Fominyh

*A. V. Fominyh, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia.

# A method for solving discontinuous systems with sliding modes ††thanks: The
main results of this paper (sections 4–6) were obtained in IPME RAS and
supported by Russian Science Foundation (project no. 20-71-10032).

A. V. Fominyh  Institute for Problems in Mechanical Engineering, Russian
Academy of Sciences, Saint Petersburg, Russia  Faculty of Applied Mathematics
and Control Processes, Saint Petersburg State University, Saint Petersburg,
Russia<EMAIL_ADDRESS>
(26 April 2016; 6 June 2016; 6 June 2016)

###### Abstract

[Summary]The paper deals with systems of ordinary differential equations
containing in the right-hand side controls which are discontinuous in phase
variables. These controls cause the occurrence of sliding modes. If one uses
one of the well-known definitions of the solution of discontinuous systems,
then the motion of an object in a sliding mode can be described in terms of
differential inclusions. With the help of the previously developed apparatus
for solving differential inclusions, a method is constructed for finding the
trajectories of a system moving in a sliding mode. Since some of frequently
used discontinuous controls contain nonsmooth functions of phase variables,
the paper pays special attention to study the differential properties of such
systems. At the end of the paper controls of a slightly different, in contrast
to the classical, type are considered which have useful differential
properties, and a method is constructed for solving systems with such controls
considered both before hitting the discontinuity surface and moving in its
vicinity.

###### keywords:

sliding mode, discontinuous system, differential inclusion,
G${\rm\hat{a}}$teaux gradient, support function, gradient descent method

††articletype: Research Article

## Introduction

In many practical problems, when trying to construct mathematical models of
real physical processes, discontinuous control is used; therefore, the right-
hand sides of the systems of differential equations describing the process
under study are discontinuous functions of the state vector. For example, the
control may be an $m$-dimensional vector function of the following form

$u_{i}(x,t)=\left\\{\begin{array}[]{ll}u_{i}^{+}(x,t),\ s_{i}(x)>0,\\\
u_{i}^{-}(x,t),\ s_{i}(x)<0,\end{array}\right.$ (1)

where $t$ belongs to the time interval on which the motion of the system is
considered, the functions $u_{i}^{+}(x,t)$, $u_{i}^{-}(x,t)$,
$i=\overline{1,m}$, are continuous, and $s_{i}(x)=0$ are discontinuity
surfaces ($s_{i}(x)$, $i=\overline{1,m}$, are some continuous functions of the
phase vector $x$). Put $s(x)=(s_{1}(x),\dots s_{m}(x))^{\prime}$. Those cases,
in which the discontinuity points are isolated, belong to classical control
theory and have been studied more fully than the cases where the set of
discontinuity points constitutes a set of nonzero measure in time.

In such systems the state vector may stay on one of the discontinuity surfaces
(or on some their intersection) over a period of time of nonzero measure.
Motion along the discontinuity surfaces or along some their intersection is
called a sliding mode. Since the system trajectories belonging to the set of
discontinuity points do not coincide with any of the system trajectories
resulting from various combinations of continuous controls $u_{i}^{+}(x,t)$,
$u_{i}^{-}(x,t)$, $i=\overline{1,m}$, they are singular. Therefore it is
required to introduce a corresponding definition in case when the sliding mode
occurs. Let us give one of the classical variants of such a definition 1,
which will be used in this paper. On the finite time interval $[0,T]$ consider
the system $\dot{x}=f(x,u_{1}(x,t),\dots,u_{r}(x,t),t)$, 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 convex. If the control is of form (1), then it is natural to
consider $U_{i}(x,t)=\mathrm{co}\\{u_{i}^{-}(x,t),u_{i}^{+}(x,t)\\}$,
$i=\overline{1,m}$, as such sets. Note that there exist some other common
definitions of discontinuous systems solutions 1, 2.

It was noticed 1, 2 that when the system state is in a sliding mode, new
useful properties are observed that are absent when the system moves in a
“normal” mode. For example, it is precisely such motions that are optimal in
the sense of some criterion in the optimal control theory. The artificial
introduction of a sliding mode into the system is also used in order to
stabilize it, as well as to get rid of unwanted disturbances acting on the
object under consideration. The sliding modes have numerous applications. For
example, in work 3 a sliding mode is used for trajectory tracking of
autonomous surface vessels. In paper 4 a sliding mode was used in order to
stabilize a wind turbine in spite of model uncertainties. Herewith, in the
most of the papers listed both first-order and second-order sliding surfaces
are implemented. It was noted that higher-order sliding modes preserve or
generalize the main properties of the standard ones and remove dangerous
chattering effect, so they are also widely used in the literature (see, e. g.,
5, 6). In some papers (7, 8) sliding modes were applied to discrete systems;
herewith the control designs were developed avoiding undesirable chattering
and high-frequency switching between different values of the control signal.
Note also paper 9 where a nontrivial and an effective control design is
constructed for a class of underactuated systems. Some of the papers aim at
constructing a continuous control (see section 6), for example, 10 (in
application to rigid robots), 11 (for the missile-interceptor guidance system
against targets performing evasive maneuvers) which reduce control efforts in
the transient state.

Therefore, it makes sense to pose the problem of choosing a control that after
a finite time interval transfers the system trajectory from an arbitrary
initial state to a small neighborhood of the discontinuity surface in which
the system continues to move for the rest of time. This is the essence of
providing the stability of the sliding mode of the system. Moreover, if one
additionally requires the fulfillment of the condition $s(x(t))\rightarrow{\bf
0_{m}}$ at $t\rightarrow\infty$ (where $s(x)={\bf 0_{m}}$ is a discontinuity
surface), then it is the stability “in big”. The given definition is of an
quality nature; a strict definition of the sliding mode stability may be found
in 2. A very powerful tool for studying the stability of sliding modes is the
theory of Lyapunov and Barbashin–Krasovskii which originates from classical
stability theory. In this paper a feedback control will be constructed for the
class of systems under consideration that endows the Lyapunov function with
the required properties, thereby ensuring the stability of sliding modes.

As one will see from the strict statement of the problem, the system motion
along the discontinuity surface can be described by some interval system. As
is shown in work 12, an interval system may be written down in the form of a
differential inclusion of a certain structure. So, if we use the above
definition of the discontinuous system solution, then after hitting the
discontinuity surface (we recall that the control used sets itself this goal,
as well as the goal of keeping the trajectory on this surface), the system
trajectory motion is described by a differential inclusion.

This paper aims at the problem of finding a solution of a differential
inclusion describing the system motion in a sliding mode. Herewith, we will
additionally impose some restrictions on the desired trajectory, for example,
hitting a certain point at the final moment of time (when moving along the
discontinuity surface). For solving this problem we will use the previously
developed apparatus for solving interval systems considered as differential
inclusions 12, 13, 14. As will be seen below, this inclusion in some cases has
the following feature: due to the special control structure the right-hand
sides of the differential inclusion can be nonsmooth functions of the phase
coordinates. Therefore, the paper also discusses the conditions for the
appearance of nonsmoothness in the right-hand side of a differential inclusion
and investigates the differential properties of such systems. Finally, in the
final part of the paper we discuss a somewhat different, in contrast to the
classical, feedback control structure which allows one to preserve some useful
differential properties of the right-hand side of the system, while at the
same time (just like the classical control type) ensuring that the system gets
into a small enough neighborhood of the discontinuity surface; although in the
general case it does not ensure the stability “in big” of the sliding mode.
The control structure proposed allows one to develop a method for searching
for the system trajectory moving both in the vicinity of the discontinuity
surface and before hitting it. Sometimes, in order to fulfill some
restrictions on the trajectory, it is necessary to choose an appropriate
structure for the sliding mode itself; this problem is also being solved in
this paper.

##  1 Basic definitions and notations

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

In the paper we assume that each trajectory $x(t)$ is a piecewise continuously
differentiable vector-function with bounded in its domain derivative. Let
$t_{0}\in[0,T)$ be a point of nondifferentiability of the vector-function
$x(t)$, then for definiteness we assume that $\dot{x}(t_{0})$ is a right-hand
derivative of the vector-function $x(t)$ at the point $t_{0}$. Similarly, we
assume that $\dot{x}(T)$ is a left-hand derivative of the vector-function
$x(t)$ at the point $T$. With the assumptions and the notations made 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 the arbitrary set $F\subset R^{n}$ let us define a 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
$a,b\in R^{n}$. Let also $S_{n}$ be a unit sphere in $R^{n}$ with the center
in the origin, let $B_{r}(c)$ be a ball with the radius $r\in R$ and the
center $c\in R^{n}$, and the vectors $\bf{e_{i}}$, $i=\overline{1,n}$, form
the standard basis in $R^{n}$. $0_{n}$ denotes a zero element of a functional
space of some $n$-dimensional vector-functions, and $\bf{0_{n}}$ — a zero
element of the space $R^{n}$. Let $E_{m}$ denote an identity matrix and
${\bf{O}_{m}}$ — a zero matrix in the space $R^{m}\times R^{m}$, let also
$\mathrm{diag[P,Q]}$ denote a diagonal matrix with the blocks $P$ and $Q$
(where $P$ and $Q$ are the matrices of some dimensions). Denote
$|x|=\sum_{i=1}^{n}|x_{i}|$, where $x\in R^{n}$.

