# A method for finding a solution to the nonsmooth differential inclusion of a
special structure

Alexander Fominyh  Institute of Problems in Mechanical Engineering, Russian
Academy of Sciences, 61, Bolshoy pr. V.O., St. Petersburg, 199178, Russian
Federation,
St. Petersburg University, 7–9, Universitetskaya nab., St. Petersburg, 199034,
Russian Federation

###### Abstract.

The paper explores the differential inclusion of a special form. It is
supposed that the support function of the set in the right-hand side of an
inclusion may contain the maximum of the finite number of continuously
differentiable (in phase coordinates) functions. It is required to find a
trajectory that would satisfy the differential inclusion with the boundary
conditions prescribed and simultaneously lie on the surface given. Such
problems arise while practical modeling of discontinuous systems and in other
applied problems. The initial problem is reduced to a variational one. It is
proved that the resulting functional to be minimized is superdifferentiable.
The necessary minimum conditions in terms of superdifferential are formulated.
The superdifferential (or the steepest) descent method in a classical form is
then applied in order to find stationary points of this functional. Herewith,
the functional is constructed in a such a way that one can verify whether the
stationary point constructed is indeed a global minimum point of the problem.
The convergence of the method proposed is proved. The method constructed is
illustrated by examples.

###### Key words and phrases:

differential inclusion, support function, superdifferential

###### 1991 Mathematics Subject Classification:

34A60, 49J52, 49K05

The main results of this paper (Sections 6, 7, 8, 9) were obtained in IPME RAS
and supported by Russian Science Foundation (grant 20-71-10032).

##  1\. Introduction

Differential inclusions are a powerful instrument for modeling dynamical
systems. In this paper the differential inclusions of some special structure
are explored. More specific: the support function of the set in the right-hand
side of a differential inclusion contains the maximum functions of the finite
number of continuously differentiable (in phase coordinates) functions. Such
differential inclusions arise from systems of differential equations with
discontinuous right-hand sides (when considering such systems moving in a
sliding mode) and from some other practical problems. So it is required to
find a trajectory of such a differential inclusion which simultaneously would
satisfy the boundary conditions and lie on the “discontinuity” surface.

Note that the majority of methods in literature consider only differential
inclusions with a free right endpoint and use some classical approaches as
Euler and Runge-Kutta schemes, various finite differences methods etc. (see,
e. g., [1], [2], [3], [4], [5]). A survey of difference methods for
differential inclusions can be found in [6]. Note one paper [7] where an
algorithm to solve boundary value problems for differential inclusions was
constructed. Let us also give some references [8], [9], [10], [11] with
optimality conditions in problems with differential inclusion. In these papers
differential inclusions of a rather general form are considered. There are
cases of phase constraints as well as nonsmooth and nonconvex ones. On the
other hand, the works listed are more of theoretical significance and some
results seem hard to be employed in practice.

The apparatus used for solving the problem in the paper is based on the ideas
developed in the author previous works. The initial problem is reduced to a
variational one, and the resulting functional to be minimized occurs to be
nonsmooth (but only superdifferentiable). So the methods of nondifferentiable
optimization are required to explore this problem. In paper [12] a method for
solving the classical nonsmooth (but only subdifferential) variational problem
is proposed based on the idea of considering phase trajectory and its
derivative as independent variables (and taking the natural connection between
these variables into account via the special penalty term). Reducing solving
differential inclusions to a variational problem via corresponding support
functions was carried out in papers [13], [14], [15]. Some other nonsmooth
problems of optimal control and variational calculus were considered via
similar methods in papers [16], [17]. The paper also uses some ideas of V. F.
Demyanov scientific school on nondifferentiable minimization. For example, in
book [18] a modified subdifferentiable (steepest) descent method for
minimizing the maximum of the finite number of continuously differentiable
functions in the finite dimensional space is justified.

In the paper presented some basic facts from the control of discontinuous
systems of differential equations and sliding modes theory is used (see
Section 2). It is well known that when the system state is in a sliding mode,
new useful properties of the model are observed. For example such motions may
be optimal in the sense of some criterion in the optimal control theory. The
sliding modes are also used in order to stabilize the system, as well as to
get rid of unwanted external disturbances. In some cases the system motion in
a sliding mode may be described via differential inclusions with the special
structure, and this is a motivation of considering such differential
inclusions in this paper. One can get in touch with the basics required as
well as with the statement problem motivation regarding sliding modes in works
[19], [20], [21].

##  2\. Basic definitions and notations

In this paper we use the following notations. Let $C_{n}[0,T]$ be the space of
$n$-dimensional continuous on $[0,T]$ vector-functions. Let also $P_{n}[0,T]$
be the space of piecewise continuous and bounded on $[0,T]$ $n$-dimensional
vector-functions. We also require the space $L^{2}_{n}[0,T]$ of square-
summable on $[0,T]$ $n$-dimensional vector-functions. If $X$ is some normed
space, then $||\cdot||_{X}$ denotes its norm and $X^{*}$ denotes the space
conjugate to the space given.

We will assume that each trajectory $x(t)$ is a piecewise continuously
differentiable vector-function. Let $t_{0}\in[0,T)$ be a point of
nondifferentiability of the vector-function $x(t)$, then we assume that
$\dot{x}(t_{0})$ is a right-hand derivative of the vector-function $x(t)$ at
the point $t_{0}$ for definiteness. Similarly, we assume that $\dot{x}(T)$ is
a left-hand derivative of the vector-function $x(t)$ at the point $T$. Now
with the assumptions and the notations taken we can suppose that the vector
function $x(t)$ belongs to the space $C_{n}[0,T]$ and that the vector function
$\dot{x}(t)$ belongs to the space $P_{n}[0,T]$.

For some arbitrary set $F\subset R^{n}$ define the support function of the
vector $\psi~{}\in~{}R^{n}$ as $c(F,\psi)=\sup\limits_{f\in F}\langle
f,\psi\rangle$ where $\langle a,b\rangle$ is a scalar product of the vectors
$a,b\in R^{n}$. Denote $S_{n}$ a unit sphere in $R^{n}$ with the center in the
origin, also let $B_{r}(c)$ be a ball with the radius $r\in R$ and the center
$c\in R^{n}$. Let the vectors $\bf{e_{i}}$, $i=\overline{1,n}$, form the
standard basis in $R^{n}$. The sum $P+Q$ of the sets $P,Q\subset R^{n}$ is
their Minkowski sum, while $\lambda P$ with $\lambda\in R$ is the Minkowski
product; if $q$ is some vector from $R^{n}$, then $Pq=\\{\langle p,q\rangle\
|\ p\in P\\}$. Let $0_{n}$ denote a zero element of a functional space of some
$n$-dimensional vector-functions and $\bf{0_{n}}$ denote a zero element of the
space $R^{n}$. Let $E_{m}$ be an identity matrix and ${\bf{O}_{n,m}}$ — a zero
matrix in the space $R^{n}\times R^{m}$. If
$\varphi(x)=\min\limits_{i=\overline{1,M}}f_{i}(x)$, where
$f_{i}(x):R^{n}\rightarrow R$, $i=\overline{1,M}$, are some functions, then we
call the function $f_{\overline{i}}(x)$, $\overline{i}\in\\{1,\dots,M\\}$, an
active one at the point $x_{0}\in R^{n}$, if $\overline{i}\in
R(x_{0})=\left\\{i\in\\{1,\dots,M\right\\}\ |\
f_{i}(x_{0})=\varphi(x_{0})\\}$.

In the paper we will use both superdifferentials of functions in a finite-
dimensional space and superdifferentials of functionals in a functional space.
Despite the fact that the second concept generalizes the first one, for
convenience we separately introduce definitions for both of these cases and
for those specific functions (functionals) and their variables and spaces
which are considered in the paper.

Consider the space $R^{n}\times R^{n}$ with the standard norm. Let
$d=[d_{1},d_{2}]\in R^{n}\times R^{n}$ be an arbitrary vector. Suppose that at
the point $(x,z)$ there exists such a convex compact set
$\overline{\partial}h(x,z)$ $\subset R^{n}\times R^{n}$ that

$\frac{\partial h(x,z)}{\partial d}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}h(x+\alpha d_{1},z+\alpha
d_{2})-h(x,z)\big{)}=\min_{w\in\overline{\partial}h(x,z)}\langle w,d\rangle.$
(1)

In this case the function $h(x,z)$ is called superdifferentiable at the point
$(x,z)$, and the set $\overline{\partial}h(x,z)$ is called the
superdifferential of the function $h(x,z)$ at the point $(x,z)$.

From expression (1) one can see that the following formula

$h(x+\alpha d_{1},z+\alpha d_{2})=h(x,z)+\alpha\frac{\partial h(x,z)}{\partial
d}+o(\alpha,x,z,d),$ $\quad\frac{o(\alpha,x,z,d)}{\alpha}\rightarrow 0,\
\alpha\downarrow 0,$

holds true.

If the function $\varsigma(\xi)$ is differentiable at the point $\xi_{0}\in
R^{\ell}$, then its superdifferential at this point is represented in the form

$\overline{\partial}\varsigma(\xi_{0})=\\{\varsigma^{\prime}(\xi_{0})\\}$ (2)

where $\varsigma^{\prime}(\xi_{0})$ is a gradient of the function
$\varsigma(\xi)$ at the point $\xi_{0}$. Note also that the superdifferential
of the finite sum of superdifferentiable functions is the sum of the
superdifferentials of summands, i. e. if the functions $\varsigma_{k}(\xi)$,
$k=\overline{1,r}$, are superdifferentiable at the point $\xi_{0}\in
R^{\ell}$, then the function $\varsigma(\xi)=\sum_{k=1}^{r}\varsigma_{k}(\xi)$
superdifferential at this point is calculated by the formula

$\overline{\partial}\varsigma(\xi_{0})=\sum_{k=1}^{r}\overline{\partial}\varsigma_{k}(\xi_{0}).$
(3)

Consider the space $C_{n}[0,T]\times P_{n}[0,T]$ with the norm
$L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]$. Let $g=[g_{1},g_{2}]\in
C_{n}[0,T]\times P_{n}[0,T]$ be an arbitrary vector-function. Suppose that at
the point $(x,z)$ there exists such a convex weakly∗ compact set
$\overline{\partial}{I(x,z)}\subset\big{(}C_{n}[0,T]\times P_{n}[0,T],$
$||\cdot||_{L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]}\big{)}^{*}$ that

$\frac{\partial I(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}I(x+\alpha g_{1},z+\alpha
g_{2})-I(x,z)\big{)}=\min_{w\in\overline{\partial}I(x,z)}w(g).$ (4)

In this case the functional $I(x,z)$ is called superdifferentiable at the
point $(x,z)$ and the set $\overline{\partial}{I(x,z)}$ is called a
superdifferential of the functional $I(x,z)$ at the point $(x,z)$.

From expression (4) one can see that the following formula

$I(x+\alpha g_{1},z+\alpha g_{2})=I(x,z)+\alpha\frac{\partial I(x,z)}{\partial
g}+o(\alpha,x,z,g),$ $\quad\frac{o(\alpha,x,z,g)}{\alpha}\rightarrow 0,\
\alpha\downarrow 0,$

holds true.

