’s generalized
directional derivative (c.f. [6, Proof of Thm. 2.7.3]) and a nonsmooth chain
rule permits one to derive the following estimate of the subdifferential of
$V(.)$ at $0$:

$\partial
V(0)\,\subset\,-\int_{S}^{T}\,\mbox{co}\,\partial_{u}H(p(t),\bar{u}(t))\dot{\bar{u}}(t)dt\,.$
(5.9)

in which

$H(p,u):=p^{T}f(u)\,.$

Here $\partial_{u}H(u,p)$ denotes the subdifferential w.r.t. the $u$ variable,
for fixed $p$. (We refer to the end of Section 1 for definition the
subdifferential.) $p(\cdot)$ is the solution of the adjoint system (5.4). The
right side of this relation is a set valued integral, defined in the usual way
as the collection of integrals of selectors of the set valued integrand.

Prop. 5.1 tells us, contrary to what the standard analysis leading to the
formula (5.9) might lead us to expect, the sensitivity function is actually
differentiable on a neighborhood of $0$. Indeed it tells us even more: the
sensitivity function is differentiable on a neighborhood of $0$, when $f(t,x)$
is $x$-dependent and $u(.)$ is merely a function of bounded variation
(continuous at its endpoints). This surprising regularity property, is a
consequence of the properties of functions of bounded variation, along a
specified trajectory, established in earlier sections of this paper.

Proof of Prop 5.1. For any $h$ we have:

$\displaystyle\int_{[S,T]}|f(\bar{x}(t),u^{h}(t))-f(\bar{x}(t),\bar{u}(t))|dt$
$\displaystyle\hskip 14.45377pt\leq k_{1}\left(\int_{[(S+h)\wedge
T,T]}|u(t)-u(t-h)|+\int_{[S,(S+h)\wedge T]}|u(t)-u((S))|\right)$
$\displaystyle\hskip 14.45377pt=k_{1}\left(\int_{[(S+h)\wedge
T,T]}\eta_{\bar{u}}(t)dt+\int_{[S,(S+h)\wedge
T]}\eta_{\bar{u}}(t)dt\right)\leq 2k_{1}\;\eta_{\bar{u}}(T)\;h\,.$

in which $\eta_{\bar{u}}(\cdot)$ is the cummulative variation function of
$u(\cdot)$. By Filippov’s Existence Theorem (see e.g. [13, Thm. 2.4.3]), there
exist a number $K_{1}$, independent of $h$, such that

$||x^{h}(.)-\bar{x}(.)||_{L^{\infty}}\leq K_{1}\;h\,.$ (5.10)

Under hypothesis (S2) there exists a unique solution to the differential
equation $x^{h}(.)$ for each $h$. It can be deduced from (S3) that there
exists $\bar{h}>0$ such that $m^{h}(\cdot,x)$ and $\nabla_{x}m^{h}(\cdot,x)$
have bounded variation along $x^{h}(\cdot)$, for all $h\in[-\bar{h},\bar{h}]$.
Write the cummulative variation functions $\eta^{h}(\cdot)$ and
$\tilde{\eta}^{h}(\cdot)$, and their $\delta$-perturbed versions
$\eta^{h\delta}(\cdot)$ and $\eta^{h\delta}(\cdot)$ respectively.

We now examine the one-sided differentiability properties of
$h^{\prime}\rightarrow J(u^{h^{\prime}})$ at $h=0$ and derive the given
formula for the one-sided derivative from the left. (Analogous arguments can
be used to treat the other cases.)

Take an arbitrary sequence $h_{i}\downarrow 0$. Then, for each $i$,

$\displaystyle J(u^{h_{i}}(\cdot))-J(\bar{u}(\cdot))$ $\displaystyle\hskip
21.68121pt=\;g(x^{h_{i}}(T))-g(\bar{x}(T))-\int_{[S,T]}p^{T}(t)\left[(\dot{x}^{h_{i}}(t)-\dot{\bar{x}}(t))\right.$
$\displaystyle\hskip
130.08621pt\left.-\left(f(x^{h_{i}}(t),u^{h_{i}}(t))-f(\bar{x}(t),\bar{u}(t)\right)\right]dt\,.$

A routine analysis, in which we make use of the costate equation and right
boundary condition (5.4) on $p(\cdot)$, apply integration by parts to the
integral $\int_{[S,T]}p^{T}(t)(\dot{x}^{h_{i}}(t)-\dot{\bar{x}}(t))dt$ and
consider first order Taylor expansions of $g(\cdot)$ about $\bar{x}(T)$ and of
$x\rightarrow f(x,u^{h_{i}}(t))$ about $\bar{x}(t)$, reveals that