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.

######  Definition 1.1.

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

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

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 (2) 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 for each number $\varepsilon>0$ there exist such numbers $\delta>0$ and
$\alpha_{0}>0$ that at $\overline{d}\in B_{\delta}(d)$ and at
$\alpha\in(0,\alpha_{0})$ one has
$|o(\alpha,x,z,\overline{d})|<\alpha\varepsilon$, then the function $h(x,z)$
is called uniformly superdifferentiable at the point $(x,z)$. Note 16 that if
the function $h(x,z)$ is superdifferentiable at the point $(x,z)$ and is
locally Lipschitz continuous in the vicinity of the point $(x,z)$, then it is
uniformly superdifferentiable at the point $(x,z)$.

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

$\overline{\partial}\varsigma(\xi_{0})=\\{\varsigma^{\prime}(\xi_{0})\\},$ (3)

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

$\overline{\partial}\varsigma(\xi_{0})=\sum_{k=1}^{r}\overline{\partial}\varsigma_{k}(\xi_{0}).$
(4)

######  Definition 1.2.

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

$\frac{\partial I(x)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}I(x+\alpha
g)-I(x)\big{)}=\min_{w\in\overline{\partial}I(x)}w(g).$ (5)

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

From expression (5) one can see that the following formula

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

holds true.

######  Remark 1.3.

Note the following fact. Since, as is known, the space
$\big{(}C_{n}[0,T],||\cdot||_{L_{2}^{n}[0,T]}\big{)}$ is everywhere dense in
the space $L_{2}^{n}[0,T]$, then the space
$\big{(}C_{n}[0,T],||\cdot||_{L_{2}^{n}[0,T]}\big{)}^{*}$ is isometrically
isomorphic to the space $L_{2}^{n}[0,T]$ 17; therefore, henceforth, we will
identify these spaces
($\big{(}C_{n}[0,T],||\cdot||_{L_{2}^{n}[0,T]}\big{)}^{*}$ and
$L_{2}^{n}[0,T]$).

##  2 Statement of the problem

Consider the system of differential equations

$\dot{x}=Ax+Bu$ (6)

with the initial point

$x(0)=x_{0}$ (7)

and with the desired endpoint

$x_{j}(T)={x_{T}}_{j}.$ (8)

In formula (6) $A$ is a constant $n\times n$ matrix, $B$ is a constant
$n\times m$ matrix. For simplicity we suppose that
$B=\mathrm{diag}[E_{m},{\bf{O}_{n-m}}]$. 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). Then by section 1 assumption
$x(t)$ is an $n$-dimensional continuous vector-function of phase coordinates
with a piecewise-continuous and bounded on $[-t^{*},T]$ derivative; the
structure of the $m$-dimensional control $u$ will be specified below. In
formula (7) $x_{0}\in R^{n}$ is a given vector; in formula (8) ${x_{T}}_{j}$
are given numbers, corresponding to those coordinates of the state vector,
which are fixed at the right endpoint, here $j\in J\subset\\{1..n\\}$, where
$J$ is a given index set.

Let also the discontinuity surface

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

be given, where $s(x,c)$ is a continuously differentiable vector-function. We
will assume that the general structure of the surface is given while the
parameters $c\in R^{\ell}$ are unknown. We will sometimes omit the dependence
of the surface vector-function on these variables for convenience of notation.

In practice, the general structure of the discontinuity surface is usually
known based on some physical considerations; one also can approximately know
the parameters of this surface. So it makes sense to “correct” the parameters
of this surface in such a way that the object is endowed with the desired
properties (for example, the desired condition on the right endpoint).

Let us immediately write out the explicit form of the control which is mainly
used in this paper. Let

$u_{i}=-\alpha_{i}|x|\mathrm{sign}(s_{i}(x)),$ (10)

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

In book 2 it is shown that under natural assumptions and with sufficiently
large values of the factors $\alpha_{i}$, $i=\overline{1,m}$, controls (10)
ensure system (6) hitting a small vicinity of discontinuity surface (9) from
arbitrary initial state (7) 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 (10) ensure the stability “in big” of the system (6) sliding mode. In
2 one may also find the estimates on the time moment $t^{*}$. Here we assume
that all the conditions required are already met, i. e. the numbers
$\underline{a}_{i}$, $\overline{a}_{i}$, $i=\overline{1,m}$, are taken
sufficiently large. In sections 3-5 we will be interested in the behavior of
the system on the discontinuity surface (on the time interval $[0,T]$).
Section 6 at the end of the paper also discusses the search for the trajectory
of the system both on the discontinuity surface and before hitting it;
herewith, a different (from that given in the formula (10)) control structure
will be used.

On the surfaces $s_{i}(x)=0$, $i=\overline{1,m}$, the solution of the original
discontinuous system (accordingly to the definition given in Introduction) is
a solution of the following differential inclusion:

$\dot{x}_{i}\in
A_{i}x+[\underline{a}_{i},\overline{a}_{i}]|x|[-1,1]=A_{i}x+[-\overline{a}_{i},\overline{a}_{i}]|x|,\quad
i=\overline{1,m}.$ (11) $\dot{x}_{i}=A_{i}x,\quad i=\overline{m+1,n}.$ (12)

In formulas (11), (12), $A_{i}$ is the $i$-th row of the matrix $A$,
$i=\overline{1,n}$.

######  Remark 2.1.

It is clear that in practice we believe that the trajectory is on the
discontinuity surface if it lies in a given sufficiently small neighborhood of
this surface (which can be determined in each specific case based on the
requirements for the accuracy of calculations and modeling of the process
described by the system).

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 (11); herewith $f_{i}(x)=F_{i}(x):=A_{i}x$,
$i=\overline{m+1,n}$ (see (12)), so we can rewrite the given inclusions in the
form

$\dot{x}\in F(x).$ (13)

######  Problem 2.2.

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 discontinuity surface (9) (the parameters $c^{*}\in R^{\ell}$ are
to be determined as well) while $t\in[0,T]$, satisfies differential inclusion
(13) and boundary conditions (7), (8). Assume that there exists such a
solution.

######  Remark 2.3.

Instead of trajectories from the space $C_{n}[0,T]$ with derivatives from the
space $P_{n}[0,T]$ one may consider absolutely continuous on the interval
$[0,T]$ trajectories with measurable and almost everywhere bounded on $[0,T]$
derivatives, 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.

##  3 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
(13) may be rewritten as follows 18:

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

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 18 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 (7) one has

$x(t)=x_{0}+\int_{0}^{t}z(\tau)d\tau.$ (14)

Put

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

and construct the functional

$\varphi(z)=\frac{1}{2}\int_{0}^{T}h^{2}\Big{(}x_{0}+\int_{0}^{t}z(\tau)d\tau,z(t)\Big{)}dt.$
(16)

Consider the set

$\Omega=\\{z\in P_{n}[0,T]\ |\ \varphi(z)=0\\}.$

It is not difficult to check that for functional (16) the relation

$\left\\{\begin{array}[]{ll}\varphi(z)=0\ (z\in\Omega),&\text{if}\
\dot{x}_{i}(t)\psi_{i}\leqslant c(F_{i}(x(t)),\psi_{i})\quad\forall\psi_{i}\in
S_{1},\quad\forall t\in[0,T],\quad i=\overline{1,n}.\\\ \varphi(z)>0\
(z\notin\Omega),&\text{otherwise},\end{array}\right.$ (17)

holds true, i. e. inclusion (13) takes place iff $\varphi(z)=0$.

Introduce the functional

$\chi(z)=\frac{1}{2}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right)^{2}.$ (18)

It is seen that condition (7) on the left endpoint is automatically satisfied
due to the vector-function $z(t)$ definition and condition (8) on the right
endpoint is satisfied iff $\chi(z)=0$.