##  3\. Statement of the problem

Consider the differential inclusion

$\dot{x}_{i}\in
A_{i}x+[\underline{a}_{i},\overline{a}_{i}]|x|[-1,1]=A_{i}x+[-\overline{a}_{i},\overline{a}_{i}]|x|,\quad
i=\overline{1,n},$ (5)

with the initial point

$x(0)=x_{0}$ (6)

and with the desired endpoint

$x_{j}(T)={x_{T}}_{j},\quad j\in J,$ (7)

In formula (5) $A_{i}$ is the $i$-th row of the constant $n\times n$ matrix
$A$, $i=\overline{1,n}$, and $\underline{a}_{i}$, $\overline{a}_{i}$,
$i=\overline{1,n}$, are given nonnegative numbers and
$\underline{a}_{i}=\overline{a}_{i}=0$, $i=\overline{m+1,n}$. The system is
considered on the given finite time interval $[0,T]$. We assume that $x(t)$ is
an $n$-dimensional continuous vector-function of phase coordinates with a
piecewise-continuous and bounded derivative on the segment $[0,T]$. In formula
(6) $x_{0}\in R^{n}$ is a given vector; in formula (7) ${x_{T}}_{j}$ are given
numbers corresponding to those coordinates of the state vector which are fixed
at the right endpoint, here $J\subset\\{1,\dots,n\\}$ is a given index set.

Let $F(x)=(f_{1}(x),\dots,f_{n}(x))^{\prime}$ where $f_{1}(x),\dots,f_{n}(x)$
run through the corresponding sets $F_{1}(x),\dots F_{n}(x)$ from the right-
hand sides of inclusions (5), so we can rewrite the given inclusions in the
form

$\dot{x}\in F(x).$ (8)

Let us also take into consideration the surface

$s(x)={\bf 0_{m}}$ (9)

where $s(x)$ is a known continuously differentiable $m$-dimensional vector-
function.

We formulate the problem as follows: it is required to find such a trajectory
$x^{*}\in C_{n}[0,T]$ (with the derivative $\dot{x}^{*}\in P_{n}[0,T]$) which
moves along surface (9), satisfies differential inclusion (8) while
$t\in[0,T]$ and meets boundary conditions (6), (7). Assume that there exists
such a solution.

Let us discuss one practical problem which leads to the statement of the
problem above. Consider the system

$\dot{x}=Ax+Bu$ (10)

with boundary conditions (6), (7). In formula (10) $A$ is a constant $n\times
n$ matrix, $B$ is a constant $n\times m$ matrix. For simplicity we suppose
that $B=[E_{m},{\bf{O}_{n-m,m}}]$. The system is considered on the given
finite time interval $[-t^{*},T]$ (here $T$ is a given final time moment; see
comments on the time moment $t^{*}$ below). We suppose that $x(t)$ is an
$n$-dimensional continuous vector-function of phase coordinates with a
piecewise-continuous and bounded derivative on $[-t^{*},T]$; the structure of
the $m$-dimensional control $u$ will be specified below.

Let also “discontinuity” surface (9) be given.

Consider the following form of controls:

$u_{i}=-\alpha_{i}|x|\mathrm{sign}(s_{i}(x)),\quad i=\overline{1,m}$ (11)

where $\alpha_{i}\in[\underline{a}_{i},\overline{a}_{i}]$, $i=\overline{1,m}$,
are some positive numbers which are sometimes called gain factors.

In book [20] it is shown that if surface (9) is a hyperplane, then under
natural assumptions and with sufficiently big values of the factors
$\alpha_{i}$, $i=\overline{1,m}$, controls (11) ensure system (10) hitting a
small vicinity of this surface $s(x)={\bf 0_{m}}$ from arbitrary initial state
(6) in the finite time $t^{*}$ and further staying in this neighborhood with
the fulfillment of the condition $s_{i}(x(t))\rightarrow 0$,
$i=\overline{1,m}$, at $t\rightarrow\infty$, i. e. controls (11) ensure the
stability “in big” of the system (10) sliding mode. In [20] one may also find
the estimates on the time moment $t^{*}$. Here we assume that all the
conditions required are already met, and the numbers $\underline{a}_{i}$,
$\overline{a}_{i}$, $i=\overline{1,m}$, are taken sufficiently big. Now we are
interested in the behavior of the system on the “discontinuity” surface (on
the time interval $[0,T]$).

We see that the right-hand sides of the first $m$ differential equations in
system (10) with controls (11) are discontinuous on the surfaces $s_{i}(x)=0$,
$i=\overline{1,m}$. So one has to use one of the known definitions of a
discontinuous system solution.

Let us use one of the classical variants of such a definition [19], [23]. The
essence of this definition is as follows. On the finite time interval $[0,T]$
consider the system $\dot{x}=f(x,u_{1}(x,t),\dots,u_{m}(x,t),t)$, in which the
vector-function $f(x,u_{1},\dots,u_{m},t)$ is continuous in all its arguments,
and the vector-functions $u_{i}(x,t)$, $i=\overline{1,m}$, are discontinuous
on the sets $s_{i}(x)=0$, $i=\overline{1,m}$, respectively. At every point
$(x,t)$ of discontinuity of the vector-function $u_{i}(x,t)$,
$i=\overline{1,m}$, a closed set $U_{i}(x,t)$, $i=\overline{1,m}$, must be
defined. It is a set of possible values of the variable $u_{i}$ of the
function $f(x,u_{1},\dots,u_{m},t)$. Denote $F(x,t)=f(x,u_{1},...,u_{m},t)$
the set of the function $f(x,u_{1},...,u_{m},t)$ values at the fixed variables
$x,t$, and while $u_{1},\dots,u_{m}$ run through the sets
$U_{1}(x,t),\dots,U_{m}(x,t)$ respectively. Then the solutions of this
differential inclusion are taken as solutions of the original differential
equation with a discontinuous right-hand side. In physical systems the sets
$U_{i}(x,t)$, $i=\overline{1,m}$, usually correspond to different blocks and
are assumed to be convex. At every point $(x,t)$ of discontinuity of the
vector-function $u_{i}(x,t)$, $i=\overline{1,m}$, the set $U_{i}(x,t)$ must
also contain all the limit points of all sequences $v_{k}\in
U_{i}(x_{k},t_{k})$, where $x_{k}\rightarrow x$ and $t_{k}\rightarrow t$ if
$k\rightarrow\infty$. If the control is of form (11), then it is natural to
consider
$U_{i}(x,t)=\mathrm{co}\\{-\overline{a}_{i}|x|,\overline{a}_{i}|x|\\}$,
$i=\overline{1,m}$, as such sets.

Thus, accordingly to this definition a solution of the discontinuous system
considered satisfies differential inclusion (5) above. Note that the more
detailed version of the definition given is presented in [23]. It has a strict
and rather complicated form so we don’t consider the details here. For our
purposes it is sufficient to postulate that the system (10) solutions with
controls (11) employed are the solutions of inclusion (5) by definition.

Instead of trajectories from the space $C_{n}[0,T]$ with derivatives from the
space $P_{n}[0,T]$ one may consider absolutely continuous trajectories on the
interval $[0,T]$ with measurable and almost everywhere bounded derivatives on
$[0,T]$ what is more natural for differential inclusions. The choice of the
solution space in the paper is explained by the possibility of its practical
construction.

##  4\. Reduction to a variational problem

We will sometimes write $F$ instead of $F(x)$ for brevity. Insofar as $\forall
x\in R^{n}$ the set $F(x)$ is a convex compact set in $R^{n}$, then inclusion
(8) may be rewritten as follows [24]:

$\dot{x}_{i}(t)\psi_{i}(t)\leq
c(F_{i}(x(t)),\psi_{i}(t))\quad\forall\psi_{i}(t)\in S_{1},\quad\forall
t\in[0,T],\quad i=\overline{1,n}.$

Calculate the support function of the set $F_{i}$. For this note that the set
$F_{i}$ is a one-dimensional “ball” with the the center

$c_{i}(x)=A_{i}x,\quad i=\overline{1,n},$

and with the “radius”

$r_{i}(x)=\overline{a}_{i}|x|,\quad i=\overline{1,m},$ $r_{i}(x)=0,\quad
i=\overline{m+1,n}.$

So the support function of the set $F_{i}$ can be expressed [24] by the
formula

$c(F_{i}(x),\psi_{i})=\psi_{i}A_{i}x+\overline{a}_{i}|x||\psi_{i}|,\quad
i=\overline{1,m},$ $c(F_{i}(x),\psi_{i})=\psi_{i}A_{i}x,\quad
i=\overline{m+1,n}.$

We see that the support function of the set $F_{i}$ is continuously
differentiable in the phase coordinates $x$ if $x_{i}\neq 0$,
$i=\overline{1,n}$.

Denote $z(t)=\dot{x}(t)$, $z\in P_{n}[0,T]$, then from (6) one has

$x(t)=x_{0}+\int_{0}^{t}z(\tau)d\tau.$ (12)

Let us now realize the following idea. “Forcibly” consider the points $z$ and
$x$ to be “independent” variables. Since, in fact, there is relationship (12)
above between these variables (which naturally means that the vector-function
$z(t)$ is a derivative of the vector-function $x(t)$), let us take this
restriction into account by using the functional

$\upsilon(x,z)=\frac{1}{2}\int_{0}^{T}\left(x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau\right)^{2}dt.$
(13)

Besides it is seen that condition (6) on the left endpoint is also satisfied
if $v(x,z)=0$.

For $i=\overline{1,n}$ put

$\ell_{i}(\psi_{i},x,z)=\langle z_{i},\psi_{i}\rangle-c(F_{i}(x),\psi_{i}),$
$h_{i}(x,z)=\max_{\psi_{i}\in S_{1}}\max\\{0,\ell_{i}(\psi_{i},x,z)\\},$

then put

$h(x,z)=(h_{1}(x,z),\dots,h_{n}(x,z))^{\prime}$

and construct the functional

$\varphi(x,z)=\frac{1}{2}\int_{0}^{T}h^{2}\big{(}x(t),z(t)\big{)}dt.$ (14)

It is not difficult to check that for functional (14) the relation

$\left\\{\begin{array}[]{ll}\varphi(x,z)=0,&\text{if}\
\dot{x}_{i}(t)\psi_{i}(t)\leq
c(F_{i}(x(t)),\psi_{i}(t))\quad\forall\psi_{i}(t)\in S_{1},\quad\forall
t\in[0,T],\quad i=\overline{1,n}.\\\
\varphi(x,z)>0,&\text{otherwise},\end{array}\right.$

holds true, i. e. inclusion (8) takes place iff $\varphi(x,z)=0$.

Introduce the functional

$\chi(z)=\frac{1}{2}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right)^{2}.$ (15)

We see that if $v(x,z)=0$, then condition (7) on the right endpoint is
satisfied iff $\chi(z)=0$.

Introduce the functional

$\omega(x)=\frac{1}{2}\int_{0}^{T}s^{2}\big{(}x(t)\big{)}dt.$ (16)

Obviously, the trajectory $x(t)$ belongs to surface (9) at each $t\in[0,T]$
iff $\omega(x)=0$.

Finally construct the functional

$I(x,z)=\varphi(x,z)+\chi(z)+\omega(x)+\upsilon(x,z).$ (17)

So the original problem has been reduced to minimizing functional (17) on the
space $C_{n}[0,T]\times P_{n}[0,T]$. Denote $z^{*}$ a global minimizer of this
functional. Then

$x^{*}(t)=x_{0}+\int_{0}^{t}z^{*}(\tau)d\tau$

is a solution of the initial problem.