$\displaystyle h_{i}^{-1}\left(J(u^{h_{i}}(\cdot))-J(\bar{u}(\cdot))\right)$
(5.11) $\displaystyle\hskip
21.68121pt=\;h_{i}^{-1}\int_{[S,T]}p^{T}(t)\left[f(\bar{x}(t),u^{h_{i}}(t))-f(\bar{x}(t),\bar{u}(t)\right]dt+e(h_{i})$
$\displaystyle\hskip
21.68121pt=\;h_{i}^{-1}\int_{[S,T]}p^{T}(t)\left[m^{0}((t-h_{i})\vee
S),\bar{x}(t))-m^{0}(t,\bar{x}(t))\right]dt+e(h_{i})\,,$

in which the ‘error term’ $e(h_{i})$ satisfies
$|e(h_{i})|\leq||p(.)||_{L^{\infty}}\;(\theta(K_{1}h_{i})\;K_{1}+\theta_{g}(K_{1}h)K_{1})$.
($\theta(.)$ is the continuity modulus of (S2) and $\theta_{g}(.)$ is a
‘second order’ continuity modulus for $g(\cdot)$, i.e. a function such that
$|g(x)-g(x^{\prime})-\nabla
g(x^{\prime})|\leq\theta_{g}(|x-x^{\prime}|)|x-x^{\prime}|$ for all
$x,x^{\prime}\in\bar{x}(T)+\delta\mathbb{B}$.) We see

$e(h_{i})\rightarrow 0\mbox{ as }i\rightarrow\infty\,.$ (5.12)

Take any $\delta>0$ such that $\tilde{\eta}^{\delta}(T)<\infty$. Since
$\eta^{\delta}(.)$ is a monotone function, there exists a subset ${\cal
I}\subset(S,T)$, of full Lebesgue measure, on which $\eta^{\delta}(.)$ is
continuous. Prop. 4.3 tells us that, for each $t\in{\cal I}$ and $i$
sufficiently large,

$\displaystyle m^{0}((t-h_{i})\vee S),\bar{x}(.))-m(t,\bar{x}(t))$
$\displaystyle\qquad=\;-\int_{[(t-h_{i})\vee
S,t]}d_{s}m^{0}(s,\bar{x}(s))+e_{1}(t,h_{i})$ (5.13)

in which the ‘error term’ $e_{1}(t,h_{i})$ satisfies

$\displaystyle|e_{1}(t,h_{i})|$ $\displaystyle\leq$
$\displaystyle\theta_{\bar{x}}(|h_{i}|)\left(\tilde{\eta}^{\delta}(t)-\tilde{\eta}^{\delta}((t-h_{i})\vee
S)\right)\,.$ (5.14)

It follows that

$\displaystyle h_{i}^{-1}\int_{[S,T]}|p^{T}(t)e_{2}(t,h_{i})|dt$
$\displaystyle\quad\leq\,h_{i}^{-1}\theta_{\bar{x}}(|h_{i}|)\;||p(.)||_{L^{\infty}}\;\int_{[S,T]}\left(\tilde{\eta}^{\delta}(t)-\tilde{\eta}^{\delta}((t-h_{i})\vee
S)\right)dt$
$\displaystyle\quad=\;\theta_{\bar{x}}(|h_{i}|)\;||p(.)||_{L^{\infty}}\;\left(h^{-1}_{i}\int_{[T-h_{i},T]}\tilde{\eta}^{\delta}(t)dt+\tilde{\eta}(S)\right)\;\rightarrow\;0\,,$
(5.15)

as $i\rightarrow\infty$. In consequence of Fubini’s Theorem

$\displaystyle h^{-1}_{i}\int_{[S,T]}\int_{[(t-h_{i})\vee
S,t]}p^{T}(t)d_{s}m^{0}(s,\bar{x}(s))dt\,=\,\int_{[S,T]}p_{i}^{T}(s)d_{s}m^{0}(s,\bar{x}(s))ds\,,$

in which

$p_{i}(s):=h_{i}^{-1}\int_{[s,(s+h_{i})\wedge T]}p(t)dt\quad\mbox{for
}s\in[S,T]\,.$

Since $p(.)$ is continuous, $p_{i}(t)\rightarrow\tilde{p}(t)$ for all
$t\in[S,T]$, where