As it was noted above, the search for a solution of a differential inclusion
is carried out on the discontinuity surfaces $s_{i}(x)=0$, $i=\overline{1,m}$,
so also introduce the functional

$\omega(z,c)=\frac{1}{2}\int_{0}^{T}s^{2}\Big{(}x_{0}+\int_{0}^{t}z(\tau)d\tau,c\Big{)}dt.$
(19)

Construct the functional

$I(z,c)=\varphi(z)+\chi(z)+\omega(z,c).$ (20)

So the original problem has been reduced to minimizing functional (20) on the
space $P_{n}[0,T]\times R^{\ell}$. Denote $z^{*},c^{*}$ 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 (and the vector $c^{*}$ defines the
discontinuity surface structure).

######  Remark 3.1.

The structure of the functional $\varphi(z)$ 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 (16) 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 (18), (19)
structures is obvious.

##  4 Necessary minimum conditions of the functional ${I(z,c)}$
in a particular case

It is obvious that the point $x^{*}$ is Problem 1 solution iff the functional
$I(z,c)$ vanishes at the corresponding point, i. e. $I(z^{*},c^{*})=0$. In
order to obtain a more constructive minimum condition (which is useful for
developing a numerical method for solving the original problem), let us study
the differential properties of the functional $I(z,c)$. Suppose that the
trajectories $x_{i}(t)$, $i=\overline{1,n}$, vanish only at isolated time
moments of the interval $[0,T]$. This assumption is natural if the
discontinuity surfaces do not contain any of these trajectories vanishing on
the interval $[0,T]$ subset of nonzero measure. Then similarly to work 12,
under the assumption made, it is proved that the functional $I(z,c)$ is
G${\rm\hat{a}}$teaux differentiable.

The proof is carried out with the help of classical variations of the
functionals $\varphi(z)$, $\chi(z)$ and $\omega(z,c)$ and uses Lebesgue’s
dominated convergence theorem, as well as such known facts as the support
function additivity in the first argument, Lagrange’s mean value theorem and
integration by parts. Let us formulate the theorem.

######  Theorem 4.1.

Let the trajectories $x_{i}(t)$, $i=\overline{1,n}$, vanish only at isolated
time moments of the interval $[0,T]$. Then the functional $I(z,c)$ is
G${\rm\hat{a}}$teaux differentiable and its gradient at the point $(z,c)$ is
expressed by the formula

$\nabla I(z,c)=\Bigg{[}\sum_{i=1}^{n}{h_{i}(x,z)}\psi_{i}^{*}(x,z){\bf
e_{i}}-\sum_{i=1}^{n}\int_{t}^{T}{h_{i}(x(\tau),z(\tau))}\frac{\partial
c_{i}(F_{i}(x(\tau)),\psi_{i}^{*}(x(\tau),z(\tau)))}{\partial x}d\tau+$
$+\sum_{j\in J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}}+\sum_{i=1}^{m}\int_{t}^{T}s_{i}(x(\tau),c)\frac{\partial
s_{i}(x(\tau),c)}{\partial x}d\tau,\
\sum_{i=1}^{m}\int_{0}^{T}s_{i}(x(\tau),c)\frac{\partial
s_{i}(x(\tau),c)}{\partial c}d\tau\Bigg{]}.$

######  Proof 4.2.

The detailed proof is carried out similarly as in 12.

Note that for the functional $I(z,c)$ G${\rm\hat{a}}$teaux differentiability
it is significantly that the vector $\psi_{i}^{*}(x,z)$ is unique in the case
$\ell_{i}(\psi_{i},x,z)>0$, $i\in\\{1..n\\}$; namely: due to the structure of
functional (15) it is easy to check that in the case
$\ell_{i}(\psi_{i},x,z)>0$ maximum of the expression
$\max\\{0,\ell_{i}({\psi_{i}},x,z)\\}=\ell_{i}({\psi_{i}},x,z)$ is achieved at
the only element $\psi_{i}^{*}(x,z)\in S_{1}$, $i\in\\{1..n\\}$. A simple
justification of this fact is carried out as in 15 and is based on the
convexity of the set $F_{i}(x)$, $i=\overline{1,n}$, at each $x$ and on the
known properties of a support function. In the case
$\ell_{i}(\psi_{i},x,z)\leq 0$ we have put $\psi_{i}^{*}(x,z)=\psi_{0}$,
$i\in\\{1..n\\}$, where $\psi_{0}\in S_{1}$ is fixed. These considerations are
valid for a more general case considered in work 12. Note that in the
particular case of this paper one can write out the explicit expression for
the vector $\psi_{i}^{*}(x,z)$ in the case $\ell_{i}(\psi_{i},x,z)>0$,
$i\in\\{1..n\\}$, as follows:

$\psi_{i}^{*}(x,z)=\mathrm{sign}\left(z_{i}-A_{i}x\right),$

note that $z_{i}-A_{i}x\neq 0$ in the case $\ell_{i}(\psi_{i},x,z)>0$.

The following theorem formulates the known minimum condition for a Gateaux
differentiable functional.

######  Theorem 4.3.

Let the trajectories $x_{i}(t)$, $i=\overline{1,n}$, vanish only at isolated
time moments of the interval $[0,T]$. In order for the point $(z^{*},c^{*})$
to minimize the functional $I(z,c)$, it is necessary that

$0_{n}\times{\bf
0_{\ell}}=\Bigg{[}\sum_{i=1}^{n}{h_{i}(x^{*},z^{*})}\psi_{i}^{*}(x^{*},z^{*}){\bf
e_{i}}-\sum_{i=1}^{n}\int_{t}^{T}{h_{i}(x^{*}(\tau),z^{*}(\tau))}\frac{\partial
c_{i}(F_{i}(x^{*}(\tau)),\psi_{i}^{*}(x^{*}(\tau),z^{*}(\tau)))}{\partial
x}d\tau+$ $+\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z^{*}_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}}+\sum_{i=1}^{m}\int_{t}^{T}s_{i}(x^{*}(\tau),c^{*})\frac{\partial
s_{i}(x^{*}(\tau),c^{*})}{\partial x}d\tau,\
\sum_{i=1}^{m}\int_{0}^{T}s_{i}(x^{*}(\tau),c^{*})\frac{\partial
s_{i}(x^{*}(\tau),c^{*})}{\partial c}d\tau\Bigg{]}$

where $0_{n}$ is a zero element of the space $P_{n}[0,T].$

It is apparent, that $I(z^{*},c^{*})=0$ is necessary and sufficient minimum
condition for this functional and in this case the equality of Theorem 2 is
automatically satisfied.

Example 1. Consider the following system

$\dot{x}_{1}(t)\in[-1,1](|x_{1}(t)|+|x_{2}(t)|+|x_{3}(t)|),\quad\dot{x}_{2}(t)=x_{1}(t)-x_{2}(t),\quad\dot{x}_{3}(t)=x_{2}(t)$

with the boundary conditions

$x_{1}(0)=-3,\ x_{2}(0)=4,\ x_{3}(0)=6,\quad x_{1}(1)=0.$

Let the discontinuity surface in this example be of the form

$s(x,c)=x_{1}+c_{1}x_{2}-c_{2}=0.$

A simplest steepest descent method (in the functional space) 19 with some
modifications (see Remark 5, b) below) was used in order to minimize the
functional $I(z,c)$ in this problem.

The point $(z_{\\{0\\}},c_{\\{0\\}})=(0,0,0,1,1)^{\prime}$ was taken as the
initial one (note that here $(0,0,0)^{\prime}$ is a zero point in the
functional space $P_{3}[0,1]$ and the point $(1,1)^{\prime}$ belongs to the
space $R^{2}$). Herewith $I(z_{\\{0\\}},c_{\\{0\\}})=37$. At the 60-th
iteration the point $(z_{\\{60\\}},c_{\\{60\\}})^{\prime}$ was constructed and
we approximately put $(z^{*},c^{*})=(z_{\\{60\\}},c_{\\{60\\}})$. Herewith
$c^{*}\approx(0.98467,0.93868)^{\prime}$, $I(z^{*},c^{*})\approx 0.00015$ and
$||\nabla I(z^{*},c^{*})||_{L^{2}_{3}[0,1]\times R^{2}}\approx 0.02009$. The
point $z^{*}$ is not given here for two reasons: 1) for brevity (as it is
constructed in the form of a rather bulky piecewise continuous vector-
function) and 2) because only the parameters $c^{*}$ are finally used to
estimate the result obtained (see the next paragraph).