The structure of the functional $\varphi(x,z)$ is natural as the value
$h_{i}(x(t),z(t))$, $i=\overline{1,n}$, at each fixed $t\in[0,T]$ is the
Euclidean distance from the point $z_{i}(t)$ to the set $F_{i}(x(t))$;
functional (14) is half the sum of squares of the deviations in
$L^{2}_{n}[0,T]$ norm of the trajectories $z_{i}(t)$ from the sets $F_{i}(x)$,
$i=\overline{1,n}$, respectively. The meaning of functionals (13), (15), (16)
structures is obvious.

Despite the fact that the dimension of functional $I(x,z)$ arguments is $n$
more the dimension of the initial problem (i. e. the dimension of the desired
point $x^{*}$), the structure of its superdifferential (in the space
$C_{n}[0,T]\times P_{n}[0,T]$ as a normed space with the norm
$L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]$), as will be seen from what follows, has
a rather simple form. This fact will allow us to construct a numerical method
for solving the original problem.

##  5\. Minimum conditions of the functional $I(x,z)$

In this section referring to superdifferential calculus rules (2), (3), we
mean their known analogues in a functional space [26].

Using classical variation it is easy to prove the Gateaux differentiability of
the functional $\chi(z)$, we have

$\nabla\chi(z)=\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right){\bf e_{j}}.$

By superdifferential calculus rule (2) one may put

$\overline{\partial}\chi(z)=\left\\{\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}}\right\\}.$ (18)

Using classical variation it is easy to prove the Gateaux differentiability of
the functional $\omega(x)$, we have

$\nabla\omega(x)=\sum_{i=1}^{m}s_{i}\big{(}x(t)\big{)}\frac{\partial
s_{i}\big{(}x(t)\big{)}}{\partial x}.$

By superdifferential calculus rule (2) one may put

$\overline{\partial}\omega(x)=\left\\{\sum_{i=1}^{m}s_{i}\big{(}x(t)\big{)}\frac{\partial
s_{i}\big{(}x(t)\big{)}}{\partial x}\right\\}.$ (19)

Using classical variation and integration by parts it is also not difficult to
check the Gateaux differentiability of the functional $\upsilon(x,z)$, we
obtain

$\nabla\upsilon(x,z,t)=\begin{pmatrix}\displaystyle{x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau}\\\
\displaystyle{-\int_{t}^{T}\left(x(\tau)-x_{0}-\int_{0}^{\tau}z(s)ds\right)d\tau}\end{pmatrix}.$

By superdifferential calculus rule (2) one may put

$\overline{\partial}\upsilon(x,z)=\left\\{\begin{pmatrix}\displaystyle{x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau}\\\
\displaystyle{-\int_{t}^{T}\left(x(\tau)-x_{0}-\int_{0}^{\tau}z(s)ds\right)d\tau}\end{pmatrix}\right\\}.$
(20)

Explore the differential properties of the functional $\varphi(x,z)$. For
this, first give the following formulas for calculating the superdifferential
$\overline{\partial}h^{2}(x,z)$ at the point $(x,z)$. At $i=\overline{1,m}$
one has

$\overline{\partial}\left({\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\right)=h_{i}(x,z)\bigg{(}\psi^{*}_{i}{\bf
e_{i+n}}-\psi^{*}_{i}[A^{\prime}_{i},{\bf
0_{n}}]+\sum_{j=1}^{n}\overline{\partial}\left(-\overline{a}_{i}|x_{j}||\psi_{i}^{*}|\right)\bigg{)},$
(21)

where at $j=\overline{1,n}$ we have

$\overline{\partial}(-\overline{a}_{i}|x_{j}||\psi_{i}^{*}|)=\left\\{\begin{array}[]{lll}-\overline{a}_{i}|\psi_{i}^{*}|{\bf
e_{j}},&\text{if}\ x_{j}>0,\\\ \overline{a}_{i}|\psi_{i}^{*}|{\bf
e_{j}},&\text{if}\ x_{j}<0,\\\
\mathrm{co}\big{\\{}-\overline{a}_{i}|\psi_{i}^{*}|{\bf
e_{j}},\overline{a}_{i}|\psi_{i}^{*}|{\bf e_{j}}\big{\\}},&\text{if}\
x_{j}=0.\end{array}\right.$

At $i=\overline{m+1,n}$ one has

$\overline{\partial}\left({\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\right)=h_{i}(x,z)\big{(}\psi^{*}_{i}{\bf
e_{i+n}}-\psi^{*}_{i}[A^{\prime}_{i},{\bf 0_{n}}]\big{)}.$

In the formulas given the value $\psi^{*}_{i}$ is such that
$\max\\{0,\ell_{i}(\psi_{i}^{*}(x,z),x,z)\\}=\max_{\psi_{i}\in
S_{1}}\max\\{0,\ell_{i}(\psi_{i},x,z)\\}$, $i=\overline{1,n}$. In [16] it is
shown that if $h_{i}(x,z)>0$ then $\psi_{i}^{*}(x,z)$ is unique and continuous
in $(x,z)$. In [25] it is shown that the functional $\varphi(x,z)$ is
superdifferentiable and that its superdifferential is determined by the
corresponding integrand superdifferential.

Let the interval $[0,T]$ may be divided into a finite number of intervals, in
every of which each phase trajectory is either identically equal to zero or
retains a certain sign. Then the functional $\varphi(x,z)$ is
superdifferentiable, i. e.

$\frac{\partial\varphi(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}\varphi(x+\alpha g_{1},z+\alpha
g_{2})-\varphi(x,z)\big{)}=\min_{w\in\overline{\partial}\varphi(x,z)}\int_{0}^{T}\langle
w(t),g(t)\rangle dt,$

where $g=[g_{1},g_{2}]\in C_{n}[0,T]\times P_{n}[0,T]$ and the set
$\overline{\partial}\varphi(x,z)$ is defined as follows

$\overline{\partial}\varphi(x,z)=\Big{\\{}w=[w_{1},w_{2}]\in
L_{n}^{\infty}[0,T]\times L_{n}^{\infty}[0,T]\ \big{|}$ (22)
$[w_{1}(t),w_{2}(t)]\in\overline{\partial}\left({\textstyle\frac{1}{2}}\,h^{2}(x(t),z(t))\right)\quad
for\ a.e.\ t\in[0,T]\Big{\\}}.$

Using formulas (17), (18), (19), (20), (22) obtained and superdifferential
calculus rule (3) we have the following final formula for calculating the
superdifferential of the functional $I(x,z)$ at the point $(x,z)$

$\overline{\partial}I(x,z)=\overline{\partial}\varphi(x,z)+\overline{\partial}\overline{\chi}(x,z)+\overline{\partial}\overline{\omega}(x,z)+\overline{\partial}\upsilon(x,z),$
(23)

where formally $\overline{\chi}(x,z):=\chi(z)$,
$\overline{\omega}(x,z):=\omega(x)$.

Using the known minimum condition [26] of the functional $I(x,z)$ at the point
$(x^{*},z^{*})$ in terms of superdifferential, we conclude that the following
theorem is true.

Let the interval $[0,T]$ may be divided into a finite number of intervals, in
every of which each phase trajectory $x^{*}_{i}(t)$, $i=\overline{1,n}$, is
either identically equal to zero or retains a certain sign. In order for the
point $(x^{*},z^{*})$ to minimize the functional $I(x,z)$, it is necessary to
have

$\overline{\partial}I(x^{*}(t),z^{*}(t))=\\{\bf 0_{2n}\\}$ (24)

at each $t\in[0,T]$ where the expression for the superdifferential
$\overline{\partial}I(x,z)$ is given by formula (23). If one has
$I(x^{*},z^{*})$, then condition (24) is also sufficient.

Theorem 5 already contains a constructive minimum condition, since on its
basis it is possible to construct the steepest (the superdifferential) descent
direction, and for solving each of the subproblems arising during this
construction there are known efficient algorithms for solving them (see
Section 6 below). Once the steepest descent direction is constructed, one is
able to use it in order to apply some of nonsmooth optimization methods (see
Section 7 below). Note once again that without “separation” of the variables
$x$ and $z$ proposed the considered functional superdifferential would have a
very complicated structure. That would make constructing the superdifferential
descent direction of this functional a significantly difficult (and as it
seems, practically impossible) problem.

##  6\. A more general case of the set $F(x)$ support function

Consider now a more general case when the support functions of the
corresponding sets $F_{i}(x)$, $i=\overline{1,n}$, are of the form

$c(F_{i}(x),\psi_{i})=\psi_{i}A_{i}x+\sum_{j=1}^{r}\overline{a}_{ij}\max\big{\\{}f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{k(j)}}(x)\psi_{i}\big{\\}},$

where $f_{i,j_{1}}(x),\dots,f_{i,j_{k(j)}}(x)$, $i=\overline{1,n}$,
$j=\overline{1,r}$ (for simplicity of presentation we suppose that $r$ is the
same for each $i=\overline{1,n}$), are continuously differentiable functions
and $\overline{a}_{ij}$, $i=\overline{1,n}$, $j=\overline{1,r}$, are some
nonnegative numbers. (We also still consider only convex and compact sets
$F_{1}(x),\dots F_{n}(x)$ at every $x\in R^{n}$).

This case indeed is more general as we have
$|x_{i}||\psi_{i}|=|x_{i}\psi_{i}|=\max\\{x_{i}\psi_{i},-x_{i}\psi_{i}\\}$.

One can also give a practical problem which leads to such a system. Let from
some physical considerations the “velocity” $\dot{x}_{1}$ of an object lie in
the range $[\min\\{x_{1},x_{2},x_{3}\\},\max\\{x_{1},x_{2},x_{3}\\}]$ of the
“coordinates” $x_{1}$, $x_{2}$, $x_{3}$. The segment given may be written down
as
${\mathrm{co}\\{x_{1},x_{2},x_{3}\\}=\mathrm{co}\bigcup\limits_{i=1}^{3}\\{x_{i}\\}}$.
The support function of this set is [24]
$\max\\{x_{1}\psi_{1},x_{2}\psi_{1},x_{3}\psi_{1}\\}$.

For simplicity consider the case $n=2$, $r=1$, $k(1)=2$, and only the
functions $\ell_{1}(\psi_{1},x,z)$ and $h_{1}(x,z)$ (here we denote them
$\ell(\psi_{1},x_{1},x_{2},z_{1})$ and $h(x_{1},x_{2},z_{1})$ respectively)
and the time interval $[t_{1},t_{2}]\subset[0,T]$ of nonzero length; the
general case is considered in a similar way. Then we have
$\ell(\psi_{1},x_{1},x_{2},z_{1})=z_{1}\psi_{1}-a_{1}x_{1}\psi_{1}-$
$-a_{2}x_{2}\psi_{1}-b\max\\{f_{1}(x)\psi_{1},f_{2}(x)\psi_{1}\\}$, where
$a_{1}:=a_{1,1}$, $a_{2}:=a_{1,2}$, $b:=\overline{a}_{1,1}>0$,
$f_{1}:=f_{1,1_{1}}$, $f_{2}:=f_{1,1_{2}}$. Fix some point $(x_{1},x_{2})\in
R^{2}$. Let $f_{1}(x(t))\psi_{1}(t)=f_{2}(x(t))\psi_{1}(t)$ at
$t\in[t_{1},t_{2}]$; other cases may be studied in a completely analogous
fashion.

a) Suppose that $h_{1}(x,z)>0$, i. e. $h_{1}(x,z)=\max_{\psi_{1}\in
S_{1}}\ell_{1}(\psi_{1},x,z)>0$.

