#  Method of Quasidifferential Descent in the Problem of
Bringing a Nonsmooth System from One Point to Another

A. V. Fominyh a The work was supported by the Russian Science Foundation
(project no. 21-71-00021) aSt. Petersburg State University, St. Petersburg,
Russia.

###### Abstract

The paper considers the problem of constructing program control for an object
described by a system with nonsmooth (but only quasidifferentiable) right-hand
side. The goal of control is to bring such a system from a given initial
position to a given final state in certain finite time. The admissible
controls are piecewise continuous and bounded vector-functions with values
from some parallelepiped. The original problem is reduced to unconditional
minimization of some penalty functional which takes into account constraints
in the form of differential equations, constraints on the initial and the
final positions of the object as well as constraints on controls. Moreover, it
is known that this functional vanishes on the solution of the original problem
and only on it. The quasidifferentiability of this functional is proved,
necessary and sufficient conditions for its minimum are written out in terms
of quasidifferential. Further, in order to solve the obtained minimization
problem in the functional space the method of quasidifferential descent is
applied. The algorithm developed is demonstrated by examples.

###### keywords:

Nonsmooth right hand-side; program control; quasidifferential;
quasidifferential descent

##  1 Introduction

Despite the rich arsenal of methods accumulated over the more than 60-year
history of the development of optimal control theory, most of them deal with
classical systems whose right-hand sides are continuously differentiable
functions of their arguments. There are approaches that do not require these
smoothness conditions on systems. However, they usually use direct
discretization or some kind of “smoothing” process; both of these approaches
lead to losing some of the information about “behavior” of the system as well
as to finite-dimensional problems of huge dimensions. The paper presented is
aimed at solving the problem of bringing a nonsmooth (but only
quasidifferentiable) system from one point to another. The relevance of
considering such systems is due to their ability to more accurately and more
fully describe the “behavior” of an object in many cases.

In order to solve the problem of this paper, we will use a combination of
reducing the original problem to the problem of minimizing a functional in
some functional space as well as the apparatus of quasidifferential calculus.
The concept of “quasidifferential” was introduced by V. F. Demyanov. A rich
and constructive calculus has been developed for this nonsmooth optimization
object (see Demyanov & Rubinov (1977)). In a finite-dimensional case
quasidifferntiable functions are those whose directional derivative may be
represented as a sum of the maximum of the scalar product of the direction and
the vector from a convex compact set (called a subdifferential) and of the
minimum of the scalar product of the direction and the vector from a convex
compact (called a superdifferential). The pair of a subdifferential and a
superdifferential is called a quasidifferential. (See the strict definition of
quasidifferential applied to the right-hand sides of the system below). The
class of quasidifferentiable functions is wide. In particular, it includes all
the functions that can be represented as a superposition of the finite number
of maxima and minima of continuously differentiable functions. The concept of
quasidifferential was generalized onto functional spaces in the works of M. V.
Dolgopolik (see, e. g. Demyanov & Dolgopolik (2013), Dolgopolik (2014)).

Let us make a brief review of some papers on nonsmooth control problems. Such
works as Vinter & Cheng (1998), Vinter (2005), Frankowska (1984), Mordukhovich
(1989), Ioffe (1984), Shvartsman (2007) are devoted to classical necessary
optimality conditions in the form of the maximum principle for nonsmooth
control problems in various formulations (including the case of the presence
of phase constraints). In paper Ito & Kunisch (2011) the minimum conditions in
the form of Karush—Kuhn—Tucker are obtained for nonsmooth problems of
mathematical programming in a general problem statement with applications to
nonsmooth problems of optimal control. In paper De Oliveira & Silva (2013) on
the basis of “maximum principle invexity” some sufficient conditions are
obtained for nonsmooth control problems. The author of the paper presented
also constructed some theoretical results in the problem of program control in
systems whose right hand-sides contain modules of linear functions; in this
problem the necessary minimum conditions are obtained in terms of
quasidifferentials (see Fominyh (2019)). For the first time, quasidifferential
(in the finite-dimensional case) was used to study nonsmooth control problems
in work Demyanov, Nikulina & Shablinskaya (1986). The works listed are mainly
of theoretical nature; and it is difficult to apply rather complex minimum
conditions obtained there to specific control problems with systems with
nonsmooth right-hand sides. Let us mention some works devoted directly to the
construction of numerical methods for solving a problem similar to that
considered in this paper. The author of this paper used the methods of
subdifferential and hypodifferential descents earlier to construct optimal
control with a subdifferentiable cost functional and the system with a
continuously differentiable right-hand side (Fominyh, Karelin & Polyakova
(2018)) as well as to solve the problem of bringing a continuously
differentiable system from one point to another (in paper Fominyh (2017)). In
work Fominyh (2021) a finite-dimensional version of the quasidifferential
descent method was applied to the optimal control problem in Mayer form with
smooth right-hand side of a system and with a nondifferentiable objective
functional. In work Outrata (1983) the optimal control problem is considered
which assumes, roughly speaking, the subdifferentiability of an objective
functional and continuous differentiability of right-hand sides of a system.
The approach of this work is based on minimization of the discretized
augmented cost functional via bundle methods. In papers Gorelik & Tarakanov
(1992), Gorelik & Tarakanov (1989) the minimax control problem is considered;
with the help of a specially constructed smooth penalty function it is reduced
to a classical continuously differentiable problem. In work Morzhin (2009) a
subdifferentiable penalty function is constructed in order to take the
constraints on control into account; after that the subdifferential smoothing
process is also used. In papers Mayne & Polak (1985), Mayne & Smith (1988) the
exact penalty function constructed (in order to take phase constraints into
account) is also subdifferentiable; an algorithm for minimizing the derivative
of this function with respect to the direction is considered. After
“transition” to continuously differentiable problems a widely developed
arsenal for solving classical control problems can be used in order to solve
them. In works Noori Skandari, Kamyad & Effati (2015), Noori Skandari, Kamyad
& Effati (2013) more general problems are considered in which the
nonsmoothness of right-hand sides of the system describing a controlled object
is allowed. Here with the help of basis functions (Fourier series) the
smoothing of system right-hand sides is also carried out, after which the
Chebyshev pseudospectral method is used to solve the problem. In paper Ross &
Fahroo (2004) the direct method for optimizing trajectories of nonsmooth
optimal control problems is proposed based on the Legendre’s pseudospectral
method.