Having obtained the parameters $c^{*}$, put
$x_{1}(t):=-c_{1}^{*}x_{2}(t)+c_{2}^{*}$ and substitute them into the last two
equations of the system given. Integrate this closed-loop system with one of
the known numerical methods (for this example the Runge-Kutta 4-5-th order
method was used), then check that the first inclusion is also satisfied and
finally have $x_{1}(1)\approx-0.00431$, so we see that the desired value on
the right endpoint is achieved with an error of order $5\times 10^{-3}$.
Picture 1 demonstrates the results of calculations. The black lines denote the
curves which were obtained via the method of the paper, while the dashed red
lines denote the curves obtained via integration of the closed-loop system
(and via the relation $x_{1}(t)=-c_{1}^{*}x_{2}(t)+c_{2}^{*}$).

Figure 1: Example 1. The trajectories $x_{1}^{*}(t)$, $x_{2}^{*}(t)$,
$x_{3}^{*}(t)$.

######  Remark 4.4.

a) In practice differential inclusions (11) are satisfied at most iterations,
since with $i=\overline{1,m}$ the function $z_{i}(t)-A_{i}x(t)$ values are
usually not very large, so they are inside the interval
$[-\overline{a}_{i},\overline{a}_{i}]|x(t)|$ during the interval $[0,T]$
(recall that the gain factor $\overline{a}_{i}$ is sufficiently large). This
fact significantly simplifies calculations insofar as in the case
$\varphi(z)>0$ (see (17)) the functional $\varphi(z)$ gradient is the most
complicated (comparatively to the other functionals considered).

b) During computation procedure it was noted that $z_{i}$, $i=\overline{1,n}$,
are so-called “fast” variables while $c_{k}$, $k=\overline{1,\ell}$, are so
called “slow” variables. This means that the trajectories may change
relatively significantly with the fixed parameters while the variables
$c_{k}$, $k=\overline{1,\ell}$, change slightly with the fixed variables
$z_{i}$, $i=\overline{1,n}$, throughout iterations. So in practice it was
reasonable to fix the parameters $c_{k}$, $k=\overline{1,\ell}$, and to make
one iteration with a relatively large descent step (in order to minimize the
functional in the variables $z_{i}$, $i=\overline{1,n}$) and then to fix the
trajectories $z_{i}$, $i=\overline{1,n}$, and to make several iterations with
small descent steps (in order to minimize the functional in the variables
$c_{k}$, $k=\overline{1,\ell}$) and to repeat this process.

##  5 Differential properties of the functional $I(x,c)$
in a more general case

In this section, it will be more convenient for presentation not to go to the
space of derivatives but to work with the following functional (we keep the
previous notation for it):

$I(x,c)=\varphi(x)+\chi(x)+\omega(x,c),$

where as previously

$\varphi(x)=\int_{0}^{T}h^{2}\big{(}x(t),\dot{x}(t)\big{)}dt,$ (21)
$\chi(x)=\frac{1}{2}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}\dot{x}_{j}(t)dt-{x_{T}}_{j}\right)^{2},$
$\omega(x,c)=\frac{1}{2}\int_{0}^{T}s^{2}(x(t),c)dt.$

In section 4 we have shown that the functional $I(z,c)$ is
G${\rm\hat{a}}$teaux differentiable, and according to the assumption made
there, the case was excluded when at least one of the phase variables would be
identically equal to zero on some time interval of the set $[0,T]$ of nonzero
length. Let us now assume that some of the trajectories can be identically
equal to zero on some nonzero time interval of the segment $[0,T]$ and study
the differential properties of the functional $I(z,c)$ in this case.

For simplicity consider the case $n=2$ and only the functions
$l_{1}(\psi_{1},x,z)$ and $h_{1}(x,z)$ (here we denote them
$l(\psi_{1},x_{1},x_{2},z_{1})$ 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
$\ell(\psi_{1},x_{1},x_{2},z_{1})=z_{1}\psi_{1}-a_{1}x_{1}\psi_{1}-a_{2}x_{2}\psi_{1}-b|x_{1}||\psi_{1}|-b|x_{2}||\psi_{1}|$,
where $a_{1}:=a_{1,1}$, $a_{2}:=a_{1,2}$, $b:=\overline{a}_{1}$. Fix some
point $(x_{1},x_{2})\in R^{2}$. Let $x_{1}(t)=0$, $z_{1}(t)=0$, $x_{2}(t)>0$
at $t\in[t_{1},t_{2}]$; other cases are studied in a completely analogous
fashion.

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

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 $(x_{1},x_{2},z_{1})$ in the vicinity of the point $(0,x_{2},0)$ the
level set

$\mathrm{lev}_{\beta}\ell(\cdot,x_{1},x_{2},z_{1})=\\{\psi_{1}\in S_{1}\ |\
-\ell(\psi_{1},x_{1},x_{2},z_{1})\leqslant\beta\\}$

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 $(0,x_{2},0)$;
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

$\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{-\ell(\psi_{1_{n}},0+\gamma_{n}d_{1,1},x_{2}+\gamma_{n}d_{1,2},0+\gamma_{n}d_{2})-(-\ell(\psi_{1_{n}},0,x_{2},0))}{\gamma_{n}}\geqslant\frac{\partial(-\ell(\overline{\psi}_{1},0,x_{2},0))}{\partial
d},$

where
$\displaystyle{\frac{\partial\ell(\overline{\psi}_{1},0,x_{2},0)}{\partial
d}}$ is the derivative of the function
$\ell(\overline{\psi}_{1},x_{1},x_{2},z_{1})$ at the point $(0,x_{2},0)$ in
the direction $d$.

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

In order to verify condition iii), it suffices to note that since $b>0$, then
at the fixed $\psi_{1}\in S_{1}$ the function $-b|x_{1}||\psi_{1}|$ is
superdifferentiable (and hence is differentiable in directions) at the point
$0$, herewith, its superdifferential at the point $0$ is the segment
$\mathrm{co}\\{-b|\psi_{1}|,b|\psi_{1}|\\}$. An explicit expression for the
derivative of this function at the point $0$ in the direction $d_{1,1}$ is
$-b|d_{1,1}||\psi_{1}|$.

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 $C$.
Calculate

$\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{\ell(\psi_{1_{n}},0+\gamma_{n}d_{1,1},x_{2}+\gamma_{n}d_{1,2},0+\gamma_{n}d_{2})-\ell(\psi_{1_{n}},0,x_{2},0)}{\gamma_{n}}=$
$=\mathrm{lim}\sup\limits_{n\rightarrow\infty}\frac{\gamma_{n}d_{2}\psi_{1_{n}}-\gamma_{n}a_{1}d_{1,1}\psi_{1_{n}}-\gamma_{n}a_{2}d_{1,2}\psi_{1_{n}}-b|\gamma_{n}d_{1,1}||\psi_{1_{n}}|-b\gamma_{n}d_{1,2}|\psi_{1_{n}}|}{\gamma_{n}}=$
$=\mathrm{lim}\sup\limits_{n\rightarrow\infty}\left(d_{2}\psi_{1_{n}}-a_{1}d_{1,1}\psi_{1_{n}}-a_{2}d_{1,2}\psi_{1_{n}}-b|d_{1,1}||\psi_{1_{n}}|-bd_{1,2}|\psi_{1_{n}}|\right)$

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

$\frac{\partial\ell(\overline{\psi}_{1},0,x_{2},0)}{\partial
d}=d_{2}\overline{\psi}_{1}-a_{1}d_{1,1}\overline{\psi}_{1}-a_{2}d_{1,2}\overline{\psi}_{1}-b|d_{1,1}||\overline{\psi}_{1}|-bd_{1,2}|\overline{\psi}_{1}|.$

From last two equalities 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 $(0,x_{2},0)$ 20, and its
derivative in the direction $d$ at this point is expressed by the formula

$\frac{\partial h(0,x_{2},0)}{\partial d}=\sup_{\psi_{1}\in
S(0,x_{2},0)}\frac{\partial\ell(\psi_{1},0,x_{2},0)}{\partial d},$

where $S(0,x_{2},0)=\mathrm{arg}\max_{\psi_{1}\in
S_{1}}\ell(\psi_{1},0,x_{2},0)$. However, as shown above, in the considered
problem the set $S(0,x_{2},0)$ consists of the only element
$\psi_{1}^{*}(0,x_{2},0)$, hence

$\frac{\partial h(0,x_{2},0)}{\partial
d}=\frac{\partial\ell(\psi_{1}^{*}(0,x_{2},0),0,x_{2},0)}{\partial d},$

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

$\frac{\partial\ell(\psi_{1}^{*}(0,x_{2},0),0,x_{2},0)}{\partial
d}=d_{2}\psi_{1}^{*}-a_{1}d_{1,1}\psi_{1}^{*}-a_{2}d_{1,2}\psi_{1}^{*}-b|d_{1,1}||\psi_{1}^{*}|-bd_{1,2}|\psi_{1}^{*}|,$

where we have put $\psi^{*}_{1}:=\psi_{1}^{*}(0,x_{2},0)$.