Our aim is to apply the corresponding theorem on a directional
differentiability from [27]. The theorem of this book considers the inf-
functions so we will apply this theorem to the function
$-\ell(\psi_{1},x_{1},x_{2},z_{1})$. For this, check that the function
$h(x_{1},x_{2},z_{1})$ satisfies the following conditions:
i) the function $\ell(\psi_{1},x_{1},x_{2},z_{1})$ is continuous on
$S_{1}\times R^{2}\times R$;
ii) there exist a number $\beta$ and a compact set $C\in R$ such that for
every $(\overline{x}_{1},\overline{x}_{2},\overline{z}_{1})^{\prime}$ in the
vicinity of the point $(x_{1},x_{2},z_{1})^{\prime}$ the level set

${lev}_{\beta}(-\ell(\cdot,\overline{x}_{1},\overline{x}_{2},\overline{z}_{1}))=\\{\psi_{1}\in
S_{1}\ |\
-\ell(\psi_{1},\overline{x}_{1},\overline{x}_{2},\overline{z}_{1})\leq\beta\\}$

is nonempty and is contained in the set $C$;
iii) for any fixed $\psi_{1}\in S_{1}$ the function
$\ell(\psi_{1},\cdot,\cdot,\cdot)$ is directionally differentiable at the
point $(x_{1},x_{2},z_{1})^{\prime}$;
iv) if $d=[d_{1},d_{2}]\in R^{2}\times R$, $\gamma_{n}\downarrow 0$ and
$\psi_{1_{n}}$ is a sequence in $C$, then $\psi_{1_{n}}$ has a limit point
$\overline{\psi}_{1}$ such that

$\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{-\ell(\psi_{1_{n}},x+\gamma_{n}d_{1},z+\gamma_{n}d_{2})-(-\ell(\psi_{1_{n}},x,z))}{\gamma_{n}}\geq\frac{\partial(-\ell(\overline{\psi}_{1},x,z))}{\partial
d}$

where
$\displaystyle{\frac{\partial\ell(\overline{\psi}_{1},x_{1},x_{2},z_{1})}{\partial
d}}$ is the derivative of the function
$\ell(\overline{\psi}_{1},x_{1},x_{2},z_{1})$ at the point
$(x_{1},x_{2},z_{1})^{\prime}$ in direction $d$.

The verification of conditions i), ii) is obvious.

In order to verify the condition iii), it is sufficient to observe that since
$b>0$, then for the fixed $\psi_{1}\in S_{1}$ the function
$-b\max\\{f_{1}(x)\psi_{1},f_{2}(x)\psi_{1}\\}$ is superdifferentiable [22]
(hence, it is differentiable in directions) at the point
$(x_{1},x_{2})^{\prime}$; herewith, its superdifferential at this point is $b\
\mathrm{co}\left\\{\left(\psi_{1}\frac{\partial f_{1}(x)}{\partial
x_{1}},\psi_{1}\frac{\partial f_{1}(x)}{\partial x_{2}}\right)^{\prime},\
\left(\psi_{1}\frac{\partial f_{2}(x)}{\partial x_{1}},\psi_{1}\frac{\partial
f_{2}(x)}{\partial x_{2}}\right)^{\prime}\right\\}$. An explicit expression of
this function derivative at the point $(x_{1},x_{2})^{\prime}$ in the
direction $d_{1}$ is $-b\max\left\\{\left\langle\psi_{1}\frac{\partial
f_{1}(x)}{\partial x},d_{1}\right\rangle,\left\langle\psi_{1}\frac{\partial
f_{2}(x)}{\partial x},d_{1}\right\rangle\right\\}$.

Finally, check condition iv). Let $[d_{1},d_{2}]\in R^{2}\times R$,
$\gamma_{n}\downarrow 0$ and $\psi_{1_{n}}$ is some sequence from the set $C$.
Calculate

$\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{-\ell(\psi_{1_{n}},x_{1}+\gamma_{n}d_{1,1},x_{2}+\gamma_{n}d_{1,2},z_{1}+\gamma_{n}d_{2})-(-\ell(\psi_{1_{n}},x_{1},x_{2},z_{1}))}{\gamma_{n}}=$
$=\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{1}{\gamma_{n}}\Big{(}-(z_{1}+\gamma_{n}d_{2})\psi_{1_{n}}+a_{1}(x_{1}+\gamma_{n}d_{1,1})\psi_{1_{n}}+a_{2}(x_{2}+\gamma_{n}d_{1,2})\psi_{1_{n}}+$
$+b\max\big{\\{}f_{1}(x_{1}+\gamma_{n}d_{1,1},x_{2}+\gamma_{n}d_{1,2})\psi_{1_{n}},f_{2}(x_{1}+\gamma_{n}d_{1,1},x_{2}+\gamma_{n}d_{1,2})\psi_{1_{n}}\big{\\}}+$
$+z_{1}\psi_{1_{n}}-a_{1}x_{1}\psi_{1_{n}}-a_{2}x_{2}\psi_{1_{n}}-b\max\big{\\{}f_{1}(x_{1},x_{2})\psi_{1_{n}},f_{2}(x_{1},x_{2})\psi_{1_{n}}\big{\\}}\Big{)}=$

$=\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{1}{\gamma_{n}}\bigg{(}-\gamma_{n}d_{2}\psi_{1_{n}}+\gamma_{n}a_{1}d_{1,1}\psi_{1_{n}}+\gamma_{n}a_{2}d_{1,2}\psi_{1_{n}}+$
$+b\max\left\\{\left[f_{1}(x)+\gamma_{n}\bigg{\langle}\frac{\partial
f_{1}(x)}{\partial
x},d_{1}\bigg{\rangle}+o_{1}(\gamma_{n},x,d)\right]\psi_{1_{n}},\left[f_{2}(x)+\gamma_{n}\left\langle\frac{\partial
f_{2}(x)}{\partial
x},d_{1}\right\rangle+o_{2}(\gamma_{n},x,d)\right]\psi_{1_{n}}\right\\}-$
$-b\max\big{\\{}f_{1}(x_{1},x_{2})\psi_{1_{n}},f_{2}(x_{1},x_{2})\psi_{1_{n}}\big{\\}}\bigg{)}\geq$
$\geq\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{1}{\gamma_{n}}\bigg{(}-\gamma_{n}d_{2}\psi_{1_{n}}+\gamma_{n}a_{1}d_{1,1}\psi_{1_{n}}+\gamma_{n}a_{2}d_{1,2}\psi_{1_{n}}+$
$+\gamma_{n}b\max\left\\{\bigg{\langle}\frac{\partial f_{1}(x)}{\partial
x},d_{1}\bigg{\rangle}\psi_{1_{n}},\left\langle\frac{\partial
f_{2}(x)}{\partial
x},d_{1}\right\rangle\psi_{1_{n}}\right\\}+b\min\big{\\{}o_{1}(\gamma_{n},x,d)\psi_{1_{n}},o_{2}(\gamma_{n},x,d)\psi_{1_{n}}\big{\\}}\bigg{)}$

where ${o_{q}(\gamma_{n},x,d)}/{\gamma_{n}}\rightarrow 0,\
\gamma_{n}\downarrow 0$, $q=\overline{1,2}$, and the last inequality follows
from the assumption $f_{1}(x)\psi_{1}=f_{2}(x)\psi_{1}$ (on the time interval
considered) made and from the corresponding property of the maximum of two
functions [18] when this assumption is satisfied.

Let $\overline{\psi}_{1}$ be a limit point of the sequence $\psi_{1_{n}}$.
Then by the directional derivative definition we have

$\frac{\partial(-\ell(\overline{\psi}_{1},x,z))}{\partial
d}=-d_{2}\overline{\psi}_{1}+a_{1}d_{1,1}\overline{\psi}_{1}+a_{2}d_{1,2}\overline{\psi}_{1}+b\max\left\\{\left\langle\overline{\psi}_{1}\frac{\partial
f_{1}(x)}{\partial
x},d_{1}\right\rangle,\left\langle\overline{\psi}_{1}\frac{\partial
f_{2}(x)}{\partial x},d_{1}\right\rangle\right\\}.$

From last two relations one obtains that condition iv) is fulfilled.

Thus, the function $h(x_{1},x_{2},z_{1})$ satisfies conditions i)-iv), so it
is differentiable in directions at the point $(x_{1},x_{2},z_{1})$ [27], and
its derivative in the direction $d$ at this point is expressed by the formula

$\frac{\partial h(x_{1},x_{2},z_{1})}{\partial
d}=\sup_{\psi_{1}\in\mathcal{S}(x_{1},x_{2},z_{1})}\frac{\partial\ell(\psi_{1},x_{1},x_{2},z_{1})}{\partial
d}$

where $\mathcal{S}(x_{1},x_{2},z_{1})=\mathrm{arg}\max_{\psi_{1}\in
S_{1}}\ell(\psi_{1},x_{1},x_{2},z_{1})$. However, as shown above, in the
problem considered the set $\mathcal{S}(x_{1},x_{2},z_{1})$ consists of the
only element $\psi_{1}^{*}(x_{1},x_{2},z_{1})$, hence

$\frac{\partial h(x_{1},x_{2},z_{1})}{\partial
d}=\frac{\partial\ell(\psi_{1}^{*}(x_{1},x_{2},z_{1}),x_{1},x_{2},z_{1})}{\partial
d}.$

Finally, recall that by the directional derivative definition one has the
equality

$\frac{\partial(-\ell(\psi_{1}^{*},x,z))}{\partial
d}=-d_{2}\psi_{1}^{*}+a_{1}d_{1,1}\psi_{1}^{*}+a_{2}d_{1,2}\psi_{1}^{*}+b\max\left\\{\left\langle\psi_{1}^{*}\frac{\partial
f_{1}(x)}{\partial
x},d_{1}\right\rangle,\left\langle\psi_{1}^{*}\frac{\partial
f_{2}(x)}{\partial x},d_{1}\right\rangle\right\\}$

where we have put $\psi^{*}_{1}:=\psi_{1}^{*}(x_{1},x_{2},z_{1})$.

From last two expressions we finally obtain that the function
$h(x_{1},x_{2},z_{1})$ is superdifferentiable at the point
$(x_{1},x_{2},z_{1})^{\prime}$ but it is also positive in the case considered,
so the function $h^{2}(x_{1},x_{2},z_{1})$ is superdifferentiable at the point
$(x_{1},x_{2},z_{1})^{\prime}$ as well as the square of a superdifferentiable
positive function (see [28]).

b) In the case $h_{1}(x,z)=0$ it is obvious that the function
$h^{2}(x_{1},x_{2},z_{1})$ is differentiable at the point
$(x_{1},x_{2},z_{1})^{\prime}$ and its gradient vanishes at this point.