$\tilde{p}(t)\,\rightarrow\,\left\\{\begin{array}[]{ll}p(t)&\mbox{if
}t\in[S,T)\\\ 0&\mbox{if }t=T\;.\end{array}\right.$

By the Dominated Convergence Theorem

$\displaystyle h^{-1}_{i}\int_{[S,T]}\int_{[(t-h_{i})\vee
S,t]}p^{T}(t)d_{s}m^{0}(s,\bar{x}(s))dt$ (5.16) $\displaystyle\hskip
36.135pt\rightarrow$
$\displaystyle\,\int_{[S,T]}\tilde{p}^{T}(s)d_{s}m^{0}(s,\bar{x}(s))ds$
$\displaystyle\hskip 36.135pt=$
$\displaystyle\,\int_{[S,T)}p^{T}(s)d_{s}m^{0}(s,\bar{x}(s))ds\,.$

Combining relations (5.11) - (5.15)), we arrive at

$h_{i}^{-1}\,\left(J(u^{h_{i}}(\cdot))-J(\bar{u}(\cdot))\right)\,\rightarrow\,-\int_{[S,T)}p^{T}(s)d_{s}m^{0}(s,\bar{x}(s))ds\,.$

We have confirmed formula (5.6) and the existence of the limit.

We now attend to the final assertion of the proposition. Suppose then that
$\bar{u}(\cdot)$ is continuous at $S$ and $T$. For $h$ sufficiently small,
$m^{h}(.,x)$ is continuous at $t=S$ and $t=T$, uniformly as $x$ ranges over
neighborhoods of $x^{h}(S)$ and $x^{h}(T)$ . It can be deduced from relation
(3.9) in Prop. 3.3 that $\eta(.)$ is right and left continuous at $S$ and $T$.
Then, by Prop. 4.3, $d_{t}m^{h}(\cdot,x^{h}(t))$ has no atom at either $S$ or
$T$. The differentiability of $h^{\prime}\rightarrow J(u^{h^{\prime}}(\cdot))$
and the formula (5.8) now follow from (5.6) and (5.7), since the integrals in
the latter two formulae, over $[S,T)$ and $(S,T]$ respectively, are the same.
$\square$

Appendix: Proof of Prop. 3.3

Take $\bar{\epsilon}>0$ such that
$\eta^{\bar{\delta}}_{\bar{\epsilon}}(T)<\infty$ and
$\delta\in(0,\bar{\delta})$. To begin, we verify the following assertion: for
any $\delta^{\prime}\in(0,\delta)$

$\eta^{\delta^{\prime}}(T)\,\leq\,\underset{T^{\prime}\uparrow
T}{\lim}\,\eta^{\delta}(T^{\prime})+\underset{x\in\bar{x}(T)+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a)),$ (5.17)

Take any $\delta^{\prime\prime}\in(\delta^{\prime},\delta)$. Since
$\bar{x}(.)$ is continuous, we can choose $T_{1}\in(S,T)$ such that

$\displaystyle\bar{x}([T_{1},T])+\delta^{\prime}\mathbb{B}$
$\displaystyle\subset$
$\displaystyle\bar{x}(T)+\delta^{\prime\prime}\mathbb{B},$ (5.18)
$\displaystyle\bar{x}([T_{1},T])+\delta^{\prime\prime}\mathbb{B}$
$\displaystyle\subset$ $\displaystyle\bar{x}(T)+\delta\mathbb{B}$ (5.19)

For any $T_{2}\in(T_{1},T]$ define

$G_{\delta^{\prime\prime}}(T_{1},T_{2})\,:=\,\sup\,\left\\{\sum_{i=0}^{N}\underset{x\in\bar{x}(T)+\delta^{\prime\prime}\mathbb{B},a\in
A}{\sup}d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\right\\}\,,$

in which the ‘outer’ supremum is taken over all partitions $\\{t_{i}\\}$ of
$[T_{1},T_{2}]$. Notice that the ‘inner’ suprema are all taken over the same
set $(\bar{x}(T)+\delta\mathbb{B})\times A$. It follows that the value of
$G_{\delta^{\prime\prime}}(T_{1},T_{2})$ is unchanged if we restrict the
magnitude of the diameters of the partitions considered in the definition;
that is, for any $\epsilon>0$ we have