From last two equalities we finally obtain that the function
$h(x_{1},x_{2},z_{1})$ is superdifferentiable at the point $(0,x_{2},0)$, but
it is also positive in this case, hence the function
$h^{2}(x_{1},x_{2},z_{1})$ is superdifferentiable at the point $(0,x_{2},0)$
as a square of a superdifferentiable positive function 21.

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 $(0,x_{2},0)$ and
its gradient vanishes at this point.

######  Remark 5.1.

It is easy to see that the proof above may be significantly simplified if one
uses the fact that $\psi_{1},\psi_{1_{n}},\overline{\psi}_{1},\psi_{1}^{*}\in
S_{1}$. However, since the statement about the superdifferentiability of
functions having a structure similar to that of the function $h_{1}(x,z)$ is
of independent interest, so such a way of proof is used, which may be applied
to the more general case when $\psi_{1}$ belongs to an arbitrary compact
subset of the space $R$ (therefore, we consciously did not put
$|\psi_{1}|=|\psi_{1_{n}}|=|\overline{\psi}_{1}|=|\psi_{1}^{*}|=1$ in the
proof).

Above we have considered a particular case. Dividing the interval $[0,T]$ into
the segments in which some of the phase trajectories are identically equal to
zero and some retain a definite sign (in this section we suppose that there is
a finite number of such segments) and arguing in each of these intervals
similarly to the case considered and also using the superdifferential calculus
rules (3), (4), write down the superdifferential of the function $h_{i}(x,z)$.
With $i=\overline{1,m}$ one has

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

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

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

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

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

Thus, in a more general case considered in this section, when the segment
$[0,T]$ 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, the integrand of functional (21) is superdifferentiable as a square of a
superdifferentiable nonnegative function 21. It turns out that this fact
allows us to conclude that functional (21) itself is superdifferentiable (in
the case considered).

######  Theorem 5.2.

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 $\varphi(x)$ is
superdifferentiable, i. e.

$\frac{\partial\varphi(x)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}\varphi(x+\alpha
g)-\varphi(x)\big{)}=\min_{w\in\overline{\partial}\varphi(x)}w(g)$ (22)

Here the set $\overline{\partial}\varphi(x)$ is of the following form

$\overline{\partial}\varphi(x)=\Big{\\{}w\in\left(C_{n}[0,T],||\cdot||_{L_{2}^{n}[0,T]}\right)^{*}\
\big{|}\ w_{1}(t),w_{2}(t)\in L^{n}_{\infty}[0,T],$ (23)
$[w_{1}(t),w_{2}(t)]\in\overline{\partial}h^{2}(x(t),\dot{x}(t))\quad for\
a.e.\ t\in[0,T]\Big{\\}},$

herewith, $\forall g\in C_{n}[0,T]$ and $\forall\dot{g}\in P_{n}[0,T]$ one has

$w(g)=\int_{0}^{T}\langle w_{1}(t),g(t)\rangle dt+\int_{0}^{T}\langle
w_{2}(t),\dot{g}(t)\rangle dt.$

######  Proof 5.3.

In accordance with definition (5) of a superdifferentiable functional, in
order to prove the theorem, one has to check that

1) the derivative of the functional $\varphi(x)$ in the direction $g$ is
actually of form (22),

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

Let us prove statement 1).

At first show that the following relation is true.

$\lim_{\alpha\downarrow 0}\frac{1}{\alpha}\Big{|}\varphi(x+\alpha
g)-\varphi(x)-\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}dt\Big{|}=0.$ (24)

Denote

$f(t,\alpha)=\frac{1}{\alpha}\Big{(}h^{2}(x(t)+\alpha
g(t),\dot{x}(t)+\alpha\dot{g}(t))-h^{2}(x(t),\dot{x}(t))\Big{)}-$ (25)
$-\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),g(t)\rangle+\langle w_{2}(t),\dot{g}(t)\rangle\big{)}.$

Our aim is to prove relation (24) via Lebesgue’s dominated convergence theorem
applied to the function $f(t,\alpha)$ (at $\alpha\downarrow 0$).

At first note that by superdifferential definition (2) 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 $f(t,\alpha)\rightarrow 0$ when
$\alpha\downarrow 0$.

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

Insofar as $x,g\in C_{n}[0,T]$, $z,\dot{g}\in P_{n}[0,T]$ and the function
$h^{2}(x,z)$ is continuous in its variables due to its structure 22, we obtain
that for each $\alpha>0$ the functions $t\rightarrow h^{2}(x(t),z(t))$ and
$t\rightarrow h^{2}(x(t)+\alpha g(t),z(t)+\alpha\dot{g}(t))$ belong to the
space $L_{\infty}^{1}[0,T]$.

Due to the upper semicontinuity of a superdifferential mapping 16 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 23. Then due to the continuity of
the function $g(t)$, the piecewise continuity of the function $\dot{g}(t)$ and
due to the continuity of the scalar product in its variables we obtain that
for each $\alpha>0$ the mapping

$t\rightarrow\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x(t),z(t))}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}$
(26)

is upper semicontinuous 24, and then is also measurable 23. 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 $g(t)$ and the piecewise continuity of the
function $\dot{g}(t)$ it is easy to check that for each $\alpha>0$ mapping
(26) is also bounded uniformly in $t\in[0,T]$. So we finally have that for
each $\alpha>0$ mapping (26) belongs to the space $L_{\infty}^{1}[0,T]$.

Now we prove that the function $f(t,\alpha)$ is dominated by some integrable
function for all sufficiently small $\alpha>0$. On the segments where the
phase trajectories retain their signs this is done via standard technique
using Lagrange’s mean value theorem. Consider the interval
$[t_{1},t_{2}]\subset[0,T]$ of nonzero length. For simplicity consider the
case $n=2$ and only the functions $\ell_{1}(\psi_{1},x,z)$ and $h_{1}(x,z)$
(here denote them $\ell(\psi_{1},x_{1},x_{2},z_{1})$ and
$h(x_{1},x_{2},z_{1})$ respectively); the general case is considered in a
similar way. Suppose that $x_{2}(t)=0$ and $x_{1}(t)>0$ at
$t\in[t_{1},t_{2}]$; other cases are studied in a completely analogous
fashion. In the case $h_{1}(x,z)=0$ the proof is elementary. So suppose
$h_{1}(x,z)>0$, i. e. $h_{1}(x,z)=\max_{\psi_{1}\in
S_{1}}\ell(\psi_{1},x,z)=\ell(\psi_{1}^{*}(x,z),x,z)$. As is shown in the
previous paragraph, the second summand in (25) is integrable functions. So it
remains to consider for sufficiently small $\alpha>0$ the first summand in
(25), i.e. (in the considered case) the function

$\frac{1}{\alpha}\Big{(}h_{1}^{2}(x+\alpha
g,\dot{x}+\alpha\dot{g})-h_{1}^{2}(x,\dot{x})\Big{)}=$
$\frac{1}{\alpha}\Big{(}\Big{[}(z_{1}+\alpha\dot{g}_{1})\psi_{1}^{*}(\alpha)-a_{1}(x_{1}+\alpha
g_{1})\psi_{1}^{*}(\alpha)-a_{2}\alpha
g_{2}\psi_{1}^{*}(\alpha)-b(x_{1}+\alpha g_{1})|\psi_{1}^{*}(\alpha)|-b|\alpha
g_{2}||\psi_{1}^{*}(\alpha)|\Big{]}^{2}-$
$-\big{[}z_{1}\psi_{1}^{*}-a_{1}x_{1}\psi_{1}^{*}-bx_{1}|\psi_{1}^{*}|\big{]}^{2}\Big{)},$
(27)

where we have put $a_{1}:=a_{1,1}$, $a_{2}:=a_{1,2}$, $b:=\overline{a}_{1}$,
$x_{1}:=x_{1}(t)$, $x_{2}:=x_{2}(t)$, $z_{1}:=z_{1}(t)$, $g:=g(t)$,
$\dot{g}:=\dot{g}(t)$, $\psi_{1}^{*}:=\psi_{1}^{*}(x,z)$,
$\psi_{1}^{*}(\alpha):=\psi_{1}^{*}(x+\alpha g,z+\alpha\dot{g})$ for brevity.