Now explore the differential properties of the functional $\varphi(x,z)$. For
this, first give the following formulas for calculating the superdifferential
$\overline{\partial}h^{2}(x,z)$ at the point $(x,z)$. At $i=\overline{1,n}$
one has

$\overline{\partial}\left({\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\right)=h_{i}(x,z)\bigg{(}\psi^{*}_{i}{\bf
e_{i+n}}-\psi^{*}_{i}[A^{\prime}_{i},{\bf
0_{n}}]+\sum_{j=1}^{r}\overline{\partial}\left(-\overline{a}_{ij}\max\big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{k(j)}}(x)\psi_{i}^{*}\big{\\}}\right)\bigg{)}$
(25)

and at $j=\overline{1,r}$ we have

$\overline{\partial}\left(-\overline{a}_{ij}\max\big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{k(j)}}(x)\psi_{i}^{*}\big{\\}}\right)=\overline{a}_{ij}\
\mathrm{co}\left\\{\left[\psi_{i}^{*}\frac{\partial f_{i,j_{p}}(x)}{\partial
x},\bf{0_{n}}\right]\right\\},\quad j_{p}\in R_{ij}(x),$ (26)
$R_{ij}(x)=\Big{\\{}j_{p}\in\\{j_{1},\dots,j_{k(j)}\\}\ \big{|}\
f_{i,j_{p}}(x)\psi_{i}^{*}=\max\big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{k(j)}}(x)\psi_{i}^{*}\big{\\}}\Big{\\}}.$

Calculate the derivative of the function
${\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)$ in the direction $g=[g_{1},g_{2}]\in
R^{n}\times R^{n}$. By virtue of formulas (25), (26) and superdifferential
calculus rules [18] we have

$\frac{\partial\left({\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\right)}{\partial
g}=h_{i}(x,z)\bigg{(}\psi^{*}_{i}{{g_{2}}_{i}}-\left\langle\psi^{*}_{i}A_{i},g_{1}\right\rangle+\sum_{j=1}^{r}\min_{j_{p}\in
R_{ij}(x)}\left\langle-\overline{a}_{ij}\psi_{i}^{*}\frac{\partial
f_{i,j_{p}}(x)}{\partial x},g_{1}\right\rangle\bigg{)}.$ (27)

Show that the functional $\varphi(x,z)$ is superdifferentiable and that its
superdifferential is determined by the corresponding integrand
superdifferential.

Let the interval $[0,T]$ may be divided into a finite number of intervals, in
every of which one (several) of the functions
$\big{\\{}f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{k(j)}}(x)\psi_{i}\big{\\}}$,
$i=\overline{1,n}$, $j=\overline{1,r}$, is (are) active. Then the functional
$\varphi(x,z)$ is superdifferentiable, i. e.

$\frac{\partial\varphi(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}\varphi(x+\alpha g_{1},z+\alpha
g_{2})-\varphi(x,z)\big{)}=\min_{w\in\overline{\partial}\varphi(x,z)}\int_{0}^{T}\langle
w(t),g(t)\rangle dt$ (28)

where $g=[g_{1},g_{2}]\in C_{n}[0,T]\times P_{n}[0,T]$ and the set
$\overline{\partial}\varphi(x,z)$ is defined as follows

$\overline{\partial}\varphi(x,z)=\Big{\\{}w=[w_{1},w_{2}]\in
L_{n}^{\infty}[0,T]\times L_{n}^{\infty}[0,T]\ \big{|}$ (29)
$[w_{1}(t),w_{2}(t)]\in\overline{\partial}\left({\textstyle\frac{1}{2}}\,h^{2}(x(t),z(t))\right)\quad
for\ a.e.\ t\in[0,T]\Big{\\}}.$

###### Proof.

In accordance with definition (4) of a superdifferentiable functional, in
order to prove the theorem, one has to check that

1) the derivative of the functional $\varphi(x,z)$ in the direction $g$ is
actually of form (28);

2) herewith, the set $\overline{\partial}\varphi(x,z)$ is convex and weakly*
compact subset of the space $\left(C_{n}[0,T]\times
P_{n}[0,T],||\cdot||_{L^{2}_{n}[0,T]\times L^{2}_{n}[0,T]}\right)^{*}$.

Let us prove statement 1).

At first show that the following relations are true.

$\lim_{\alpha\downarrow 0}\frac{1}{\alpha}\bigg{|}\varphi(x+\alpha
g_{1},z+\alpha
g_{2})-\varphi(x,z)-\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}\left(\frac{1}{2}h^{2}(x,z)\right)}\big{(}\langle
w_{1}(t),\alpha g_{1}(t)\rangle+\langle w_{2}(t),\alpha
g_{2}(t)\rangle\big{)}dt\bigg{|}=0$ (30)

or

$\lim_{\alpha\downarrow 0}\frac{1}{\alpha}\bigg{|}\varphi(x+\alpha
g_{1},z+\alpha
g_{2})-\varphi(x,z)-\int_{0}^{T}\frac{\partial\left({\textstyle\frac{1}{2}}\,h^{2}(x(t),z(t))\right)}{\partial
g}dt\bigg{|}=0.$

Denote at $t\in[0,T]$

$\Phi(t,\alpha)=\frac{1}{\alpha}\Big{(}h^{2}(x(t)+\alpha g_{1}(t),z(t)+\alpha
g_{2}(t))-h^{2}(x(t),z(t))\Big{)}-$ (31)
$-\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),g_{1}(t)\rangle+\langle w_{2}(t),g_{2}(t)\rangle\big{)}.$

Our aim is to prove relation (30) via Lebesgue’s dominated convergence theorem
applied to the function $\Phi(t,\alpha)$ (at $\alpha\downarrow 0$).

At first note that by superdifferential definition (1) and by the
superdifferentiability of the function $h^{2}(x,z)$ (proved at the beginning
of this section) for each $t\in[0,T]$ we have $\Phi(t,\alpha)\rightarrow 0$
when $\alpha\downarrow 0$.

In the following two paragraphs we show that for every $\alpha>0$ one has
$\Phi(t,\alpha)\in L_{1}^{\infty}[0,T]$.

Insofar as $x$, $g_{1}\in C_{n}[0,T]$, $z$, $g_{2}\in P_{n}[0,T]$ and the
function $h^{2}(x,z)$ is continuous in its variables due to its structure
[18], we obtain that for each $\alpha>0$ the functions
$t~{}\rightarrow~{}h^{2}(x(t),z(t))$ and $t\rightarrow h^{2}(x(t)+\alpha
g_{1}(t)$, $z(t)+\alpha g_{2}(t))$ belong to the space $L^{\infty}_{1}[0,T]$.

Due to the upper semicontinuity of a superdifferential mapping [22] and the
structure of the superdifferential $\overline{\partial}h^{2}(x,z)$, it is easy
to check that the mapping $t\rightarrow\overline{\partial}h^{2}(x(t),z(t))$ is
upper semicontinuous, so it is measurable (see [24]). Then due to the
continuity of the function $g_{1}(t)$, the piecewise continuity of the
function $g_{2}(t)$ and due to the continuity of the scalar product in its
variables we obtain that the mapping

$t\rightarrow\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x(t),z(t))}\big{(}\langle
w_{1}(t),g_{1}(t)\rangle+\langle w_{2}(t),g_{2}(t)\rangle\big{)}$ (32)

is upper semicontinuous [29], and then is also measurable [24]. While proving
statement 2) it will be shown that under the assumptions made the set
$\overline{\partial}h^{2}(x,z)$ is bounded uniformly in $t\in[0,T]$, then by
the continuity of the function $g_{1}(t)$ and the piecewise continuity of the
function $g_{2}(t)$ it is easy to check that mapping (32) is also bounded
uniformly in $t\in[0,T]$. So we finally have that mapping (32) belongs to the
space $L^{\infty}_{1}[0,T]$.

Now we prove that the function $\Phi(t,\alpha)$ is dominated by some
integrable function on $[0,T]$ for all sufficiently small $\alpha>0$. As is
shown in the previous paragraph, the second summand in (31) is an integrable
function. So it remains to consider for sufficiently small $\alpha>0$ the
first summand in (31). From the mean value theorem [30] (applied to the
superdifferential) at each $t\in[0,T]$ and at each $\alpha>0$ one has

$\frac{1}{\alpha}\Big{(}h^{2}(x(t)+\alpha g_{1}(t),z(t)+\alpha
g_{2}(t))-h^{2}(x(t),z(t))\Big{)}\in\overline{\partial}h^{2}\big{(}v_{1}(\alpha,t),v_{2}(\alpha_{2},t)\big{)}g(t)$

where $v_{1}(\alpha,t)\in\mathrm{co}\big{\\{}x(t),x(t)+\alpha
g_{1}(t)\big{\\}}$, $v_{2}(\alpha,t)\in\mathrm{co}\big{\\{}z(t),z(t)+\alpha
g_{2}(t)\big{\\}}$. While proving statement 2) it will be shown that under the
assumptions made the set $\overline{\partial}h^{2}(x,z)$ is bounded uniformly
in $t\in[0,T]$, then by the continuity of the function $g_{1}(t)$ and the
piecewise continuity of the function $g_{2}(t)$ the first summand in (31) is
dominated by a piecewise continuous function for all sufficiently small
$\alpha>0$.

In [12] it is shown that

$\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}\left(\frac{1}{2}h^{2}(x,z)\right)}\big{(}\langle
w_{1}(t),g_{1}(t)\rangle+\langle
w_{2}(t),g_{2}(t)\rangle\big{)}dt=\min_{w\in\overline{\partial}\varphi(x,z)}\int_{0}^{T}\langle
w_{1}(t),g_{1}(t)\rangle+\langle w_{2}(t),g_{2}(t)\rangle dt.$ (33)

From relations (30), (33) one obtains expression (28).

Prove statement 2): the corresponding proof may be found in [25].

The theorem is proved. ∎

Let the interval $[0,T]$ may be divided into a finite number of intervals, in
every of which one (several) of the functions
$\big{\\{}f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{k(j)}}(x)\psi_{i}\big{\\}}$,
$i=\overline{1,n}$, $j=\overline{1,r}$, is (are) active. In order for the
point $(x^{*},z^{*})$ to minimize the functional $I(x,z)$, it is necessary to
have

$\overline{\partial}I(x^{*}(t),z^{*}(t))=\\{\bf 0_{2n}\\}$ (34)

at each $t\in[0,T]$, where the expression for the superdifferential
$\overline{\partial}I(x,z)$ is given by formula (23) (where we take formula
(29) for the superdifferential $\overline{\partial}\varphi(x,z)$). If one has
$I(x^{*},z^{*})$, then condition (34) is also sufficient.

##  7\. Constructing the superdifferential descent direction
of the functional $I(x,z)$

In this section we consider only the points $(x,z)$ which do not satisfy
minimum condition in Theorem 6. Our aim here is to find the superdifferential
(or the steepest) descent direction of the functional $I(x,z)$ at the point
$(x,z)$. Denote this direction $G(x,z)$. Herewith $G=[G_{1},G_{2}]\in
L^{2}_{n}[0,T]\times L^{2}_{n}[0,T]$. In order to construct the vector-
function $G(x,z)$, consider the problem

$\max_{w\in\overline{\partial}I(x,z)}||w||^{2}_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}=\max_{w\in\overline{\partial}I(x,z)}\int_{0}^{T}w^{2}(t)dt.$
(35)

Denote $\overline{w}$ the solution of this problem (below we will see that
such a solution exists). The vector-function $\overline{w}$, of course,
depends on the point $(x,z)$ but we omit this dependence in the notation for
brevity. Then one can check that the vector-function

$G\big{(}x(t),z(t),t\big{)}=-\frac{\overline{w}\big{(}x(t),z(t),t\big{)}}{||\overline{w}||_{L^{2}_{2n}[0,T]}}$
(36)

is a superdifferential descent direction of the functional $I(x,z)$ at the
point $(x,z)$ (cf. (43), (44)). Recall that we are seeking the direction
$G(x,z)$ in the case when the point $(x,z)$ does not satisfy minimum condition
(34), so $||\overline{w}||_{L_{2n}^{2}[0,T]}>0$.