$G_{\delta^{\prime\prime}}(T_{1},T_{2})\,=\,\sup\,\left\\{\sum_{i=0}^{N}\underset{x\in\bar{x}(T)+\delta^{\prime\prime}\mathbb{B},a\in
A}{\sup}d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\,|\,\mbox{
diam}\,(\\{t_{i}\\})\,\leq\epsilon\right\\}\,.$ (5.20)

By considering the modification of arbitrary partitions of $[S,T]$ to include
the extra ‘grid point’ $T_{1}$ and taking account of (5.18) and (3.5), we see
that, for any $\epsilon\in(0,\bar{\epsilon}]$,

$\eta_{\epsilon}^{\delta^{\prime}}(T)\,\leq\,\eta_{\epsilon}^{\delta^{\prime}}(T_{1})+2\,\gamma(c\,\epsilon)+G_{\delta^{\prime\prime}}(T_{1},T)\,.$

Here, $c$ and $\gamma(.)$ are as in hypotheses (C1) and (C2). In the limit as
$\epsilon\downarrow 0$, we obtain

$\eta^{\delta^{\prime}}(T)\,\leq\,\eta^{\delta^{\prime}}(T_{1})+G_{\delta^{\prime\prime}}(T_{1},T)\,.$
(5.21)

Take any $\rho>0$. Then there exists a partition
$\\{t^{\rho}_{0},\ldots,t^{\rho}_{N^{\rho}}\\}$ of $[T_{1},T]$ achieving the
‘outer’ supremum defining $G_{\delta^{\prime\prime}}(T_{1},T)$, with error at
most $\rho$. From (5.19), we deduce

$G_{\delta^{\prime\prime}}(T_{1},T)\,\leq\,G_{\delta^{\prime\prime}}(T_{1},t^{\rho}_{N^{\rho}-1})+\underset{x\in\bar{x}(T)+\delta^{\prime\prime}\mathbb{B},\,a\in
A}{\sup}d_{H}(F(t^{\rho}_{N^{\rho}-1},x,a),F(T,x,a))+\rho\,.$ (5.22)

Because $\\{t^{\rho}_{i}\\}$ is a partition, $t^{\rho}_{N^{\rho}-1}<T$. In
view of (5.20) which, we recall, is valid for any $\epsilon>0$, we can arrange
that

$|T-t^{\rho}_{N^{\rho}-1}|\leq\rho\,.$ (5.23)

But, in consequence of (5.19) and (5.20), we know that, for any
$\epsilon\in(0,\bar{\epsilon}]$,

$\eta_{\epsilon}^{\delta}(t^{\rho}_{N^{\rho}-1})\,\geq\,\eta_{\epsilon}^{\delta}(T_{1})+G_{\delta^{\prime\prime}}(T_{1},t^{\rho}_{N^{\rho}-1})\,.$

In the limit, as $\epsilon\downarrow 0$, we obtain

$\eta^{\delta}(t^{\rho}_{N^{\rho}-1})\,\geq\,\eta^{\delta}(T_{1})+G_{\delta^{\prime\prime}}(T_{1},t^{\rho}_{N^{\rho}-1})\,.$
(5.24)

Combining (5.21),(5.22) and (5.24) and noting that
$\eta^{\delta^{\prime}}(T_{1})\leq\eta^{\delta}(T_{1})$, we see that

$\displaystyle\eta^{\delta^{\prime}}(T)$ $\displaystyle\leq$
$\displaystyle\eta^{\delta}(t^{\rho}_{N^{\rho}-1})+\underset{x\in\bar{x}(T)+\delta^{\prime\prime}\mathbb{B},\,a\in
A}{\sup}d_{H}(F(t^{\rho}_{N^{\rho}-1},x,a),F(T,x,a))+\rho\,.$

This relation is valid for any $\rho>0$. Passing to the limit as
$\rho\downarrow 0$, while taking account of (5.23) and using Prop. 3.1 (part
(b)) to evaluate the limit of the sup term on the right side, and noting that
$\delta^{\prime\prime}<\delta$, we deduce

$\eta^{\delta^{\prime}}(T)\,\leq\,\lim_{T^{\prime}\uparrow
T}\eta^{\delta}(T^{\prime})+\underset{x\in\bar{x}(T)+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T^{-},x,a),F(T,x,a))\,.$

This confirms relation (5.17).

Our next task will be to relate the $\delta$-perturbed cummulative variation
functions $\eta^{\delta}(\cdot)$ and $\tilde{\eta}^{\delta}(\cdot)$ of
$F(\cdot,x,a)$ and $\tilde{F}(\cdot,x,a)$, respectively, at times $t\in(S,T)$.