The method considered in this paper belongs to the so called direct methods of
the variational analysis (see Demyanov & Tamasyan (2011)). The method is also
“continuous” unlike most methods in literature as it is not based on direct
discretization of the original problem. Although similar methods have already
been applied to some problems of variational calculus and optimal control, so
far it was impossible to apply this method to nonsmooth control problems. The
main difficulty was in a too complicated form of quasidifferentials and
optimality conditions obtained. The new technical idea of the current paper is
to consider phase trajectory and its derivative as independent variables (and
to take the natural relation between these variables into account via penalty
function of a special form). To the best of the author’s knowledge, this idea
is used in literature for the first time. It allowed to simplify the
quasidifferential structure of the functional under consideration and to solve
the problem of finding the steepest descent direction of the minimized
functional.

##  2 Basic Definitions and Notation

The paper is organized as follows. Section 2 contains the notation of the
paper as well as definitions of the quasidifferentials of the functions (the
functionals) required. In Section 3 the problem statement and the main
assumptions are presented. In Section 4 the original problem is reduced to the
unconstrained minimization one. The quasidifferentiability of the main
functional is proved in Section 5; after that minimum conditions for the
unconstrained problem are obtained. The quasidifferential descent method is
described in Section 6. In this section we also discuss the methods for
solving the auxiliary problems arising during the basic algorithm
implementation. Justification of possibility of finding the steepest descent
direction at discrete time moments is carried out in the section as well.
Section 7 contains the numerical examples illustrating the method realization
with a rather detailed analysis of the problems considered. In Section 8
advantages and disadvantages of the method are discussed. Section 9 summarizes
the main results of the paper. Finally, Appendix contains some known
quasidifferential rules description applied to the specific functional
considered.

In the paper we will use the following notations. $C_{n}[0,T]$ is the space of
continuous on $[0,T]$ $n$-dimensional vector-functions; $P_{n}[0,T]$ is the
space of piecewise continuous and bounded on $[0,T]$ $n$-dimensional vector-
functions. Denote $L_{p}^{n}[0,T]$, $1\leqslant p<\infty$, the space of
measurable on $[0,T]$ $n$-dimensional vector-functions which are summable with
the degree $p$, also denote $L_{\infty}^{n}[0,T]$ the space of measurable on
$[0,T]$ and a. e. bounded $n$-dimensional vector-functions. If the function
$p(t)$ is defined on the segment $[0,T]$ and $\overline{T}$ is some subset of
this segment, then $p(t)|_{\overline{T}}$ denotes its restriction to this set.
Denote $\mathrm{co}P$ a convex hull of the set $P\subset R^{n}$. The sum $E+F$
of the sets $E,F\subset R^{n}$ is their Minkowski sum, while $\lambda E$ with
$\lambda\in R$ is the Minkowski product. Let $B_{r}(c)$ ($D_{r}(c)$) denote a
closed (an open) ball of some space with the radius $r>0$ and with the center
$c$ from this space; herewith, for some set $C$ (from the same space)
$B_{r}(C)$ ($D_{r}(C)$) denotes the union of all closed (open) balls with the
radius $r>0$ and the centers from the set $C$. Denote $\langle a,b\rangle$ the
scalar product of the vectors $a$, $b$ $\in R^{d}$. Let $X$ be a normed space,
then $||\cdot||_{X}$ denotes the norm in this space, and $X^{*}$ denotes the
space, conjugate to the space $X$. Finally, for some number $\alpha$ let
$o(\alpha)$ denote such a value that $o(\alpha)/\alpha\rightarrow 0\
\text{if}\ \alpha\rightarrow 0$.

In the paper we will use both quasidifferentials of functions in a finite-
dimensional space and quasidifferentials 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 $R^{n}\times R^{m}$ with the standard norm. Let
$g=[g_{1},g_{2}]\in R^{n}\times R^{m}$ be an arbitrary vector. Suppose that at
every time moment $t\in[0,T]$ at the point $(x,u)$ there exist such convex
compact sets $\underline{\partial}f_{i}(x,u,t)$,
$\overline{\partial}f_{i}(x,u,t)$ $\subset R^{n}\times R^{m}$,
$i=\overline{1,n}$, that

$\frac{\partial f_{i}(x,u,t)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}f_{i}(x+\alpha g_{1},u+\alpha
g_{2},t)-f_{i}(x,u,t)\big{)}=$ (1)
$=\max_{v\in\underline{\partial}f_{i}(x,u,t)}\langle
v,g\rangle+\min_{w\in\overline{\partial}f_{i}(x,u,t)}\langle w,g\rangle,\quad
i=\overline{1,n}.$

In this case the function $f_{i}(x,u,t)$, $i=\overline{1,n}$, is called
quasidifferentiable at the point $(x,u)$ and the pair
$\mathcal{D}f_{i}(x,u)=\big{[}\underline{\partial}f_{i}(x,u,t),\overline{\partial}f_{i}(x,u,t)\big{]}$
is called a quasidifferential of the function $f_{i}(x,u,t)$ (herewith, the
sets $\underline{\partial}f_{i}(x,u,t)$ and $\overline{\partial}f_{i}(x,u,t)$
are called a subdifferential and a superdifferential respectively of the
function $f_{i}(x,u,t)$ at the point $(x,u)$).

From expression (1) one can see that at each $t\in[0,T]$ the following formula

$f_{i}(x+\alpha g_{1},u+\alpha g_{2},t)=f_{i}(x,u,t)+\alpha\frac{\partial
f_{i}(x,u,t)}{\partial g}+o_{i}(\alpha,x,u,g,t),$ (2)
$\quad\frac{o_{i}(\alpha,x,u,g,t)}{\alpha}\rightarrow 0,\ \alpha\downarrow
0,\quad i=\overline{1,n},$

holds true.

If for each number $\varepsilon>0$ there exist such numbers $\delta>0$ and
$\alpha_{0}>0$ that at $\overline{g}\in B_{\delta}(g)$ and at
$\alpha\in(0,\alpha_{0})$ one has
$|o_{i}(\alpha,x,u,\overline{g},t)|<\alpha\varepsilon$, $i=\overline{1,n}$,
then the function $f_{i}(x,u,t)$, $i=\overline{1,n}$, is called uniformly
quasidifferentiable at the point $(x,u)$. Note (Demyanov & Vasil’ev (1986))
that if at each $t\in[0,T]$ the function $f_{i}(x,u,t)$, $i=\overline{1,n}$,
is quasidifferentiable at the point $(x,u)$ and is locally Lipschitz
continuous in the vicinity of the point $(x,u)$, then it is uniformly
quasidifferentiable at the point $(x,u)$. If for the uniformly
quasidifferentiable function $f_{i}(x,u,t)$, $i=\overline{1,n}$, in expression
(2) one has $\displaystyle{\frac{o_{i}(\alpha,x,u,g,t)}{\alpha}\rightarrow
0}$, $\alpha\downarrow 0$, $i=\overline{1,n}$, uniformly in $t\in[0,T]$, then
such a function is called absolutely uniformly quasidifferentiable.