Consider only the summands

$\frac{z_{1}^{2}(\psi_{1}^{*}(\alpha))^{2}-z_{1}^{2}(\psi_{1}^{*})^{2}}{\alpha}+\frac{a_{1}^{2}x_{1}^{2}(\psi_{1}^{*}(\alpha))^{2}-a_{1}^{2}x_{1}^{2}(\psi_{1}^{*})^{2}}{\alpha}+\frac{b^{2}x_{1}^{2}(\psi_{1}^{*}(\alpha))^{2}-b^{2}x_{1}^{2}(\psi_{1}^{*})^{2}}{\alpha}+\frac{-2a_{1}z_{1}x_{1}(\psi_{1}^{*}(\alpha))^{2}+2a_{1}z_{1}x_{1}(\psi_{1}^{*})^{2}}{\alpha}+$
$+\frac{-2bz_{1}|x_{1}||\psi_{1}^{*}(\alpha)|\psi_{1}^{*}(\alpha)+2bz_{1}|x_{1}||\psi_{1}^{*}|\psi_{1}^{*}}{\alpha}+\frac{2a_{1}bx_{1}|\psi_{1}^{*}(\alpha)|\psi_{1}^{*}(\alpha)-2a_{1}bx_{1}|\psi_{1}^{*}|\psi_{1}^{*}}{\alpha}$

of the right-hand side of equation (27). The first four summands in this
expression are equal to zero for all $\alpha>0$ as
$|\psi_{1}^{*}(\alpha)|=|\psi_{1}^{*}|=1$. The second two summands in this
expression are equal to zero for all sufficiently small $\alpha>0$ as the
function $\psi_{1}^{*}(x,z)$ is continuous in $(x,z)$ and also as
$|\psi_{1}^{*}(\alpha)|=|\psi_{1}^{*}|=1$.

By direct calculation one can easily check that due to the continuity of the
functions $x_{1}(t)$, $x_{2}(t)$ and due to the piecewise continuity of the
functions $z_{1}(t)$ and $\psi_{1}^{*}(x(t),z(t))$, $t\in[t_{1},t_{2}]$, other
summands of the right-hand side of equation (27) are dominated by a piecewise
continuous function for all sufficiently small $\alpha>0.$

Now consider the functional
$\displaystyle{\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}dt}$ in details. For each $\alpha>0$
and for a. e. $t\in[0,T]$ we have the obvious inequality

$\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}\leqslant\langle w_{1}(t),\alpha
g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle,$

where $[w_{1}(t),w_{2}(t)]$ is some measurable selector of the mapping
$t\rightarrow\overline{\partial}h^{2}(x(t),\dot{x}(t))$ (due to the noted
boundedness uniformly in $t\in[0,T]$ of the set
$\overline{\partial}h^{2}(x,\dot{x})$ we have $w_{1},w_{2}\in
L_{\infty}^{n}[0,T]$), then taking into account the form of formula (23) for
each $\alpha>0$ one also has the inequality

$\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}dt\leqslant\min_{w\in\overline{\partial}\varphi(x)}\int_{0}^{T}\langle
w_{1}(t),\alpha g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle dt.$

Insofar as for each $\alpha>0$ and for a.e. $t\in[0,T]$ we have

$\max_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}\in$ $\in\Big{\\{}\langle
w_{1}(t),\alpha g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle\ \big{|}\
[w_{1}(t),w_{2}(t)]\in\overline{\partial}h^{2}(x(t),\dot{x}(t))\Big{\\}},$

and the set $\overline{\partial}h^{2}(x,z)$ is closed and bounded at each
fixed $t$ by the superdifferential definition and the mapping
$t\rightarrow\overline{\partial}h^{2}(x(t),z(t))$, as noted above, is upper
semicontinuous and besides, the scalar product is continuous in its arguments
and $g\in C_{n}[0,T]$, $\dot{g}\in P_{n}[0,T]$, then due to Filippov lemma 25
there exists such measurable selector
$[\overline{w}_{1}(t),\overline{w}_{2}(t)]$ of the mapping
$t\rightarrow\overline{\partial}h^{2}(x(t),z(t))$ that for each $\alpha>0$ and
for a. e. $t\in[0,T]$ we have

$\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}=\langle w_{1}(t),\alpha
g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle,$

so we have found the element $\overline{w}$ of the set
$\overline{\partial}\varphi(x)$ which brings the equality in the previous
inequality. Thus, one finally has

$\int_{0}^{T}\min_{[w_{1},w_{2}]\in\overline{\partial}h^{2}(x,z)}\big{(}\langle
w_{1}(t),\alpha g(t)\rangle+\langle
w_{2}(t),\alpha\dot{g}(t)\rangle\big{)}dt=\min_{w\in\overline{\partial}\varphi(x)}\int_{0}^{T}\langle
w_{1}(t),\alpha g(t)\rangle+\langle w_{2}(t),\alpha\dot{g}(t)\rangle dt.$ (28)

From relations (24), (28) one obtains expression (22).

Prove statement 2).

The convexity of the set $\overline{\partial}\varphi(x)$ follows directly from
the convexity of the set $\overline{\partial}h^{2}(x,z)$ at each fixed
$t\in[0,T]$.

Prove the boundedness of the set $\overline{\partial}h^{2}(x,z)$ uniformly in
$t\in[0,T]$. Due to the upper semicontinuity of the mapping
$t\rightarrow\overline{\partial}h^{2}(x(t),z(t))$, for each $t\in[0,T]$ there
exists such number $\delta(t)$, that under the condition
$|\overline{t}-t|<\delta(t)$ the inclusion
$\overline{\partial}h^{2}(x(\overline{t}),z(\overline{t}))\subset
B_{r}(\overline{\partial}h^{2}(x(t),z(t)))$ holds true at
$\overline{t}\in[0,T]$, where $r$ is some fixed finite positive number. The
intervals $D_{\delta(t)}(t)$, $t\in[0,T]$, form an open cover of the segment
$[0,T]$, so by Heine-Borel lemma one can take a finite subcover from this
cover. Hence, there exists such number $\delta>0$ that for every $t\in[0,T]$
the inclusion
$\overline{\partial}h^{2}(x(\overline{t}),z(\overline{t}))\subset
B_{r}(\overline{\partial}h^{2}(x(t),z(t)))$ holds true once
$|\overline{t}-t|<\delta$ and $\overline{t}\in[0,T]$. This means that for the
segment $[0,T]$ there exists a finite partition
$t_{1}=0,t_{2},\dots,t_{N-1},t_{N}=T$ with the diameter $\delta$ such that
$\overline{\partial}h^{2}(x,z)\subset\bigcup\limits_{i=1}^{N}B_{r}(\overline{\partial}h^{2}(x(t_{i}),z(t_{i})))$
for all $t\in[0,T]$. It remains to notice that the set
$\bigcup\limits_{i=1}^{N}B_{r}(\overline{\partial}h^{2}(x(t_{i}),z(t_{i})))$
is bounded due to the compactness of the set $\overline{\partial}h^{2}(x,z)$
at each fixed $t\in[0,T]$.

As shown in statement 1) and at the beginning of statement 2) proof, the set
$\overline{\partial}\varphi(x)$ is convex and its elements $w$ belong to the
space $L_{\infty}^{n}[0,T]$. Then all the more the set
$\overline{\partial}\varphi(x)$ is a convex subset of the space
$L_{2}^{n}[0,T]$. Let us prove that the set $\overline{\partial}\varphi(x)$ is
closed in the weak topology of the space $L_{2}^{n}[0,T]$. Let
$\\{w_{n}\\}_{n=1}^{\infty}$ be the sequence of vector-functions from the set
$\overline{\partial}\varphi(x)$, converging to the vector-function $w^{*}$ in
the strong topology of the space $L_{2}^{n}[0,T]$. It is known 26 that this
sequence has the subsequence $\\{w_{n_{k}}\\}_{n_{k}=1}^{\infty}$ converging
pointwise to $w^{*}$ almost everywhere on $[0,T]$, i. e. there exists such
subset $T^{\prime}\subset[0,T]$ having the measure $T$ that for every point
$t\in T^{\prime}$ we have $w_{n_{k}}(t)\in\overline{\partial}h^{2}(x(t),z(t))$
and $w_{n_{k}}(t)$ converges to $w^{*}(t)$, $n_{k}=1,2,\dots$. But the set
$\overline{\partial}h^{2}(x(t),z(t))$ is closed at each $t\in[0,T]$ by the
definition of the subdifferential, hence for every $t\in T^{\prime}$ we have
$w^{*}(t)\in\overline{\partial}h^{2}(x(t),z(t))$. So the set
$\overline{\partial}\varphi(x)$ is closed in the strong topology of the space
$L_{2}^{n}[0,T]$, but it is also convex, so it is also closed in the weak
topology of the space $L_{2}^{n}[0,T]$ 27.