Note that we have the equalities

$\frac{\partial I(x,z)}{\partial
G(x,z)}=\min_{w\in\partial\overline{I}(x,z)}\int_{0}^{T}\left\langle
w(t),G(x(t),z(t),t)\right\rangle
dt=\min_{w\in\overline{\partial}I(x,z)}\int_{0}^{T}\left\langle
w(t),\frac{-\overline{w}(t)}{||\overline{w}||_{L_{2n}^{2}[0,T]}}\right\rangle
dt=$
$=\frac{-1}{||\overline{w}||_{L_{2n}^{2}[0,T]}}\left(-\min_{w\in\overline{\partial}I(x,z)}\int_{0}^{T}\left\langle-w(t),{\overline{w}}\right\rangle
dt\right)=\frac{-1}{||\overline{w}||_{L_{2n}^{2}[0,T]}}\max_{w\in\partial\overline{I}(x,z)}\int_{0}^{T}\left\langle
w(t),{\overline{w}(t)}\right\rangle dt=-||\overline{w}||_{L_{2n}^{2}[0,T]}$

which, considering (4) and the inequality
$||\overline{w}||_{L_{2n}^{2}[0,T]}>0$, implies

$I\left((x,z)+\alpha G(x,z)\right)<I(x,z)$ (37)

for sufficiently small $\alpha>0$.

It is easy to check that in this case the solution of problem (35) is such a
selector of the multivalued mapping
$t\rightarrow\overline{\partial}I\big{(}x(t),z(t),t\big{)}$ that maximizes the
distance from zero to the points of the set
$\overline{\partial}I\big{(}x(t),z(t),t)$ at each time moment $t\in[0,T]$. In
other words, to solve problem (35) means to solve the following problem

$\max_{w(t)\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)$ (38)

for each $t\in[0,T]$. Actually, for every $t\in[0,T]$ we have the obvious
inequality

$\max_{w\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)\geq w^{2}(t)$

where $w(t)$ is a measurable selector of the mapping
$t\rightarrow\overline{\partial}I\big{(}x(t),z(t),t\big{)}$ (from (22) we have
$w\in L^{\infty}_{2n}[0,T]$), then we obtain the inequality

$\int_{0}^{T}\max_{w\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)dt\geq\max_{w\in\overline{\partial}I(x,z)}\int_{0}^{T}w^{2}(t)dt.$

Insofar as for every $t\in[0,T]$ we have

$\max_{w\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)\in\Big{\\{}w^{2}(t)\
\big{|}\ w(t)\in\overline{\partial}I(x(t),z(t),t)\Big{\\}}$

as the set $\overline{\partial}I\big{(}x(t),z(t),t\big{)}$ is closed and
bounded at every fixed $t$ by definition of the superdifferential and the
mapping $t\rightarrow\overline{\partial}I\big{(}x(t),z(t),t\big{)}$ is upper
semicontinuous [22] and besides, the norm (and its square) is continuous in
its argument, then due to Filippov lemma [31] there exists such a measurable
selector $\overline{w}(t)$ of the mapping
$t\rightarrow\overline{\partial}I\big{(}x(t),z(t),t\big{)}$ that for every
$t\in[0,T]$ one obtains

$\max_{w\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)=\overline{w}^{2}(t),$

so we have found the element $\overline{w}$ of the set
$\overline{\partial}I(x,z)$ which brings the equality to the previous
inequality. Hence, finally we obtain

$\int_{0}^{T}\max_{w\in\overline{\partial}I(x(t),z(t),t)}w^{2}(t)dt=\max_{w\in\overline{\partial}I(x,z)}\int_{0}^{T}w^{2}(t)dt.$

Problem (38) at each fixed $t\in[0,T]$ is a finite-dimensional problem of
finding the maximal distance from zero to the points of the convex compact
(the superdifferential). This problem can be effectively solved; the next
paragraph describes its solution. In practice one makes a (uniform) partition
of the interval $[0,T]$, and this problem is being solved for every point of
the partition, i. e. one has to calculate
$G\big{(}x(t_{i}),z(t_{i}),t_{i}\big{)}$ where $t_{i}\in[0,T]$,
$i=\overline{1,N}$, are the points of discretization (see notation in Lemma 7
below). Under some natural additional assumption Lemma 7 below guarantees that
the vector-function obtained with the help of piecewise linear interpolation
of the superdifferential descent directions evaluated at every point of such a
partition of the interval $[0,T]$ converges in the space $L^{2}_{2n}[0,T]$ to
the vector-function $G\big{(}x(t),z(t),t\big{)}$ sought when the
discretization rank tends to infinity.

As noted in the previous paragraph, in order to construct the
superdifferential descent direction of the functional $I(x,z)$ at the point
$(x,z)$, it is required to find the maximal distance from zero to the points
of the functional $I(x(t),z(t))$ superdifferential at each moment of time of a
(uniform) partition of the interval $[0,T]$. From formula (23) (see also (25))
we see that the superdifferential $\overline{\partial}I(x(t),z(t))$ is a
convex polyhedron $P(t)\subset~{}R^{2n}$. Herewith, of course, the set $P(t)$
depends on the point $(x,z)$. We will omit this dependence in the notation in
this paragraph for simplicity. It is clear that in this case it is sufficient
to go over all the vertexes $p_{j}(t)$, $j=\overline{1,s}$ (here $s$ is a
number of vertexes of the polyhedron $P(t)$) and choose among the values
$||p_{j}(t)||_{R^{2n}}$ the greatest one. Denote the corresponding vertex
${p}_{\overline{j}}(t)$ ($\overline{j}\in\\{1,\dots,s\\}$), and if there are
several vertexes on which the maximal norm-value is achieved, then choose any
of them. Finally, put $\overline{w}(t)=p_{\overline{j}}(t)$.

In work [25] one lemma is given which, on the one hand, has rather natural for
applications conditions and, on the other hand, guarantees that the function
$L(t)$ obtained via piecewise linear interpolation of the function $v\in
L_{1}^{\infty}[0,T]$ sought converges to this function in the space
$L_{1}^{2}[0,T]$ when the rank of a (uniform) partition of the interval
$[0,T]$ tends to infinity.

Let the function $v\in L_{1}^{\infty}[0,T]$ satisfy the following condition:
for every $\overline{\delta}>0$ the function $v(t)$ is piecewise continuous on
the set $[0,T]$ with the exception of only the finite number of the intervals
$\big{(}\overline{t}_{1}(\overline{\delta}),\overline{t}_{2}(\overline{\delta})\big{)},$
$\dots$,
$\big{(}\overline{t}_{r}(\overline{\delta}),\overline{t}_{r+1}(\overline{\delta})\big{)}$
whose union length does not exceed the value $\overline{\delta}$.

Choose the (uniform) finite splitting $t_{1}=0,t_{2},\dots,t_{N-1},t_{N}=T$ of
the interval $[0,T]$ and calculate the values $v(t_{i})$, $i=\overline{1,N}$,
at these points. Let $L(t)$ be the function obtained via piecewise linear
interpolation with the nodes $(t_{i},v(t_{i}))$, $i=\overline{1,N}$. Then for
every $\varepsilon>0$ there exists such a number $\overline{N}(\varepsilon)$
that for every $N>\overline{N}(\varepsilon)$ one has
$||L-v||^{2}_{L_{1}^{2}[0,T]}\leq\varepsilon$.

At a qualitative level Lemma 7 condition means that the sought function does
not have “too many” points of discontinuity on the interval $[0,T]$. If this
condition is satisfied for the vector-function $\overline{w}(t)$ (what it
natural for applications), then this lemma justifies the approximation of the
vector-function $\overline{w}(t)$ and hence, the approximation of the vector-
function $\overline{G}\big{(}x(t),z(t),t\big{)}$, by the values
$\overline{w}(t_{i})$, $i=\overline{1,N}$, at the separate points of
discretization implemented as described above.

##  8\. On a method for finding the stationary points of the functional
$I(x,z)$

Once the steepest (the superdifferential) descent direction has been
constructed (see the previous section), one can apply some methods (based on
using this direction) of nonsmooth optimization in order to find stationary
points of the functional $I(x,z)$.

The simplest steepest (superdifferential) descent algorithm is used for
numerical simulations of the paper. In order to convey the main ideas, we
firstly consider the convergence of this method for an analogous problem in a
finite-dimensional case (which is of an independent interest); and then turn
to a more general problem considered in this paper.

First consider the problem of minimization of a function which is a minimum of
the finite number of continuously differentiable functions. So let

$\varphi(x)=\min_{i=\overline{1,M}}f_{i}(x),$

where $f_{i}(x)$, $i=\overline{1,M}$, are continuously differentiable
functions on $R^{n}$.

It is known [18] that the function $\varphi(x)$ is differentiable at every
point $x_{0}\in R^{n}$ in any direction $g\in R^{n}$, $||g||_{R^{n}}=1$, and

$\frac{\partial\varphi(x_{0})}{\partial g}=\min_{i\in
R(x_{0})}\left\langle\frac{\partial f_{i}(x_{0})}{\partial
x},g\right\rangle=\min_{w\in\overline{\partial}\varphi(x_{0})}\langle
w,g\rangle,$ (39) $R(x_{0})=\\{i\in\\{1,\dots,M\\}\ |\
f_{i}(x_{0})=\varphi(x_{0})\\},$
$\overline{\partial}\varphi(x_{0})=\mathrm{co}\left\\{\frac{\partial
f_{i}(x_{0})}{\partial x},\ i\in R(x_{0})\right\\}.$

It is also known [18] that in order for the point $x^{*}\in R^{n}$ to minimize
the function $\varphi(x)$, it is necessary to have

$\min_{||g||_{R^{n}}=1}\min_{i\in R(x^{*})}\left\langle\frac{\partial
f_{i}(x^{*})}{\partial x},g\right\rangle\geq 0.$ (40)

Denote

$\Psi(x)=\min_{||g||_{R^{n}}=1}\min_{i\in R(x)}\left\langle\frac{\partial
f_{i}(x)}{\partial x},g\right\rangle.$ (41)

Then necessary minimum condition (40) of the function $\varphi(x)$ at the
point $x^{*}$ may be rewritten as the inequality $\Psi(x^{*})\geq 0$.