Consider the space $C_{n}[0,T]\times P_{n}[0,T]\times P_{m}[0,T]$ with the
following norm: $L_{2}^{n}[0,T]\times L_{2}^{n}[0,T]\times L_{2}^{m}[0,T]$.
Let $g=[g_{1},g_{2},g_{3}]\in C_{n}[0,T]\times P_{n}[0,T]\times P_{m}[0,T]$ be
an arbitrary vector-function. Suppose that at the point $(x,z,u)$ there exist
such convex weakly* compact sets
$\underline{\partial}{I(x,z,u)},\overline{\partial}{I(x,z,u)}\subset\big{(}C_{n}[0,T]\times
P_{n}[0,T]\times P_{m}[0,T],$ $||\cdot||_{L_{2}^{n}[0,T]\times
L_{2}^{n}[0,T]\times L_{2}^{m}[0,T]}\big{)}^{*}$ that

$\frac{\partial I(x,z,u)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}I(x+\alpha g_{1},z+\alpha g_{2},u+\alpha
g_{3})-I(x,z,u)\big{)}=$ (3)
$=\max_{v\in\underline{\partial}I(x,z,u)}v(g)+\min_{w\in\overline{\partial}I(x,z,u)}w(g).$

In this case the functional $I(x,z,u)$ is called quasidifferentiable at the
point $(x,z,u)$ and the pair
$\mathcal{D}I(x,z,u)=\big{[}\underline{\partial}{I(x,z,u)},\overline{\partial}{I(x,z,u)}\big{]}$
is called a quasidifferential of the functional $I(x,z,u)$ (herewith, the sets
$\underline{\partial}{I(x,z,u)}$ and $\overline{\partial}{I(x,z,u)}$ are
called a subdifferential and a superdifferential respectively of the
functional $I(x,z,u)$ at the point $(x,z,u)$).

From expression (3) one can see that the following formula

$I(x+\alpha g_{1},z+\alpha g_{2},u+\alpha g_{3})=I(x,z,u)+\alpha\frac{\partial
I(x,z,u)}{\partial g}+o(\alpha,x,z,u,g),$ (4)
$\quad\frac{o(\alpha,x,z,u,g)}{\alpha}\rightarrow 0,\ \alpha\downarrow 0,$

holds true.

##  3 Statement of the Problem

Consider the system of ordinary differential equations

$\dot{x}(t)=f(x(t),u(t),t)$ (5)

with the initial point

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

In formula (5) $f(x,u,t)$, $t~{}\in~{}[0,T]$, is a given $n$-dimensional
vector-function; $T>0$ is a known finite time moment. In formula (6) $x_{0}\in
R^{n}$ is a given vector.

Assumption 1. The $n$-dimensional vector-function $x(t)$ of phase coordinates
is assumed to be continuous and continuously differentiable at every
$t\in[0,T]$ with the exception of, possibly, only the finite number of points
(herewith, we suppose that its derivative is bounded on its domain). The
$m$-dimensional vector-function $u(t)$ of controls is supposed to be piecewise
continuous and bounded on $[0,T]$. The vector-function $f(x,u,t)$ is supposed
to be continuous; and each of its components $f_{i}(x,u,t)$,
$i=\overline{1,n}$, — to be quasidifferentiable and locally Lipschitz
continuous in the pair $(x,u)$ of variables at each fixed $t\in[0,T]$.

Under the assumptions made for system (5), (6), the classical solution
existence and uniqueness theorems hold true, at least, in some neighborhood of
the initial point.

As noted above, in the paper we assume that each trajectory $x(t)$ is a
piecewise continuously differentiable vector-function with bounded on its
domain derivative and $u(t)$ is a piecewise continuous and bounded vector-
function. If $t_{0}\in[0,T)$ is a discontinuity point of the vector-function
$u(t)$, then for definiteness we assume that
$u(t_{0})=\lim\limits_{t\downarrow t_{0}}u(t)$. At the point $T$ put
$u(T)=\lim\limits_{t\uparrow T}u(t)$. So we also assume that $\dot{x}(t_{0})$
is a right-hand derivative of the vector-function $x(t)$ at the point $t_{0}$
and 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 $C_{n}[0,T]$, the vector-
function $\dot{x}(t)$ belongs to the space $P_{n}[0,T]$ and the vector-
function $u(t)$ belongs to the space $P_{m}[0,T]$.

Introduce the set of admissible controls

$U=\big{\\{}u\in P_{m}[0,T]\ |\ \underline{u}_{i}\leq
u_{i}(t)\leq\overline{u}_{i},\ i=\overline{1,m},\ t\in[0,T]\big{\\}}.$ (7)

Here $\underline{u}_{i},\overline{u}_{i}\in R,i=\overline{1,m}$, are given
numbers.

It is required to find such a control $u^{*}\in U$ that brings the
corresponding trajectory $x^{*}(t)$ from initial point (6) to the final state

$x(T)=x_{T},$ (8)

where $x_{T}\in R^{n}$ is a given vector.

Assumption 2. We suppose that there exists such a solution.