Recall that by virtue of Remark 1 it is sufficient to consider the space
$L_{2}^{n}[0,T]$. The weak* compactness of the set
$\overline{\partial}\varphi(x)$ in the space $L_{2}^{n}[0,T]$ follows from its
weak compactness (in $L_{2}^{n}[0,T]$) by virtue of these topologies
definitions 17. The space $L_{2}^{n}[0,T]$ is reflexive 27, so the set there
is weakly compact if and only if it is bounded in norm and weakly closed 27 in
this space. These properties required have been proved in the previous two
paragraphs. The theorem is proved.

######  Remark 5.4.

An interesting problem for future research is to make an attempt to construct
a numerical method for minimizing the functional $I(x,c)$ based on the well-
known in nonsmooth optimization optimality conditions, which are written out
using the formula for the superdifferential of the functional $\varphi(x)$
obtained in Theorem 3 (and with the formulas for the functionals $\chi(x)$,
$\omega(x,c)$ G${\rm\hat{a}}$teaux gradients).

##  6 Another type of control for providing a sliding mode

As is seen from the previous sections of the paper, the main difficulty while
operating with control (10) is caused by nondifferentiability (and even
discontinuity) of this function in the phase variables. In order to overcome
this difficulty, let us try to change the control structure in such a way that
on the one hand, it would retain the main property of this control (namely,
would ensure that the system hit some (acceptable in practice) neighborhood of
the surface $s(x)={\bf 0_{m}}$ and stay there), and on the other hand, would
provide the desired control with the required differential properties, giving
possibility (as we will see below) to use the developed “variational”
technique to solve the “full” problem of finding the trajectory of the system
both on the surface $s(x)={\bf 0_{m}}$ and before hitting it. Note that in
this section we do not use the term “discontinuity surface” (but simply talk
about the surface $s(x)={\bf 0_{m}}$) because, as will be shown below, the
right-hand side of the system with the types of control structure considered
in this section preserves the property of continuity (and even continuous
differentiability if $x_{i}\neq 0$, $i=\overline{1,n}$).

In this section we restrict ourselves to consideration of the surfaces
$s(x)={\bf 0_{m}}$ with the hyperplanes

$s(x)=Cx-b,$

where the elements of the $m\times n$ matrix $C$ and the vector $b$ of the
dimension $m$ are still to be determined (we will omit the dependence of the
surface vector-function on these variables for convenience of notation).
Regulation of the surface structure via these parameters will give more
opportunities to achieve certain goals of the object motion, for example,
hitting the required position at the given time moment. As noted in section 2,
in many practical cases it is natural to assume that these parameters are
approximately known on the basis of the physical meaning of a system. Instead
of control (10) consider two more types of control for $i=\overline{1,m}$:

$u_{i}^{[1]}(x,C,b)=-\alpha_{i}|x|s_{i}(x)\exp\Big{\\{}\mathrm{sign}(s_{i}(x))(-s_{i}(x))\Big{\\}},$
(29)

$\left\\{\begin{array}[]{lll}u_{i}^{[2]}(x,C,b)=\alpha_{i}|x|k\sqrt{-s_{i}(x)},&\text{if}\
s_{i}(x)\leqslant-\delta,\\\
u_{i}^{[2]}(x,C,b)=-\alpha_{i}|x|(es^{3}_{i}(x)+fs_{i}(x)),&\text{if}\
-\delta\leqslant s_{i}(x)\leqslant\delta,\\\
u_{i}^{[2]}(x,C,b)=-\alpha_{i}|x|k\sqrt{s_{i}(x)},&\text{if}\
s_{i}(x)\geqslant\delta,\end{array}\right.$ (30)

where $\alpha_{i}$ are still some positive numbers which are fixed in this
section (see Remark 8 below), the given number $\delta>0$ is sufficiently
small, and the parameters $k$, $e$ and $f$ are chosen based on the requirement
of continuous differentiability of the functions $u_{i}^{[2]}(x,C,b)$ (if
$x_{i}\neq 0$, $i=\overline{1,n}$). It is not difficult to check that for
every given in advance vicinity of the surface $s(x)={\bf 0_{m}}$ one can
ensure the object hitting this vicinity in a finite time via the choice of the
numbers $\alpha_{i}$, $i=\overline{1,m}$, and $\delta$. The disadvantage of
such a choice of controls is as follows: despite the fact that they transfer
the system to a small neighborhood of the chosen surface (that is, in
accordance with the definition given in Introduction, they ensure the
stability of the sliding mode), these controls do not endow the system with
the stability property “in big”, since the value of $s(x(t))$ generally does
not tend to zero as $t$ tends to infinity. Hence, “correction” of the surface
parameters has an additional sense: to more strictly satisfy the conditions on
the right endpoint (than the accuracy provided by introduced controls (29),
(30)). The advantage of such a choice is the fact that the right-hand side of
the system is a continuously differentiable function of the phase coordinates
if $x_{i}\neq 0$, $i=\overline{1,n}$ (and of the variables $C$, $b$). Note
that although controls used in works 10, 11 (mentioned in Introduction) are
continuous, they are not continuously differentiable in the phase coordinates.

Let us also note the constructive features of the controls given. The
advantage of the control $u^{[1]}$ is the relative simplicity (compared to the
control $u^{[2]}$) of defining this function. The disadvantage of the control
$u^{[1]}$ is that the necessary (in order to ensure the stability of the
sliding mode) values of the gains $\alpha_{i}$, $i=\overline{1,m}$, are too
large and are even sometimes unacceptable for practice in the case when the
initial position of the object is rather far from the surface $s(x)={\bf
0_{m}}$ (the control $u^{[2]}$ control does not have this negative feature).

######  Problem 6.1.

As previously, the problem is of finding a trajectory $x^{*}$ from the space
$C_{n}[0,T]$ (with the derivative $\dot{x}^{*}$ from the space $P_{n}[0,T]$)
described by the system

$\dot{x}_{i}=A_{i}x+u_{i}^{[1],[2]}(x,C,b),\quad i=\overline{1,m},$
$\dot{x}_{i}=A_{i}x,\quad i=\overline{m+1,n},$

with new control (29) or (30) (the parameters $C^{*}\in R^{m}\times R^{n}$ and
$b^{*}\in R^{m}$ are to be determined as well), satisfying boundary conditions
(7), (8). We assume that there exists such a solution.

Denote

$h_{i}^{[1],[2]}(x,z,C,b)=z_{i}-A_{i}x-u_{i}^{[1],[2]}(x,C,b),\quad
i=\overline{1,m},$ $h_{i}(x,z)=z_{i}-A_{i}x,\quad i=\overline{m+1,n},$

and construct the functional

$\varphi^{[1],[2]}(z,C,b)=\frac{1}{2}\sum_{i=1}^{m}\int_{0}^{T}h_{i}^{{2\,[1],[2]}}(x(t),z(t),C,b)dt+\frac{1}{2}\sum_{i=m+1}^{n}\int_{0}^{T}h_{i}^{2}(x(t),z(t))dt,$

where instead of the phase variable $x(t)$ one should write its expression via
its derivative $z(t)$ by formula (14).

Then as previously construct the functional

$I^{[1],[2]}(z,C,b)=\varphi^{[1],[2]}(z,C,b)+\chi(z),$ (31)

in which the function $\chi(z)$ is given by formula (18).

Consider the problem of minimizing functional (31) on the space
$P_{n}[0,T]\times R^{m}\times R^{n}\times R^{m}$. Denote $(z^{*},C^{*},b^{*})$
a global minimizer of this functional. Then

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

is a solution of the original problem (and the matrix $C^{*}$ and the vector
$b^{*}$ define the considered surface structure).

It is obvious that the point $x^{*}$ is a solution of Problem 2 iff the
functional $I^{[1],[2]}(z,C,b)$ vanishes at the corresponding point, i. e.
$I^{[1],[2]}(z^{*},C^{*},b^{*})=0$. In order to obtain a more constructive
minimum condition, which is useful for developing a numerical method for
solving the original problem, let us study the differential properties of the
functional $I^{[1],[2]}(z,C,b)$. Suppose that the trajectories $x_{i}(t)$,
$i=\overline{1,n}$, vanish only at isolated time moments of the segment
$[0,T]$. This assumption is natural if before the transition to the sliding
mode there are no trajectories which are in zero position and have zero
“speed” on some subspace of the interval $[0,T]$ of nonzero measure and if the
discontinuity surfaces do not contain any of these trajectories vanishing on
the interval $[0,T]$ subset of nonzero measure. With the help of the classical
variation, Lagrange’s mean value theorem and integration by parts it is easy
to check the G${\rm\hat{a}}$teaux differentiability of the given functional.