From formulas (39), (41) we see that by definition the vector $g_{0}\in
R^{n}$, $||g_{0}||_{R^{n}}=1$, is the steepest descent direction of the
function $\varphi(x)$ at the point $x^{0}$ if we have

$\min_{i\in R(x_{0})}\left\langle\frac{\partial f_{i}(x_{0})}{\partial
x},g_{0}\right\rangle=\Psi(x_{0}).$ (42)

In terms of superdifferential (see (39)) a steepest descent direction [22] is
the vector

$g_{0}=-\frac{w_{0}}{||w_{0}||_{R^{n}}}$ (43)

where $w_{0}$ is a solution of the problem

$\max_{w\in\overline{\partial}\varphi(x_{0})}||w||=||w_{0}||.$ (44)

Apply the superdifferential (the steepest) descent method to the function
$\varphi(x)$ minimization. Describe the algorithm as applied to the function
and the space under consideration. Fix an arbitrary initial point $x_{1}\in
R^{n}$. Suppose that the set

$lev_{\varphi(x_{1})}\varphi(\cdot)=\\{x\in R^{n}\ |\
\varphi(x)\leq\varphi(x_{1})\\}$

is bounded (due to the arbitrariness of the initial point, in fact, one must
assume that the set $lev_{\varphi(x_{1})}\varphi(\cdot)$ is bounded for every
initial point taken). Due to the function $\varphi(x)$ continuity [22] the set
$lev_{\varphi(x_{1})}\varphi(\cdot)$ is also closed. Let the point $x_{k}\in
R^{n}$ be already constructed. If $\Psi(x_{k})\geq 0$ (in practice we check
that this condition is satisfied only with some fixed accuracy
$\overline{\varepsilon}$, i. e. $\Psi(x_{k})\geq-\overline{\varepsilon}$),
then the point $x_{k}$ is a stationary point of the function $\varphi(x)$ and
the process terminates. Otherwise, construct the next point according to the
rule

$x_{k+1}=x_{k}+\alpha_{k}g_{k}$

where the vector $g_{k}$ is a steepest (superdifferential) descent of the
function $\varphi(x)$ at the point $x_{k}$ (see (43), (44)) and the value
$\alpha_{k}$ is a solution of the following one-dimensional minimization
problem

$\min_{\alpha\geq 0}\varphi(x_{k}+\alpha
g_{k})=\varphi(x_{k}+\alpha_{k}g_{k}).$

Then $\varphi(x_{k+1})<\varphi(x_{k})$.

Under the assumptions made one has the inequality

$\underline{\lim}_{k\rightarrow\infty}\Psi(x_{k})\geq 0$ (45)

for the sequence built according to the algorithm above.

###### Proof.

The following proof is partially based on the ideas of an analogous one in
book [18].

Assume the contrary. Then there exist such a subsequence
$\\{x_{k_{j}}\\}_{j=1}^{\infty}$ and such a number $b>0$ that for each
$j\in\mathcal{N}$ we have the inequality

$\Psi(x_{k_{j}})\leq-b.$ (46)

Note that for every $j\in\mathcal{N}$ the set $R(x_{k_{j}})$ is nonempty.

From the steepest descent direction definition (see (42)) it follows that for
every $j\in\mathcal{N}$ there exists an index $\overline{i}\in R(x_{k_{j}})$
such that for each $\alpha>0$ the relation

$f_{\overline{i}}(x_{k_{j}}+\alpha
g_{k_{j}})=f_{\overline{i}}(x_{k_{j}})+\alpha\Psi(x_{k_{j}})+o_{\overline{i}}(x_{k_{j}},g_{k_{j}},\alpha)$
(47)

holds true.

Recall that $f_{\overline{i}}(x_{k_{j}})=\varphi(x_{k_{j}})$ for each index
$\overline{i}$ from the set $R(x_{k_{j}})$ by this index set definition. Then
from inequality (46) since $f_{\overline{i}}(x)$ is a continuously
differentiable function by assumption, there exists [18] such
$\overline{\alpha}>0$, which does not depend on the number $k_{j}$, such that
for $\overline{i}\in R(x_{k_{j}})$ satisfying (47) and for each
$\alpha\in(0,\overline{\alpha}]$ one has the inequality

$f_{\overline{i}}(x_{k_{j}}+\alpha
g_{k_{j}})\leq\varphi(x_{k_{j}})-\frac{1}{2}\alpha b.$

Using the fact that $R(x_{k_{j}})\neq\emptyset$ for all $j\in\mathcal{N}$, we
finally have

$\varphi(x_{k_{j}}+\overline{\alpha}g_{k_{j}})\leq\varphi(x_{k_{j}})-\frac{1}{2}\overline{\alpha}b=\varphi(x_{k_{j}})-\frac{1}{2}\beta$
(48)

uniformly in $j\in\mathcal{N}$.

This inequality leads to a contradiction. Indeed, the sequence
$\\{\varphi(x_{k})\\}_{k=1}^{\infty}$ is monotonically decreasing and bounded
below by the number $\min\limits_{x\in
lev_{\varphi(x_{(1)})}\varphi(\cdot)}\varphi(x)$, hence it has a limit:

$\\{\varphi(x_{k})\\}\rightarrow\varphi^{*}\ \mathrm{at}\ k\rightarrow\infty,$

herewith, at each $k\in\mathcal{N}$ one has

$\varphi(x_{k})\geq\varphi^{*}.$ (49)

Now choose such a big number $\overline{j}$ that

$\varphi(x_{k_{\overline{j}}})<\varphi^{*}+\frac{1}{2}\beta.$

Due to (48) we have

$\varphi(x_{k_{\overline{j}}+1})\leq\varphi(x_{k_{\overline{j}}}+\overline{\alpha}g_{k_{\overline{j}}})\leq\varphi^{*}-\frac{1}{2}\beta$

what contradicts (49). ∎

Now turn back to the problem of functional $I(x,z)$ minimization. Denote (see
formulas (35), (36))

$\Psi(x,z)=\min_{||g||_{L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]}=1}\frac{\partial
I(x,z)}{\partial g}=\frac{\partial I(x,z)}{\partial G}.$ (50)

Then necessary minimum condition of the functional $I(x,z)$ at the point
$(x^{*},z^{*})$ may be written [32] as the inequality $\Psi(x^{*},z^{*})\geq
0$.

Apply the superdifferential (the steepest) descent method to the functional
$I(x,z)$ minimization. Describe the algorithm as applied to the functional and
the space under consideration. Fix an arbitrary initial point
$(x_{(1)},z_{(1)})\in C_{n}[0,T]\times P_{n}[0,T]$. Suppose that the set

$lev_{I(x_{(1)},z_{(1)})}I(\cdot,\cdot)=\left\\{(x,z)\in C_{n}[0,T]\times
P_{n}[0,T]\ |\ I(x,z)\leq I(x_{(1)},z_{(1)})\right\\}$

is bounded in $L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]$-norm (due to the
arbitrariness of the initial point, in fact, one must assume that the set
$lev_{I(x_{(1)},z_{(1)})}I(\cdot,\cdot)$ is bounded for every initial point
taken). Let the point $(x_{(k)},z_{(k)})\in C_{n}[0,T]\times P_{n}[0,T]$ be
already constructed. If $\Psi(x_{(k)},z_{(k)})\geq 0$ (in practice we check
that this condition is satisfied only with some fixed accuracy
$\overline{\varepsilon}$, i. e.
$\Psi(x_{(k)},z_{(k)})\geq-\overline{\varepsilon}$) (in other words, if
minimum condition (34) is satisfied (in practice with some fixed accuracy
$\overline{\varepsilon}$, i. e.
$||\overline{w}\big{(}x_{k}(t),z_{k}(t),t\big{)}||_{L_{2n}^{2}[0,T]}\leq\overline{\varepsilon}$)),
then the point $(x_{(k)},z_{(k)})$ is a stationary point of the functional
$I(x,z)$ and the process terminates. Otherwise, construct the next point
according to the following rule

$(x_{(k+1)},z_{(k+1)})=(x_{(k)},z_{(k)})+\alpha_{(k)}G\big{(}x_{(k)},z_{(k)}\big{)}$

where the vector-function $G\big{(}x_{(k)},z_{(k)}\big{)}$ is a
superdifferential descent direction of the functional $I(x,z)$ at the point
$(x_{(k)},z_{(k)})$ (see (35), (36)), and the value $\alpha_{(k)}$ is a
solution of the following one-dimensional minimization problem

$\min_{\alpha\geq 0}I\Big{(}(x_{(k)},z_{(k)})+\alpha
G\big{(}x_{(k)},z_{(k)}\big{)}\Big{)}=I\Big{(}(x_{(k)},z_{(k)})+\alpha_{(k)}G\big{(}x_{(k)},z_{(k)}\big{)}\Big{)}.$

Then according to (37) one has

$I\big{(}x_{(k+1)},z_{(k+1)}\big{)}<I\big{(}x_{(k)},z_{(k)}\big{)}.$

Introduce now the set family $\mathcal{I}$. At first define the functional
$I_{q}$, $q=\overline{1,\left(\prod_{j=1}^{r}{k(j)}\right)^{n}}$ as follows.
Its integrand is the same, as the functional $I$ one, but the maximum function
$\max\big{\\{}f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{k(j)}}(x)\psi_{i}\big{\\}}$,
$j=\overline{1,r}$, is substituted for each $i=\overline{1,n}$ by only the one
of the functions $f_{i,j_{1}}\psi_{i},\dots,f_{i,j_{k(j)}}\psi_{i}$,
$j\in\\{1,\dots,r\\}$. Let the family $\mathcal{I}$ consist of the sums of the
integrals over the intervals of the time interval $[0,T]$ splitting for all
possible such finite splittings. Herewith, the integrand of each summand in
the sum taken is the same, as some functional $I_{q}$ one,
$q\in\left\\{1,\dots,\left(\prod_{j=1}^{r}{k(j)}\right)^{n}\right\\}$.

Let for every point constructed by the method described the following
assumption is valid: the interval $[0,T]$ may be divided into a finite number
of intervals, in every of which for each $i=\overline{1,n}$ either
$h_{i}(x_{(k)},z_{(k)})=0$, or one (several) of the functions
$\left\langle-\overline{a}_{ij}\psi_{i}^{*}\frac{\partial
f_{i,j_{p}}(x_{(k)})}{\partial x},G_{1}(x_{(k)},z_{(k)})\right\rangle$,
$j=\overline{1,r}$, $p=\overline{1,k(j)}$, is (are) active.