Fix $t\in(S,T)$ and take $\epsilon\in(0,\bar{\epsilon}]$. Let ${\cal
T}=\\{t_{0}=S,\ldots,t_{N}=t\\}$ be an arbitrary partition of $[S,t]$ with
diam$({\cal T})\leq\epsilon$. Take an arbitrary sequence $s_{j}\downarrow S$.
Then, for $j$ sufficiently large, $s_{j}<t_{1}$ and

$\displaystyle\eta^{\delta}_{\epsilon}(t)\,\geq\,\sup\left\\{d_{H}(F(s_{j},x,a),F(S,x,a))\;|\;x\in\bar{x}([S,s_{j}])+\delta
B,a\in A\right\\}$ $\displaystyle\hskip
72.26999pt+\sup\left\\{d_{H}(F(s_{j},x,a),F(t_{1},x,a))\;|\;x\in\bar{x}([s_{j},t_{1}])+\delta
B,\,a\in A\right\\}$ $\displaystyle\hskip
72.26999pt+\sum_{i=1}^{N-1}\sup\left\\{d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\;|\;x\in\bar{x}([t_{i},t_{i+1}])+\delta
B,a\in A\right\\}\;.$

In view of Prop. 3.1, we may pass to the limit as $j\rightarrow\infty$ in this
relation to obtain:

$\displaystyle\eta^{\delta}_{\epsilon}(t)\,\geq\,\sup\left\\{d_{H}(F(S^{+},x,a),F(S,x,a))\;|\;x\in\bar{x}(S)+\delta
B,\,a\in A\right\\}$ $\displaystyle\hskip
72.26999pt+\sup\left\\{d_{H}(F(S^{+},x,a),F(t_{1},x,a))\;|\;x\in\bar{x}([S,t_{1}])+\delta
B,\,a\in A\right\\}$ $\displaystyle\hskip
72.26999pt+\sum_{i=1}^{N-1}\sup\left\\{d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\;|\;x\in\bar{x}([t_{i},t_{i+1}])+\delta
B,\,a\in A\right\\}\;.$

Since ${\cal T}$ was an arbitrary partition with diam$({\cal T})\leq\epsilon$,
it follows that

$\eta^{\delta}_{\epsilon}(t)\geq\sup\left\\{d_{H}(F(S^{+},x,a),F(S,x,a))\;|\;x\in\bar{x}(S)+\delta
B,\,a\in A\right\\}+\tilde{\eta}^{\delta}_{\epsilon}(t)\,.$ (5.25)

Take any partition ${\cal T}=\\{t_{0}=S,\ldots,t_{N}=t\\}$ of $[S,t]$ with
diam$({\cal T})\leq\epsilon$. Then

$\displaystyle\tilde{\eta}^{\delta}_{\epsilon}(t)\,\geq\,\sup\left\\{d_{H}(F(S^{+},x,a),F(t_{1},x,a))\;|\;x\in\bar{x}([S,t_{1}])+\delta
B,a\in A\right\\}\qquad\qquad$
$\displaystyle+\sum_{i=1}^{N-1}\sup\left\\{d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\;|\;x\in\bar{x}([t_{i},t_{i+1}])+\delta
B,a\in A\right\\}.$ (5.26)

By the triangle inequality we have, for each
$x\in\bar{x}([S,t_{1}])+\delta\mathbb{B}$ and $a\in A$,

$d_{H}(F(S^{+},x,a),F(t_{1},x,a))\,\geq\,d_{H}(F(S,x,a),F(t_{1},x,a))-d_{H}(F(S^{+},x,a),F(S,x,a))\;.$

Furthermore,

$\displaystyle\sup d_{H}(F(S^{+},x,a),F(t_{1},x,a))\geq$ $\displaystyle\hskip
36.135pt\sup
d_{H}(F(S,x,a),F(t_{1},x,a))-\sup\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,,$

where, in each term, the sup is taken over
$(x,a)\in(\bar{x}([S,t_{1}])+\delta\mathbb{B})\times A$. Since ${\cal T}$ was
an arbitrary partition such that diam$({\cal T})\leq\epsilon$, we deduce from
(5.26) that

$\displaystyle\tilde{\eta}^{\delta}_{\epsilon}(t)\,\geq\,\eta^{\delta}_{\epsilon}(t)-\sup\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,|\,(x,a)\in(\bar{x}([S,t_{1}])+\delta\mathbb{B})\times
A\\}\,.$