##  4 Reduction to an Unconstrained Minimization Problem

Construct the functional taking into account different constraints on the
object and on control which are given in the statement of the problem. Let
$z(t)=\dot{x}(t)$ (under the assumptions made, $z\in P_{n}[0,T]$), then
according to (6) (where the initial state of the system is given) we have

$\displaystyle{x(t)=x_{0}+\int_{0}^{t}z(\tau)d\tau}.$ (9)

Construct the following functional on the space $P_{n}[0,T]\times P_{m}[0,T]$:

$\overline{I}(z,u)=\sum_{i=1}^{n}\int_{0}^{T}\Big{|}z_{i}(t)-f_{i}\Big{(}x_{0}+\int_{0}^{t}z(\tau)d\tau,u(t),t\Big{)}\Big{|}dt+\frac{1}{2}\Big{(}x_{0}+\int_{0}^{T}z(t)dt-
x_{T}\Big{)}^{2}+$
$+\sum_{i=1}^{m}\int_{0}^{T}\max\big{\\{}\underline{u}_{i}-u_{i}(t),0\big{\\}}dt+\sum_{i=1}^{m}\int_{0}^{T}\max\big{\\{}{u}_{i}(t)-\overline{u}_{i}(t),0\big{\\}}dt.$

In the functional $\overline{I}(z,u)$ the first summand (which is a sum) takes
into account differential constraint (5), the second summand takes into
account constraint (8) on the final state of the system, the third summand
(consisting of two sums) takes into account constraint (7) on control. Note
that this functional is nonnegative for any of its arguments and
$\overline{I}(z^{*},u^{*})=0$ iff the pair $(x^{*},u^{*})\in C_{n}[0,T]\times
P_{m}[0,T]$ is a solution of the original problem, i. e. the control $u^{*}$
belongs to the set $U$ of admissible controls and brings the corresponding
trajectory $\displaystyle{x^{*}(t)=x_{0}+\int_{0}^{t}z^{*}(\tau)d\tau}$ from
the given initial position $x_{0}$ to the given final state $x_{T}$ in the
time $T$.

Transition to the “space of derivatives” $\left(z\in P_{n}[0,T]\right)$ has
been used in many works of V. F. Demyanov and his students to study various
variational and control problems. Under some natural additional assumptions
one can prove the quasidifferentiability of the functional $\overline{I}(z,u)$
in the space $P_{n}[0,T]\times P_{m}[0,T]$ as a normed space with the norm
$L_{2}^{n}[0,T]\times L_{2}^{m}[0,T]$. However, the quasidifferential of this
functional has a rather complicated structure which makes it practically
unsuitable for constructing numerical methods. Therefore, in this paper it is
proposed to consider some modification of this functional, “forcibly”
considering the points $z$ and $x$ to be “independent” variables. Since, in
fact, there is relationship (9) between these variables (which naturally means
that the vector-function $z(t)$ is a derivative of the vector-function
$x(t)$), let us take it into account by adding the corresponding (last) term
when constructing the new functional on the space $C_{n}[0,T]\times
P_{n}[0,T]\times P_{m}[0,T]$:

$I(x,z,u)=I_{1}(x,z,u)+I_{2}(z)+I_{3}(u)+I_{4}(x,z)=$
$=\sum_{i=1}^{n}\int_{0}^{T}\big{|}z_{i}(t)-f_{i}(x(t),u(t),t)\big{|}dt+\frac{1}{2}\Big{(}x_{0}+\int_{0}^{T}z(t)dt-
x_{T}\Big{)}^{2}+$ (10)
$+\sum_{i=1}^{m}\int_{0}^{T}\max\\{\underline{u}_{i}-u_{i}(t),0\\}dt+\sum_{i=1}^{m}\int_{0}^{T}\max\\{{u}_{i}(t)-\overline{u}_{i},0\\}dt+$
$+\frac{1}{2}\int_{0}^{T}\Big{(}x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau\Big{)}^{2}dt.$

Note that this functional is also nonnegative for any of its arguments and
$I(x^{*},z^{*},u^{*})=0$ iff the pair $(x^{*},u^{*})\in C_{n}[0,T]\times
P_{m}[0,T]$ is a solution of the original problem, i. e. the control $u^{*}$
belongs to the set $U$ of admissible controls and brings the corresponding
trajectory $\displaystyle{x^{*}(t)=x_{0}+\int_{0}^{t}z^{*}(\tau)d\tau}$ from
the given initial position $x_{0}$ to the given final state $x_{T}$ in the
time $T$. It is obvious that if some of the right endpoint coordinates of an
object are free, then we put the corresponding summands of the functional
$I_{2}(z)$ equal to zero. It is also obvious that if some of the restrictions
on controls are absent, one has to remove the corresponding summands from the
functional $I_{3}(u)$. In both these cases we keep for the functional
$I(x,z,u)$ its notation.

Despite the fact that the dimension of functional $I(x,z,u)$ arguments is $n$
more the dimension of functional $\overline{I}(z,u)$ arguments, the structure
of its quasidifferential (in the space $C_{n}[0,T]\times P_{n}[0,T]\times
P_{m}[0,T]$ as a normed space with the norm $L_{2}^{n}[0,T]\times\times
L_{2}^{n}[0,T]\times L_{2}^{m}[0,T]$), as will be seen from what follows, is
much simpler than the structure of the functional $\overline{I}(z,u)$
quasidifferential. This will allow us to construct a numerical method for
solving the original problem.

Thus, the initial problem has been reduced to finding an unconstrained global
minimum point of the functional $I(x,z,u)$ on the space

$X=\Big{(}C_{n}[0,T]\times P_{n}[0,T]\times
P_{m}[0,T],||\cdot||_{L_{2}^{n}[0,T]\times L_{2}^{n}[0,T]\times
L_{2}^{m}[0,T]}\Big{)}.$

######  Remark 1.

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 ${L_{2}^{n}[0,T]}$ and the space
$\Big{(}P_{n}[0,T],||\cdot||_{L_{2}^{n}[0,T]}\Big{)}$ is also everywhere dense
in the space ${L_{2}^{n}[0,T]}$, then the space $X^{*}$ conjugate to the space
$X$ introduced in the previous paragraph is isometrically isomorphic to the
space ${L_{2}^{n}[0,T]\times L_{2}^{n}[0,T]\times L_{2}^{m}[0,T]}$ (see
Kolmogorov & Fomin (1999)).

##  5 Minimum Conditions of the Functional $I(x,z,u)$

Let us formulate a necessary and sufficient minimum condition for the
functional $I(x,z,u)$ that follows from its construction.

######  Theorem 5.1.

Let Assumptions 1, 2 be satisfied. In order for the point
$(x^{*},z^{*},u^{*})$ to minimize the functional $I(x,z,u)$, it is necessary
and sufficient to have $I(x^{*},z^{*},u^{*})=~{}0$.

In order to obtain a more constructive (than that given in Theorem 5.1)
minimum condition useful for constructing numerical methods for solving the
problem posed, first, let us investigate the differential properties of the
functional $I(x,z,u)$.