Let us illustrate this assumption by an example. Consider the following
simplest functional (whose structure, however, preserves the basic features of
the general case)

$\int_{0}^{1}\frac{1}{2}\mathrm{min}^{2}\\{x(t)+1,-x(t)+1\\}dt.$

Consider the point $x_{(1)}=0$ (i. e. $x_{(1)}(t)=0$ for all $t\in[0,1]$). In
order to find the steepest (the superdifferential) descent direction of the
functional at this point one has, according to the theory described, to
minimize the directional derivative (calculated at this point), i. e. to find
such a function $G\in L_{1}^{2}[0,T]$,
$\displaystyle{\int_{0}^{1}G^{2}(t)}dt=1$, that minimizes the functional

$\int_{0}^{1}\min\\{x_{(1)}(t)+1,-x_{(1)}(t)+1\\}\min\\{g(t),-g(t)\\}dt.$

Here $g\in L_{1}^{2}[0,T]$, $\displaystyle{\int_{0}^{1}g^{2}(t)}dt=1$. Take

$G(t)=\left\\{\begin{array}[]{ll}-1,&\text{if}\ t\in[0,0.5],\\\ 1,&\text{if}\
t\in(0.5,1]\end{array}\right.$

as one of obvious solutions. Herewith,
$\Psi(x_{(1)})=\displaystyle\int_{0}^{1}-1dt=-1$. We see that the assumption
made is satisfied. Take

$\hat{I}(x)=\int_{0}^{0.5}\frac{1}{2}(x(t)+1)^{2}dt+\int_{0.5}^{1}\frac{1}{2}(-x(t)+1)^{2}dt.$

Then we have

$\hat{I}\left(x_{(1)}+\alpha
G(x_{(1)})\right)=\hat{I}(x_{(1)})+\alpha\Psi(x_{(1)})+o(x_{(1)},G(x_{(1)}),\alpha)$

since

$\nabla\hat{I}(x_{(1)})=\left\\{\begin{array}[]{ll}1,&\text{if}\
t\in[0,0.5],\\\ -1,&\text{if}\ t\in(0.5,1],\end{array}\right.$

i. e. $\nabla\hat{I}(x_{(1)})G(x_{(1)})=\displaystyle\int_{0}^{1}-1dt=-1$. It
is obvious that $\hat{I}\in\mathcal{I}$. Note that with $\alpha_{(1)}=1$ one
gets the point

$x_{(2)}(t)=\left\\{\begin{array}[]{ll}-1,&\text{if}\ t\in[0,0.5],\\\
1,&\text{if}\ t\in(0.5,1],\end{array}\right.$

which delivers the global minimum to the functional considered.

For functionals from the family $\mathcal{I}$ we make the following additional
assumption. Let there exists such a finite number $L$ that for every
$\hat{I}\in\mathcal{I}$ and for all
$\overline{x},\overline{z},\overline{\overline{x}},\overline{\overline{z}}$
from a ball with the center in the origin and with some finite radius
$R^{\prime}+\hat{\alpha}$ (here $R^{\prime}>\sup\limits_{(x,z)\in
lev_{I(x_{(1)},z_{(1)})}I(\cdot,\cdot)}||(x,z)||_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}$ and $\hat{\alpha}$ is some positive number) one has

$||\nabla\hat{I}(\overline{x},\overline{z})-\nabla\hat{I}(\overline{\overline{x}},\overline{\overline{z}})||_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}\leq
L||({\overline{x}},{\overline{z}})^{\prime}-(\overline{\overline{x}},\overline{\overline{z}})^{\prime}||_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}.$ (51)

At first glance it may seem that the Lipschitz constant $L$ existence for all
$\hat{I}\in\mathcal{I}$ simultaneously in the assumption made is too
burdensome. However, if one remembers that on each of the finite number of the
interval $[0,T]$ segments the functional $\hat{I}\in\mathcal{I}$ integrand
coincides with the integrand of the functional $I_{q}$,
$q\in\left\\{1,\dots,\left(\prod_{j=1}^{r}{k(j)}\right)^{n}\right\\}$, by
construction (see the set $\mathcal{I}$ definition), then this assumption is
natural if we suppose the Lipschitz-continuity of every of the gradients
$\nabla I_{q}$, $q=\overline{1,\left(\prod_{j=1}^{r}{k(j)}\right)^{n}}$; and
this gradient Lipschitz-continuity condition is a common assumption for
justifying classical optimization methods for differentiable functionals.

Let condition (51) be satisfied. Then for each functional $I\in\mathcal{I}$
and for all $(x,z)\in lev_{I(x_{(1)},z_{(1)})}I(\cdot,\cdot)$, $G\in
C_{n}[0,T]\times P_{n}[0,T]$, $||G||_{L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]}=1$,
$\alpha\in R$, $0<\alpha\leq\hat{\alpha}$ one has the inequality

$I\left((x,z)+\alpha G\right)\leq I(x,z)+\alpha\left\langle\nabla
I(x,z),G\right\rangle+\alpha^{2}\frac{L}{2}.$

###### Proof.

The proof can be carried out with obvious modifications in a similar way as
for the analogous statement in [33]. ∎

We suppose that during the method realization for each $k\in\mathcal{N}$ one
has $\alpha_{(k)}<\hat{\alpha}$ where $\hat{\alpha}$ is a number from Lemma 8
(see also the assumption before Remark 8).

Under the assumptions made one has the inequality

$\underline{\lim}_{k\rightarrow\infty}\Psi(x_{k},z_{k})\geq 0$ (52)

for the sequence built according to the rule above.

###### Proof.

Assume the contrary. Then there exist such a subsequence
$\\{(x_{k_{j}},z_{k_{j}})\\}_{j=1}^{\infty}$ and such a number $b>0$ that for
each $j\in\mathcal{N}$ we have the inequality

$\Psi(x_{k_{j}},z_{k_{j}})\leq-b.$ (53)

From the steepest descent direction (see (50)) and the set $\mathcal{I}$
definitions (see also (27)) it follows that for every $j\in\mathcal{N}$ there
exists a functional $\hat{I}$ from the family $\mathcal{I}$ such that for each
$\alpha>0$ the relation

$\hat{I}\left((x_{k_{j}},z_{k_{j}})+\alpha
G(x_{k_{j}},z_{k_{j}})\right)=\hat{I}(x_{k_{j}},z_{k_{j}})+\alpha\Psi(x_{k_{j}},z_{k_{j}})+o\left((x_{k_{j}},z_{k_{j}}),G(x_{k_{j}},z_{k_{j}}),\alpha\right)$
(54)

holds; herewith,

$\Psi(x_{k_{j}},z_{k_{j}})=\left\langle\nabla\hat{I}(x_{k_{j}},z_{k_{j}}),G(x_{k_{j}},z_{k_{j}})\right\rangle.$

Recall that $\hat{I}(x_{k_{j}},z_{k_{j}})=I(x_{k_{j}},z_{k_{j}})$ for each
functional $\hat{I}$ from the family $\mathcal{I}$ by this set definition.
Then from inequality (53) by virtue of Lemma 8 there exists
$\overline{\alpha}>0$, which does not depend on the number $k_{j}$, such that
for $\hat{I}\in\mathcal{I}$ satisfying (54) and for each
$\alpha\in(0,\overline{\alpha}]$ one has the inequality

$\hat{I}\left((x_{k_{j}},z_{k_{j}})+\alpha G(x_{k_{j}},z_{k_{j}})\right)\leq
I(x_{k_{j}},z_{k_{j}})-\frac{1}{2}\alpha b.$

Using the set $\mathcal{I}$ definition, we finally have

$I\left((x_{k_{j}},z_{k_{j}})+\alpha G(x_{k_{j}},z_{k_{j}})\right)\leq
I(x_{k_{j}},z_{k_{j}})-\frac{1}{2}\overline{\alpha}b=I(x_{k_{j}},z_{k_{j}})-\frac{1}{2}\beta$
(55)

uniformly in $j\in\mathcal{N}$.

This inequality leads to a contradiction. Indeed, the sequence
$\left\\{I(x_{(k)},z_{(k)})\right\\}_{k=1}^{\infty}$ is monotonically
decreasing and bounded below by zero (recall that the functional $I(x,z)$ is
nonnegative by construction), hence, it has a limit:

$\left\\{I(x_{(k)},z_{(k)})\right\\}\rightarrow I^{*}\ \mathrm{at}\
k\rightarrow\infty,$ (56)

herewith, at each $k\in\mathcal{N}$ one has $\\{I(x_{(k)},z_{(k)})\\}\geq
I^{*}$.

Now choose such a big number $\overline{j}$ that

$I(x_{(k_{\overline{j}})},z_{(k_{\overline{j}})})<I^{*}+\frac{1}{2}\beta.$

Due to (55) we have

$I(x_{(k_{\overline{j}}+1)},z_{(k_{\overline{j}}+1)})\leq
I(x_{(k_{\overline{j}})},z_{(k_{\overline{j}})})+\overline{\alpha}G_{k_{\overline{j}}})\leq
I^{*}-\frac{1}{2}\beta$

what contradicts (56). ∎

It is easy to show that, in fact, in formulas (45), (52) the lower limit can
be substituted by the “ordinary” limit and the inequality can be substituted
by the equality in the cases considered.

##  9\. Numerical examples

In this section we give two illustrative examples of the described algorithm
implementation.

Consider the differential inclusion

$\dot{x}_{1}\in x_{2}+[-2,2](|x_{1}|+|x_{2}|),\quad\dot{x}_{2}\in
x_{1}+[-2,2](|x_{1}|+|x_{2}|)$

on the time interval $[0,1]$ with the boundary conditions

$x_{1}(0)=0,\,x_{2}(0)=1,\quad x_{1}(1)=0,\,x_{2}(1)=2.$

As the phase constraint surface put

$s(x)=x_{1}=0.$

Take $(x_{(1)},z_{(1)})=(0,0,0,0)^{\prime}$ as the first approximation (i. e.
all the functions considered are identically equal to zero on the interval
$[0,1]$), then $I(x_{(1)},z_{(1)})=1$. At the end of the process the
discretization step was equal to $2\times 10^{-1}$. Figure 1 illustrates the
trajectories obtained. From the picture we see that the differential inclusion
is satisfied. The trajectory lies on the surface required as well. The
boundary values error doesn’t exceed the magnitude $5\times 10^{-3}$. In order
to obtain such an accuracy 5 iterations has been required. The functional
value on the trajectory obtained is approximately $3\times 10^{-5}$.

Figure 1. Example 1

Consider the differential inclusion

$\dot{x}_{1}\in{\mathrm{co}\\{x_{1},x_{2},x_{3}\\}},\quad\dot{x}_{2}\in{\mathrm{co}\\{-2x_{1},x_{3}\\}},\quad\dot{x}_{3}=x_{1}+x_{2}$

on the time interval $[0,1]$ with the boundary conditions

$x_{1}(0)=-1,\,x_{2}(0)=-1,\,x_{3}(0)=2.5,\quad
x_{1}(1)=1,\,x_{2}(1)=1,\,x_{3}(1)=2.5.$

As the phase constraint surface put

$s(x)=x_{1}-x_{2}=0.$

Take $(x_{(1)},z_{(1)})=(0,0,0,0,0,0)^{\prime}$ as the first approximation (i.
e. all the functions considered are identically equal to zero on the interval
$[0,1]$), then $I(x_{(1)},z_{(1)})=8.125$. At the end of the process the
discretization step was equal to $10^{-1}$. Figure 2 illustrates the
trajectories obtained. From the picture we see that the differential inclusion
is satisfied. The trajectory lies on the surface required as well. The
boundary values error doesn’t exceed the magnitude $2\times 10^{-3}$. In order
to obtain such an accuracy 66 iterations has been required. The functional
value on the trajectory obtained is approximately $2\times 10^{-5}$.

Note that we have intentionally taken such a constraint surface in order to
make the trajectories $x_{1}(t)$ and $x_{2}(t)$ coincide. So if on some
iterations the trajectory already lies on the surface $s(x)=0$ and
$h_{1}(x,z)>0$, then both of these trajectories would be active on some
nonzero time interval (if $x_{1}(t)=x_{2}(t)>x_{3}(t)$ on this interval). That
would lead to calculating the “full-fledged” superdifferential (not the one
that degenerates into a single point) on these iterations.

Figure 2. Example 2

The main results of this paper (Sections 6, 7, 8, 9) were obtained in IPME RAS
and supported by Russian Science Foundation (grant 20-71-10032).

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