This relation combines with (5.25) to yield

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\eta^{\delta}_{\epsilon}(t)-\tilde{\eta}^{\delta}_{\epsilon}(t)-\sup\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,|\,x\in\bar{x}(S)+\delta\mathbb{B},\,a\in
A\\}$ (5.27) $\displaystyle\leq$ $\displaystyle\Delta(\epsilon,\delta)\,,$

in which

$\displaystyle\Delta(\epsilon,\delta)\,:=\,\sup\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,|\,(x,a)\in(\bar{x}([S,(S+\epsilon)\wedge
T])+\delta\mathbb{B})\times A$ $\displaystyle\hskip
21.68121pt-\,\sup\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,|\,x\in\bar{x}(S)+\delta\mathbb{B},a\in
A\\}\;.$ (5.28)

Since $\bar{x}(.)$ is continuous and, as is easily shown,
$F(S^{+},\cdot,\cdot)$ has modulus of continuity $\gamma(\cdot)$ on
$\\{\bar{x}(S)+\delta\mathbb{B}\\}\times A$, where $\gamma(\cdot)$ is as in
hypothesis (C2), we have

$\lim_{\epsilon^{\prime}\downarrow 0}\Delta(\epsilon^{\prime},\delta)=0\;.$

Considering (5.27) in the limit as $\epsilon\downarrow 0$ yields

$\eta^{\delta}(t)=\tilde{\eta}^{\delta}(t)+\max\\{d_{H}(F(S^{+},x,a),F(S,x,a))\,|\,x\in\bar{x}(S)+\delta\mathbb{B},\,a\in
A\\}\;.$ (5.29)

Taking the limit $\delta\downarrow 0$ yields

$\eta(t)=\tilde{\eta}(t)+\underset{a\in
A}{\sup}\,d_{H}(F(S^{+},\bar{x}(S),a),F(S,\bar{x}(S),a))\;.$ (5.30)

We have validated the property (3.10) in the case $t\in(S,T)$.

Now we verify the right continuity of $\tilde{\eta}^{\delta}(\cdot)$ and
$\tilde{\eta}(\cdot)$ at $S$. We shall merely confirm the right continuity of
$\tilde{\eta}^{\delta}(\cdot)$ at $S$, since this will imply the right
continuity of $\tilde{\eta}(\cdot)$ at $S$, in consequence of the relation

$0\leq\
\tilde{\eta}(t)-\tilde{\eta}(S)=\tilde{\eta}(t)-0\leq\tilde{\eta}^{\delta}(t)-0=\tilde{\eta}^{\delta}(t)-\tilde{\eta}^{\delta}(S)\;\mbox{
for all }t\in(S,T].$

Let us assume, in contradiction, that $\tilde{\eta}^{\delta}(.)$ is not right
continuous at $S$. Then there exists $\alpha>0$ such that
$\tilde{\eta}^{\delta}(t)-(\tilde{\eta}^{\delta}(S)=0)\geq\alpha\mbox{ for all
$t\in(S,T]$}$. Taking account of Prop. 3.1, we can choose $\epsilon\in(0,T-S)$
such that $\tilde{\eta}^{\delta}_{\epsilon}(t)<\infty$ and

$\sup_{(x,a)\in(\bar{x}([S,t])+\delta B)\times
A}d_{H}(F(t,x,a),F(S^{+},x,a))\leq\alpha/4$ (5.31)

for all $t\in[S,S+\epsilon]$. We may now choose a partition
$\\{s_{0},...,s_{N}\\}$ of $[S,S+\epsilon]$, of diameter at most $\epsilon$,
such that

$\displaystyle\tilde{\eta}^{\delta}_{\epsilon}(S+\epsilon)$
$\displaystyle\leq$ $\displaystyle\sup_{(x,a)\in(\bar{x}([S,s_{1}])+\delta
B)\times A}d_{H}(F(s_{1},x,a),F(S^{+},x,a))+\Sigma_{2}+\alpha/4$ (5.32)
$\displaystyle\leq$
$\displaystyle\alpha/4+\Sigma_{2}+\alpha/4\,=\,\Sigma_{2}+\alpha/2\,,$

in which

$\Sigma_{2}\,:=\,\sum_{i=1}^{N-1}\sup_{(x,a)\in(\bar{x}([s_{i},s_{i+1}])+\delta\mathbb{B})\times
A}d_{H}(F(s_{i+1},x,a),F(s_{i},x,a))\,.$