Using the classical variation one can directly prove the Gateaux
differentiability of the functional $I_{2}(z)$; we have

$\nabla I_{2}(z)=x_{0}+\int_{0}^{T}z(t)dt-x_{T}.$

By quasidifferential calculus rules (Dolgopolik (2011), Example 3.1) one may
put

$\mathcal{D}\,I_{2}(z)=\big{[}\underline{\partial}I_{2}(z),\,\overline{\partial}I_{2}(z)\big{]}=\bigg{[}x_{0}+\int_{0}^{T}z(t)dt-
x_{T},\ 0_{n}\bigg{]}.$

Using the classical variation and integration by parts one can directly check
(cf., e. g. Demyanov & Tamasyan (2011), Formula (14)) the Gateaux
differentiability of the functional $I_{4}(x,z)$; we obtain

$\nabla
I_{4}(x,z,t)=\begin{pmatrix}\displaystyle{x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau}\\\
\displaystyle{-\int_{t}^{T}\Big{(}x(\tau)-x_{0}-\int_{0}^{\tau}z(s)ds\Big{)}d\tau}\end{pmatrix}.$

By quasidifferential calculus rules (Dolgopolik (2011), Example 3.1) one may
put

$\mathcal{D}\,I_{4}(x,z)=\big{[}\underline{\partial}I_{4}(x,z),\,\overline{\partial}I_{4}(x,z)\big{]}=\left[\begin{pmatrix}\displaystyle{x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau}\\\
\displaystyle{-\int_{t}^{T}\Big{(}x(\tau)-x_{0}-\int_{0}^{\tau}z(s)ds\Big{)}d\tau}\end{pmatrix},\
0_{2n}\right].$

Study now the differential properties of the functionals $I_{1}(x,z,u)$ and
$I_{3}(u)$. For this, we prove the following theorem for a functional of a
more general form.

######  Theorem 5.2.

Let the functional

$J(\xi)=\int_{0}^{T}\varphi(\xi(t),t)dt$

be given where $\xi\in P_{l}[0,T]$, the function $\varphi(\xi,t)$ is
continuous and is also absolutely uniformly quasidifferentiable, with the
quasidifferential
$[\underline{\partial}\varphi(\xi,t),\overline{\partial}\varphi(\xi,t)]$.
Suppose also that the mappings
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ and
$t\rightarrow\overline{\partial}\varphi(\xi(t),t)$ are upper semicontinuous.

Then the functional $J(\xi)$ is quasidifferentiable, i. e.

1) The derivative of the functional $J(\xi)$ in the direction $g$ exists and
is of the form

$\frac{\partial J(\xi)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\big{(}J(\xi+\alpha g)-J(\xi)\big{)}=$ (11)
$=\max_{v\in\underline{\partial}J(\xi)}\int_{0}^{T}\langle v(t),g(t)\rangle
dt+\min_{w\in\overline{\partial}J(\xi)}\int_{0}^{T}\langle w(t),g(t)\rangle
dt$

where $g\in P_{l}[0,T]$, and the sets $\underline{\partial}J(\xi)$,
$\overline{\partial}J(\xi)$ are of the form

$\underline{\partial}J(\xi)=\Big{\\{}v\in L^{l}_{\infty}[0,T]\ \big{|}\
v(t)\in\underline{\partial}\varphi(\xi(t),t)\ \forall t\in[0,T]\Big{\\}},$
(12) $\overline{\partial}J(\xi)=\Big{\\{}w\in L^{l}_{\infty}[0,T]\ \big{|}\
w(t)\in\overline{\partial}\varphi(\xi(t),t)\ \forall t\in[0,T]\Big{\\}}.$ (13)

2) The sets $\underline{\partial}J(\xi)$, $\overline{\partial}J(\xi)$ are
convex and weakly* compact subsets of the space
$\big{(}P_{l}[0,T],||\cdot||_{L_{2}^{l}[0,T]}\big{)}^{*}$.

###### Proof.

Prove statement 1).

Insofar as the function $\varphi(\xi,t)$ is quasidifferentiable by assumption,
then for every $g\in P_{l}[0,T]$ and for each $\alpha>0$ we have (see formula
(2))

$J(\xi+\alpha
g)-J(\xi)=\int_{0}^{T}\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle
v(t),\alpha g(t)\rangle
dt+\int_{0}^{T}\min_{w\in\overline{\partial}\varphi(\xi,t)}\langle w(t),\alpha
g(t)\rangle dt+$ (14)
$+\int_{0}^{T}{o(\alpha,\xi(t),g(t),t)}dt,\quad\frac{o(\alpha,\xi(t),g(t),t)}{\alpha}\rightarrow
0,\ \alpha\downarrow 0.$

At this point let us check that the integrals in the right-hand side of this
formula are correctly defined.

Insofar as $\xi,g\in P_{l}[0,T]$ and the function $\varphi(\xi,t)$ is
continuous, one has that for each $\alpha>0$ the functions
$t\rightarrow\varphi(\xi(t),t)$ and $t\rightarrow\varphi(\xi(t)+\alpha
g(t),t)$ belong to the space $L_{\infty}^{1}[0,T]$.