######  Theorem 6.2.

Let the trajectories $x_{i}(t)$, $i=\overline{1,n}$, vanish only at isolated
time moments of the interval $[0,T]$. Then the functional $I^{[1],[2]}(z,C,b)$
is G${\rm\hat{a}}$teaux differentiable and its gradient at the point $(z,C,b)$
is expressed by the formula

$\nabla
I^{[1],[2]}(z,C,b)=\sum_{i=1}^{m}\Bigg{[}h_{i}^{[1],[2]}(x(t),z(t),C,b){\bf
e_{i}}-\int_{t}^{T}h_{i}^{[1],[2]}(x(\tau),z(\tau),C,b)\left(A^{\prime}_{i}+\frac{\partial
u_{i}^{[1],[2]}(x(\tau),C,b)}{\partial x}\right)d\tau,$
$-\int_{0}^{T}h_{i}^{[1],[2]}(x(t),z(t),C,b)\frac{\partial
u_{i}^{[1],[2]}(x(t),C,b)}{\partial
C}dt,-\int_{0}^{T}h_{i}^{[1],[2]}(x(t),z(t),C,b)\frac{\partial
u_{i}^{[1],[2]}(x(t),C,b)}{\partial b}dt\Bigg{]}+$
$+\Bigg{[}\sum_{i=m+1}^{n}h_{i}(x(t),z(t)){\bf
e_{i}}-\int_{t}^{T}h_{i}(x(\tau),z(\tau))A^{\prime}_{i}\,d\tau,\,{\bf
0_{m\times n}},\,{\bf 0_{m}}\Bigg{]}+$ $+\Bigg{[}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}},\,{\bf 0_{m\times n}},\,{\bf 0_{m}}\Bigg{]}.$

Let us write down the known minimum condition for a Gateaux differentiable
functional.

######  Theorem 6.3.

Let the trajectories $x_{i}(t)$, $i=\overline{1,n}$, vanish only at the
isolated time moments of the interval $[0,T]$. In order for the point
$(z^{*},C^{*},b^{*})$ to minimize the functional $I^{[1],[2]}(z,C,b)$, it is
necessary that

$0_{n}\times{\bf 0_{n}}\times{\bf 0_{m}}\times{\bf
0_{m}}=\sum_{i=1}^{m}\Bigg{[}h_{i}^{[1],[2]}(x^{*}(t),z^{*}(t),C^{*},b^{*}){\bf
e_{i}}-\int_{t}^{T}h_{i}^{[1],[2]}(x^{*}(\tau),z^{*}(\tau),C^{*},b^{*})\left(A^{\prime}_{i}+\frac{\partial
u_{i}^{[1],[2]}(x^{*}(\tau),C^{*},b^{*})}{\partial x}\right)d\tau,$
$-\int_{0}^{T}h_{i}^{[1],[2]}(x^{*}(t),z^{*}(t),C^{*},b^{*})\frac{\partial
u_{i}^{[1],[2]}(x^{*}(t),C^{*},b^{*})}{\partial
C}dt,-\int_{0}^{T}h_{i}^{[1],[2]}(x^{*}(t),z^{*}(t),C^{*},b^{*})\frac{\partial
u_{i}^{[1],[2]}(x^{*}(t),C^{*},b^{*})}{\partial b}dt\Bigg{]}+$
$+\Bigg{[}\sum_{i=m+1}^{n}h_{i}(x^{*}(t),z^{*}(t)){\bf
e_{i}}-\int_{t}^{T}h_{i}(x^{*}(\tau),z^{*}(\tau))A^{\prime}_{i}\,d\tau,\,{\bf
0_{m\times n}},\,{\bf 0_{m}}\Bigg{]}+$ $+\Bigg{[}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z^{*}_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}},\,{\bf 0_{m\times n}},\,{\bf 0_{m}}\Bigg{]}.$

where $0_{n}$ is a zero element of the space $P_{n}[0,T]$.

It is apparent, that $I(z^{*},C^{*},b^{*})=0$ is necessary and sufficient
minimum condition for this functional and in this case the equality of Theorem
5 is automatically satisfied.

Example 2. Consider the following system

$\dot{x}_{1}(t)=x_{1}(t)-3x_{2}(t)+x_{3}(t)-50|x(t)|(c_{11}x_{1}(t)-b_{1})\exp\Big{\\{}\mathrm{sign}(c_{11}x_{1}(t)-b_{1})(-(c_{11}x_{1}(t)-b_{1}))\Big{\\}},$
$\dot{x}_{2}(t)=5x_{1}(t)+x_{2}(t)-x_{3}(t)-50|x(t)|(c_{22}x_{2}(t)-b_{2})\exp\Big{\\{}\mathrm{sign}(c_{22}x_{2}(t)-b_{2})(-(c_{22}x_{2}(t)-b_{2}))\Big{\\}},$
$\dot{x}_{3}(t)=5x_{1}(t)-x_{2}(t)+x_{3}(t)$

with the boundary conditions

$x_{1}(0)=2,\ x_{2}(0)=-2,\ x_{3}(0)=2,\quad x_{1}(0.2)=0.55,\
x_{2}(0.2)=2.5,\ x_{3}(0.2)=2.95.$

As we see, the surfaces in this example are of the form

$s_{1}(x)=c_{11}x_{1}-b_{1}=0,\quad s_{2}(x)=c_{22}x_{2}-b_{2}=0.$

A simplest steepest descent method (in the functional space) 19 with some
modifications (see Remark 5, b) above) was used in order to minimize the
functional $I^{[1]}(z,c,b)$ in this problem. We use the notation for the
parameters $c=(c_{11},c_{22})^{\prime}$ here instead of the matrix $C$ for
simplicity.

The point
$(z_{\\{0\\}},c_{\\{0\\}},b_{\\{0\\}})=(0,0,0,0.18,0.2,0.12,0.51)^{\prime}$
was taken as the initial one (note that here $(0,0,0)^{\prime}$ is a zero
point in the functional space $P_{3}[0,1]$ and the points
$(0.18,0.2)^{\prime}$ and $(0.12,0.51)^{\prime}$ belong to the spaces $R^{2}$
and $R^{2}$ respectively). Herewith
$I(z_{\\{0\\}},c_{\\{0\\}},b_{\\{0\\}})\approx 1591.75905$. At the 79-th
iteration the point $(z_{\\{79\\}},c_{\\{79\\}},b_{\\{79\\}})^{\prime}$ was
constructed and we we approximately put
$(z^{*},c^{*},b^{*})=(z_{\\{79\\}},c_{\\{79\\}},b_{\\{79\\}})$. Herewith
$c^{*}\approx(0.1836729,0.2016907)^{\prime}$,
$b^{*}\approx(0.1139969,0.4974675)^{\prime}$, $I(z^{*},c^{*},b^{*})\approx
0.00064$ and $||\nabla I(z^{*},c^{*},b^{*})||_{L^{2}_{3}[0,1]\times
R^{2}\times R^{2}}\approx 0.0308$. The point $z^{*}$ is not given here for two
reasons: 1) for brevity (as it is constructed as a rather bulky piecewise
continuous vector-function) and 2) because only the parameters $c^{*}$,
$b^{*}$ are finally used to estimate the result obtained (see the next
paragraph).

Having obtained the parameters $c^{*}$, $b^{*}$, substitute them into the
system given. Integrate the closed-loop system numerically (for this example
the Rosenbrock stiff 3-4-th order method was used) and finally have
$x_{1}(0.2)\approx 0.54513$, $x_{2}(0.2)\approx 2.50537$, $x_{3}(0.2)\approx
2.95221$, so we see that the desired value on the right endpoint is achieved
with an error of order $5\times 10^{-3}$. Picture 2 demonstrates the results
of calculations. The black lines denote the curves which were obtained via the
method of the paper, while the dashed red lines denote the curves obtained via
integration of the closed-loop system.

Figure 2: Example 2. The trajectories $x_{1}^{*}(t)$, $x_{2}^{*}(t)$,
$x_{3}^{*}(t)$.

######  Remark 6.4.

In addition to regulating the structure of the vector-function $s(x)$ (i.e.
the parameters $C$ and $b$), one may consider the gains $\alpha_{i}$,
$i=\overline{1,m}$, “free” as well and optimize them based on the desired
properties of the system according to a similar scheme.

######  Remark 6.5.

As the paper presented is at most of a theoretical nature, Examples 1, 2 are
rather simple and aim at only illustrating the approach developed, so we don’t
describe the computational aspects in detail. It is planned to consider more
complicated examples and to study some aspects of the method, as well as to
use its modifications (taking into account the specifics of the functionals
under consideration) or some other known optimization methods in functional
space in order to improve computational efficiency, in future research.

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