Now we choose a partition $\\{t_{0},\ldots,t_{M}\\}$ of $[S,s_{1}]$, with
diam$(\\{t_{i}\\})\leq\epsilon$, such that

$\alpha\leq\tilde{\eta}^{\delta}_{\epsilon}(s_{1})\leq\Sigma_{1}+\alpha/4\,,$

where

$\Sigma_{1}\,:=\,\sum_{i=0}^{N-1}\sup_{\underset{a\in
A}{x\in\bar{x}([t_{i},t_{i+1}])+\delta\mathbb{B}}}d_{H}(F(t_{i+1},x,a),F(t_{i},x,a))\,.$

It follows that

$\Sigma_{1}\geq 3\alpha/4\,.$

But since the concatenation of $\\{t_{0}\ldots,t_{M}\\}$ and
$\\{s_{1}\ldots,s_{N}\\}$ is a partition of $[S,S+\epsilon]$, of diameter no
greater than $\epsilon$, we know from the preceding inequality that

$\tilde{\eta}^{\delta}_{\epsilon}(S+\epsilon)\geq\Sigma_{1}+\Sigma_{2}\geq\Sigma_{2}+3\alpha/4\,.$

But this contradicts (5.32). We have confirmed that $\tilde{\eta}^{\delta}(.)$
(and so also $\tilde{\eta}(.)$) are continuous from the left at $S$.

Next, we shall show that

$\displaystyle\eta^{\delta}(T)\,=\,\underset{T^{\prime}\uparrow
T}{\lim}\,\eta^{\delta}(T^{\prime})+\underset{x\in\bar{x}(T)+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a))\,,$ (5.33)
$\displaystyle\eta(T)\,=\,\underset{T^{\prime}\uparrow
T}{\lim}\,\eta(T^{\prime})+\underset{a\in
A}{\sup}\;d_{H}(F(T,\bar{x}(T),a),F(T^{-},\bar{x}(T),a))\,.$ (5.34)

This will complete the proof of the remaining assertions of the proposition.
Indeed, since the multifunction $t\rightarrow\tilde{F}(t,.,.)$ is continuous
at the right end-point T, the analysis leading to (5.29) and (5.30) , but now
applied to $\tilde{F}$, yields

$\tilde{\eta}^{\delta}(T)\,=\,\underset{T^{\prime}\uparrow
T}{\lim}\,\tilde{\eta}^{\delta}(T^{\prime})+0\quad\mbox{ and
}\quad\tilde{\eta}(T)\,=\,\underset{T^{\prime}\uparrow
T}{\lim}\,\tilde{\eta}(T^{\prime})+0\,.$

This is the claimed left continuity of $\tilde{\eta}^{\delta}(\cdot)$ (and, by
implication, of $\tilde{\eta}(.)$) at $T$. On the other hand, (5.33) and
(5.34) combine with (5.29) and (5.30) to yield the representation of
$\tilde{\eta}^{\delta}(T)$ in terms of $\eta^{\delta}(T)$ in (3.10) (and the
analogous representation of $\tilde{\eta}(T)$).

To prove (5.33) and (5.34) we first note that, since
$\delta^{\prime}\rightarrow\eta^{\delta^{\prime}}(T)$ is monotone, we can find
$\delta_{1}\in(\delta,\bar{\delta})$, arbitrarily close to $\delta$, such that
$\delta^{\prime}\rightarrow\eta^{\delta^{\prime}}(T)$ is continuous at
$\delta_{1}$. But then, by (5.17),

$\displaystyle\eta^{\delta_{1}}(T)$ $\displaystyle=$
$\displaystyle\underset{\delta^{\prime}\uparrow\delta_{1}}{\lim}\eta^{\delta_{1}}(T)\,$
(5.35) $\displaystyle\leq$ $\displaystyle\underset{T^{\prime}\uparrow
T}{\lim}\,\eta^{\delta_{1}}(T^{\prime})+\underset{x\in\bar{x}(T)+\delta_{1}\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a))\,.$

By Lemma 3.2 however we have, for any $T^{\prime}\in(S,T)$,

$\eta^{\delta}(T)-\eta^{\delta}(T^{\prime})\,\leq\,\eta^{\delta_{1}}(T)-\eta^{\delta_{1}}(T^{\prime}).$
(5.36)