Under the assumption made, the mappings
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ and
$t\rightarrow\overline{\partial}\varphi(\xi(t),t)$ are upper semicontinuous
and then are also measurable (Blagodatskikh & Filippov (1986)). Then due to
the piecewise continuity and the boundedness of the function $g(t)$ and due to
continuity of the scalar product in its variables we obtain that for each
$\alpha>0$ the mappings
$t\rightarrow\max_{v(t)\in\underline{\partial}\varphi(\xi(t),t)}\langle
v(t),\alpha g(t)\rangle$ and
$t\rightarrow\min_{w(t)\in\overline{\partial}\varphi(\xi(t),t)}\langle
w(t),\alpha g(t)\rangle$ are upper semicontinuous (Aubin & Frankowska (1990))
and then are also measurable (Blagodatskikh & Filippov (1986)). During the
proof of statement 2) it will be shown that under the assumptions made, the
sets $\underline{\partial}\varphi(\xi,t)$ and
$\overline{\partial}\varphi(\xi,t)$ are bounded uniformly in $t\in[0,T]$, from
here taking into account the fact that $g\in P_{l}[0,T]$, check that for each
$\alpha>0$ the mappings
$t\rightarrow\max_{v(t)\in\underline{\partial}\varphi(\xi(t),t)}\langle
v(t),\alpha g(t)\rangle$ and
$t\rightarrow\min_{w(t)\in\overline{\partial}\varphi(\xi(t),t)}\langle
w(t),\alpha g(t)\rangle$ are also bounded uniformly in $t\in[0,T]$. Indeed,
fix some $g\in P_{l}[0,T]$ and $\alpha>0$ and for each $t\in[0,T]$ take such a
vector $\overline{v}(t)\in\underline{\partial}\varphi(\xi(t),t)$ that
$\langle\overline{v}(t),\alpha
g(t)\rangle=\max_{v(t)\in\underline{\partial}\varphi(\xi(t),t)}\langle
v(t),\alpha g(t)\rangle$ (the vector $\overline{v}(t)$ exists since for each
$t\in[0,T]$ the set $\underline{\partial}\varphi(\xi(t),t)$ is a convex
compact). Then by Cauchy–Schwarz inequality $\langle\overline{v}(t),\alpha
g(t)\rangle\leq\alpha||\overline{v}(t)||_{R^{l}}||g(t)||_{R^{l}}$, and the
value on the right-hand side is bounded (uniformly in $t\in[0,T]$) since $g\in
P_{l}[0,T]$ and since the set $\underline{\partial}\varphi(\xi(t),t)$ is
bounded uniformly in $t\in[0,T]$. (The justification regarding the mapping
$t\rightarrow\min_{w(t)\in\overline{\partial}\varphi(\xi(t),t)}\langle
w(t),\alpha g(t)\rangle$ is carried out in a completely analogous fashion.) So
we finally have that for each $\alpha>0$ the mappings
$t\rightarrow\max_{v(t)\in\underline{\partial}\varphi(\xi(t),t)}\langle
v(t),\alpha g(t)\rangle$ and
$t\rightarrow\min_{w(t)\in\overline{\partial}\varphi(\xi(t),t)}\langle
w(t),\alpha g(t)\rangle$ belong to the space $L_{\infty}^{1}[0,T]$.

Then for every $\alpha>0$ one has $t\rightarrow{o(\alpha,\xi(t),g(t),t)}\in
L_{\infty}^{1}[0,T]$ and due to the absolutely uniformly
quasidifferentiability of the function $\varphi(\xi,t)$ we have

$\frac{o(\alpha,\xi(t),g(t),t)}{\alpha}=:\frac{o(\alpha)}{\alpha}\rightarrow
0,\ \alpha\downarrow 0.$ (15)

Now our aim is to “bring the operations of taking maximum and minimum out of
the integral”, i. e. to obtain the expression in the right-hand side of
formula (11).

Consider the functional
$\displaystyle{\int_{0}^{T}\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle
v(t),\alpha g(t)\rangle dt}$ in detail. For each $\alpha>0$ and for each
$t\in[0,T]$ we have the obvious inequality

$\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle v(t),\alpha
g(t)\rangle\geqslant\langle v(t),\alpha g(t)\rangle$

where $v(t)$ is a measurable selector of the mapping
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ (due to the noted
boundedness property of the set $\underline{\partial}\varphi(\xi,t)$ uniformly
in $t\in[0,T]$ we have $v\in L_{\infty}^{l}[0,T]$) and by virtue of formula
(12) form for every $\alpha>0$ one has the inequality

$\int_{0}^{T}\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle v(t),\alpha
g(t)\rangle
dt\geqslant\max_{v\in\underline{\partial}J(\xi)}\int_{0}^{T}\langle
v(t),\alpha g(t)\rangle dt.$

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

$\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle v(t),\alpha
g(t)\rangle\in\Big{\\{}\langle v(t),\alpha g(t)\rangle\ \big{|}\
v(t)\in\underline{\partial}\varphi(\xi(t),t)\Big{\\}}$

and the set $\underline{\partial}\varphi(\xi,t)$ is closed and bounded at each
fixed $t$ by the definition of subdifferential and the mapping
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ is upper semicontinuous by
assumption and also because the scalar product is continuous in its arguments
and $g\in P_{l}[0,T]$, then due to Filippov lemma (see Filippov (1959)) there
exists such a measurable selector $\overline{v}(t)$ of the mapping
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ that for each $\alpha>0$
and for each $t\in[0,T]$ we have

$\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle v(t),\alpha
g(t)\rangle=\langle\overline{v}(t),\alpha g(t)\rangle,$

so we have found the element $\overline{v}$ from the set
$\underline{\partial}J(\xi)$ which brings the equality in the previous
inequality. Thus, finally we obtain

$\int_{0}^{T}\max_{v\in\underline{\partial}\varphi(\xi,t)}\langle v(t),\alpha
g(t)\rangle dt=\max_{v\in\underline{\partial}J(\xi)}\int_{0}^{T}\langle
v(t),\alpha g(t)\rangle dt.$ (16)

Consideration of the functional
$\displaystyle{\int_{0}^{T}\min_{w\in\overline{\partial}\varphi(\xi,t)}\langle
w(t),\alpha g(t)\rangle dt}$ is carried out in a completely analogous fashion.
Taking formula (13) form into account we have

$\int_{0}^{T}\min_{w\in\overline{\partial}\varphi(\xi,t)}\langle w(t),\alpha
g(t)\rangle dt=\min_{w\in\overline{\partial}J(\xi)}\int_{0}^{T}\langle
w(t),\alpha g(t)\rangle dt.$ (17)

From expressions (14), (15), (16), (17) follows formula (11) (see expression
(4)).

Prove statement 2).

The convexity of the sets $\underline{\partial}J(\xi)$ and
$\overline{\partial}J(\xi)$ immediately follows from the convexity at each
fixed $t\in[0,T]$ of the sets $\underline{\partial}\varphi(\xi,t)$ and
$\overline{\partial}\varphi(\xi,t)$ respectively.

Prove the boundedness of the set $\underline{\partial}\varphi(\xi,t)$
uniformly in $t\in[0,T]$. Due to the upper semicontinuity of the mapping
$t\rightarrow\underline{\partial}\varphi(\xi(t),t)$ at each $t\in[0,T]$ there
exists such a number $\delta(t)$ that under the condition
$|\overline{t}-t|<\delta(t)$ the inclusion
$\underline{\partial}\varphi(\xi(\overline{t}),\overline{t})\subset
B_{r}(\underline{\partial}\varphi(\xi({t}),{t}))$ 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 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 a number $\delta>0$ that for every $t\in[0,T]$
the inclusion
$\underline{\partial}\varphi(\xi(\overline{t}),\overline{t})\subset
B_{r}(\underline{\partial}\varphi(\xi({t}),{t}))$ 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
$\underline{\partial}\varphi(\xi,{t})\subset\bigcup\limits_{i=1}^{N}B_{r}(\underline{\partial}\varphi(\xi({t_{i}}),{t_{i}}))$
for all $t\in[0,T]$. It remains to notice that the set
$\bigcup\limits_{i=1}^{N}B_{r}(\underline{\partial}\varphi(\xi({t_{i}}),{t_{i}}))$
is bounded due to the compactness of the set
$\underline{\partial}\varphi(\xi,{t})$ at each fixed $t\in[0,T]$. The
boundedness of the set $\overline{\partial}\varphi(\xi,t)$ uniformly in
$t\in[0,T]$ may be proved similarly.