Consequently

$\eta^{\delta}(T)-\lim_{T^{\prime}\uparrow
T}\eta^{\delta}(T^{\prime})\,\leq\,\eta^{\delta_{1}}(T)-\lim_{T^{\prime}\uparrow
T}\eta^{\delta_{1}}(T^{\prime}).$

This relation combines with (5.35) to give

$\eta^{\delta}(T)\,\leq\,\underset{T^{\prime}\uparrow
T}{\lim}\,\eta^{\delta}(T^{\prime})+\underset{x\in\bar{x}(T)+\delta_{1}\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a))\,.$ (5.37)

Since $\delta_{1}$ can be chosen such that $\delta_{1}-\delta$ is arbitrarily
small and in view of the continuity properties of $F(.,.,.)$, we see that the
preceding relation is true when the supremum is taken over
$x\in\bar{x}(T)+\delta\mathbb{B}$ in place of
$x\in\bar{x}(T)+\delta_{1}\mathbb{B}$.

Take $\epsilon\in(0,T-S)$ and $\epsilon^{\prime}\in(0,\epsilon)$. Since,
$\eta_{\epsilon^{\prime}}^{\delta}(T)\leq\eta_{\epsilon}^{\delta}(T)$, we have

$\displaystyle\eta_{\epsilon}^{\delta}(T)$ $\displaystyle\geq$
$\displaystyle\eta_{\epsilon^{\prime}}^{\delta}(T-\epsilon)+\underset{x\in\bar{x}([T-\epsilon,T])+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T-\epsilon,x,a))\,.$ (5.38)

Passing to the limit in the preceding inequality, first as
$\epsilon^{\prime}\downarrow 0$ and then as $\epsilon\downarrow 0$, noting
that

$\displaystyle\underset{\epsilon\downarrow
0}{\lim}\;\left\\{\underset{x\in\bar{x}([T-\epsilon,T])+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T-\epsilon,x,a))\right\\}$ $\displaystyle\hskip
144.54pt=\;\underset{x\in\bar{x}(T)+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a))\,,$

and taking account of (5.37) (when $\delta$ replaces $\delta_{1}$), we deduce
(5.33).

Now take $\delta^{\prime}\in(0,\delta)$. From (5.38)

$\displaystyle\eta_{\epsilon}^{\delta}(T)$ $\displaystyle\geq$
$\displaystyle\eta_{\epsilon^{\prime}}^{\delta^{\prime}}(T-\epsilon)+\underset{x\in\bar{x}([T-\epsilon,T])+\delta\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T-\epsilon,x,a))\,.$

Passing to the limit, as $\epsilon^{\prime}\downarrow 0$,
$\delta^{\prime}\downarrow 0$ and $\epsilon\downarrow 0$ and $\delta\downarrow
0$ (in that order), yields

$\displaystyle\eta(T)$ $\displaystyle\geq$
$\displaystyle\underset{T^{\prime}\uparrow
T}{\lim}\eta(T^{\prime})+\underset{a\in
A}{\sup}\;d_{H}(F(T,\bar{x}(T),a),F(T^{-},\bar{x}(T),a))\,.$ (5.39)

Take $\delta_{1}\in(0,\bar{\delta})$ such that
$\delta\rightarrow\eta^{\delta}(T)$ is continuous at $\delta_{1}$. Passing to
the limit as $\delta\downarrow 0$ in (5.36), for fixed $T^{\prime}$, and then
as $T^{\prime}\rightarrow T$ yields

$\eta(T)-\lim_{T^{\prime}\uparrow
T}\eta(T^{\prime})\,\leq\,\eta^{\delta_{1}}(T)-\lim_{T\ \uparrow
T}\eta^{\delta_{1}}(T^{\prime}).$

But then from (5.35) we deduce

$\eta(T)\,\leq\,\underset{T^{\prime}\uparrow
T}{\lim}\,\eta(T^{\prime})+\underset{x\in\bar{x}(T)+\delta_{1}\mathbb{B},a\in
A}{\sup}d_{H}(F(T,x,a),F(T^{-},x,a))\,.$

Since $\delta_{1}$, in this relation can be chosen arbitrarily small, the
relation remains valid when we set $\delta_{1}=0$. Taking note also of (5.39),
we conclude (5.34). The proof is complete. $\square$

Acknowledgments. This work was co-funded by the European Union under the 7th
Framework Programme “FP7-PEOPLE-2010-ITN”, grant agreement number
264735-SADCO, and by EPSRC under the grant “Control For Energy and
Sustainability”, grant reference EP/G066477/1.

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