Prove that the set $\underline{\partial}J(\xi)$ is weakly closed. As shown in
statement 1) proof and at the beginning of statement 2) proof, the set
$\underline{\partial}J(\xi)$ is convex and its elements $v$ belong to the
space $L_{\infty}^{l}[0,T]$. Then all the more the set
$\underline{\partial}J(\xi)$ is a convex subset of the space $L_{2}^{l}[0,T]$.
Let us prove that the set $\underline{\partial}J(\xi)$ is closed in the weak
topology of the space $L_{2}^{l}[0,T]$. Let $\\{v_{n}\\}_{n=1}^{\infty}$ be
the sequence of functions from the set $\underline{\partial}J(\xi)$ converging
to the function $v^{*}$ in the strong topology of the space $L_{2}^{l}[0,T]$.
It is known (Munroe (1953)) that this sequence has the subsequence
$\\{v_{n_{k}}\\}_{n_{k}=1}^{\infty}$ converging pointwise to $v^{*}$ almost
everywhere on $[0,T]$, i. e. there exists such a subset
$T^{\prime}\subset[0,T]$ having the measure $T$ that for every point $t\in
T^{\prime}$ we have $v_{n_{k}}(t)\in\underline{\partial}\varphi(\xi(t),t)$ and
$v_{n_{k}}(t)$ converges to $v^{*}(t)$, $n_{k}=1,2,\dots$. But the set
$\underline{\partial}\varphi(\xi(t),t)$ is closed at each $t\in[0,T]$ by the
definition of subdifferential, hence for every $t\in T^{\prime}$ we have
$v^{*}(t)\in\underline{\partial}\varphi(\xi(t),t)$. So the set
$\underline{\partial}J(\xi)$ is closed in the strong topology of the space
$L_{2}^{l}[0,T]$ but it is also convex, so it is also closed in the weak
topology of the space $L_{2}^{l}[0,T]$ (Dunford & Schwartz (1958)). One can
prove that the set $\overline{\partial}J(\xi)$ is weakly closed (in
$L_{2}^{l}[0,T]$) in a similar way.

Recall that by virtue of Remark 1 it is sufficient to consider the space
$L_{2}^{l}[0,T]$. The weak* compactness of the sets
$\underline{\partial}J(\xi)$, $\overline{\partial}J(\xi)$ in the space
$L_{2}^{l}[0,T]$ follows from their weak compactness (in $L_{2}^{l}[0,T]$) by
virtue of these topologies definitions (see Kolmogorov & Fomin (1999)). The
space $L_{2}^{l}[0,T]$ is reflexive (Dunford & Schwartz (1958)), so the set
there is weakly compact if and only if it is bounded in norm and weakly closed
(Dunford & Schwartz (1958)) in this space. These required properties have been
proved in the previous two paragraphs.

The theorem is proved. ∎

Thus, as one can see from Theorem 5.2, the quasidifferentials of the
functionals $I_{1}(x,z,u)$ and $I_{3}(u)$ are completely defined by the
quasidifferentials of their integrands (at each time moment $t\in[0,T]$). The
Appendix contains the detailed description of calculating the
quasidifferentials required as well as the main quasidifferential rules.

We have the following final formula for calculating the quasidifferential of
the functional $I(x,z,u)$ at the point $(x,z,u)$

$\mathcal{D}\,I(x,z,u)=\big{[}\underline{\partial}I(x,z,u),\,\overline{\partial}I(x,z,u)\big{]}=\left[\sum_{k=1}^{4}\underline{\partial}I_{k}(x,z,u),\
\sum_{k=1}^{4}\overline{\partial}I_{k}(x,z,u)\right]$ (18)

where formally $I_{2}(x,z,u):=I_{2}(z)$, $I_{3}(x,z,u):=I_{3}(u)$,
$I_{4}(x,z,u):=I_{4}(x,z)$.

######  Remark 2.

The assumption of the absolute uniform quasidifferentiability has been made in
order to simplify the presentation. Via a special form of the mean value
theorem (Dolgopolik (2018), Pr. 2) for quasidifferentials one can show that
this assumption is actually redundant.

Using the known minimum condition (of the functional $I(x,z,u)$ at the point
$(x^{*},z^{*},u^{*})$ in this case) in terms of quasidifferential, we conclude
that the following theorem is true.

######  Theorem 5.3.

(Demyanov & Dolgopolik (2013), Th. 1)

Let Assumptions 1, 2 be satisfied. In order for the control $u^{*}\in U$ to
bring system (5) from initial point (6) to final state (8) in the time $T$, it
is necessary to have

$-\overline{\partial}I(x^{*},z^{*},u^{*})\subset\underline{\partial}I(x^{*},z^{*},u^{*})$
(19)

at each $t\in[0,T]$ where the expressions for the subdifferential
$\underline{\partial}I(x,z,u)$ and for the superdifferential
$\overline{\partial}I(x,z,u)$ are given by formula (18). If one has
$I(x^{*},z^{*},u^{*})=0$, then condition (19) is also sufficient.

Theorem 5.3 already contains a constructive minimum condition since on its
basis it is possible to construct the quasidifferential descent method; and
for solving each of the subproblems arising during realization of this method
(for a wide class of functions) there are known efficient algorithms for
solving them.

##  6 The Quasidifferential Descent Method

Let us describe the following quasidifferentiable descent method for finding
stationary points of the functional $I(x,z,u)$.

Fix an arbitrary initial point $(x_{(1)},z_{(1)},u_{(1)})\in C_{n}[0,T]\times
P_{n}[0,T]\times P_{m}[0,T]$. Let the point $(x_{(k)},z_{(k)},u_{(k)})\in
C_{n}[0,T]\times P_{n}[0,T]\times P_{m}[0,T]$ be already constructed. If for
each $t\in[0,T]$ minimum condition (19) is satisfied (in practice, with some
fixed accuracy $\overline{\varepsilon}$), then $(x_{(k)},z_{(k)},u_{(k)})$ is
a stationary point of the functional $I(x,z,u)$ and the process terminates.
Otherwise, put

$(x_{(k+1)},z_{(k+1)},u_{(k+1)})=(x_{(k)},z_{(k)},u_{(k)})+\gamma_{(k)}G(x_{(k)},z_{(k)},u_{(k)})$

where the vector-function $G(x_{(k)},z_{(k)},u_{(k)})$ is the
quasidifferential descent direction of the functional $I(x,z,u)$ at the point
$(x_{(k)},z_{(k)},u_{(k)})$ and the value $\gamma_{(k)}$ is a solution of the
following one-dimensional problem

$\min_{\gamma\geqslant 0}I\Big{(}(x_{(k)},z_{(k)},u_{(k)})+\gamma
G(x_{(k)},z_{(k)},u_{(k)})\Big{)}=I\Big{(}(x_{(k)},z_{(k)},u_{(k)})+\gamma_{(k)}G(x_{(k)},z_{(k)},u_{(k)})\Big{)}.$
(20)

Then $I(x_{(k+1)},z_{(k+1)},u_{(k+1)})\leqslant I(x_{(k)},z_{(k)},u_{(k)})$.

As seen from the algorithm described, in order to realize the $k$-th
iteration, one has to solve three subproblems. The first subproblem is to
calculate the quasidifferential of the functional $I(x,z,u)$ at the point
$(x_{(k)},z_{(k)},u_{(k)})$. With the help of quasidifferential calculus rules
the solution of this subproblem is obtained in formula (18). The second
subproblem is to find the quasidifferential descent direction
$G(x_{(k)},z_{(k)},u_{(k)})$; the following two paragraphs are devoted to
solving this subproblem. Finally, the third subproblem is one-dimensional
minimization (20); there are many effective methods (Vasil’ev (2002)) for
solving this subproblem.

In order to obtain the vector-function $G(x_{(k)},z_{(k)},u_{(k)})$, consider
the problem

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(21)

Denote $\overline{v}(t)$, $\overline{w}(t)$ its solution. (The vector-
functions $\overline{v}(t)$, $\overline{w}(t)$, of course, depend on the point
$(x_{(k)},z_{(k)},u_{(k)})$ but we omit this dependence in the notation for
brevity.) Then the vector-function
$G(x_{(k)},z_{(k)},u_{(k)})=-\big{(}\overline{v}(t)+\overline{w}(t)\big{)}$ is
a quasidifferential descent direction of the functional $I(x,z,u)$ at the
point $(x_{(k)},z_{(k)},u_{(k)})$. Note that the functional $I(x,z,u)$
quasidifferential at each time moment $t\in[0,T]$ is calculated independently
(i. e. the functional $I(x,z,u)$ quasidifferential, calculated at one time
moment, does not depend on the functional $I(x,z,u)$ quasidifferential,
calculated at some other time moment).

Now let us check that in order to solve problem (21) in this case, one has to
solve the problem

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}$
(22)

for each $t\in[0,T]$.

Indeed, let $\overline{v}$, $\overline{w}\in C_{n}[0,T]\times P_{n}[0,T]\times
P_{m}[0,T]$ be such that for each $t\in[0,T]$ we have

$\big{(}\overline{v}(t)+\overline{w}(t)\big{)}^{2}=\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}.$

Then we obtain

$\int_{0}^{T}\big{(}\overline{v}(t)+\overline{w}(t)\big{)}^{2}dt=\int_{0}^{T}\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}dt=$
$=\int_{0}^{T}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+\overline{w}(t)\big{)}^{2}dt=\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+\overline{w}(t)\big{)}^{2}dt$

where the last equality holds due to Filippov lemma (cf. formula (17)). Hence
the following inequality holds true:

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt\geq$
$\geq\int_{0}^{T}\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(23)

Now fix some $\overline{\overline{w}}\in C_{n}[0,T]\times P_{n}[0,T]\times
P_{m}[0,T]$. Again, by Filippov lemma we get

$\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+\overline{\overline{w}}(t)\big{)}^{2}dt=\int_{0}^{T}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+\overline{\overline{w}}(t)\big{)}^{2}dt\leq$
$\leq\int_{0}^{T}\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}dt.$

Since the vector function $\overline{\overline{w}}(t)$ was chosen arbitrarily,
we obtain the inequality

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt\leq$
$\leq\int_{0}^{T}\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(24)

From inequalities (23) and (24) we finally get the equality

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt=$
$=\int_{0}^{T}\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)},u_{(k)})}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(25)

The equality (25) justifies that in order to solve problem (21) it is
sufficient to solve problem (22) for each time moment $t\in[0,T]$. Once again
we emphasize that this statement holds true due to the special structure of
the quasidifferential which in turn takes place due to the separation
implemented of the vector functions $x(t)$ and $\dot{x}(t)$ into “independent”
variables.

Problem (22) at each fixed $t\in[0,T]$ is a finite-dimensional problem of
finding the Hausdorff deviation of one convex compact set (a minus
superdifferential) from another convex compact set (a subdifferential). This
problem may be effectively solved for a rich class of functions; its solution
is described in the next paragraph. In practice, one makes a (uniform)
partition of the interval $[0,T]$ and this problem is being solved for each
point of the partition, i. e. one calculates
$G(t_{i},(x_{(k)},z_{(k)},u_{(k)}))$ where $t_{i}\in[0,T]$,
$i=\overline{1,N}$, are discretization points (see notation of Lemmas 6.1, 6.2
below). Under additional natural assumption Lemma 6.1 below guarantees that
the vector-function obtained via piecewise-linear interpolation of the
quasidifferential descent directions calculated at each point of such a
partition of the interval $[0,T]$ converges in the space $L_{2}^{2n+m}[0,T]$
(as the discretization rank $N$ tends to infinity) to the vector-function
$G(x_{(k)},z_{(k)},u_{(k)})$ sought.

As noted in the previous paragraph, during the algorithm realization it is
required to find the Hausdorff deviation of the minus superdifferential from
the subdifferential of the functional $I(x(t),z(t),u(t))$ at each time moment
of a (uniform) partition of the interval $[0,T]$. In this paragraph we
describe in detail a solution (for a rich class of functions) of this
subproblem for some fixed value $t\in[0,T]$. It is known (Demyanov & Rubinov
(1977)) that in many practical cases the subdifferential
$\underline{\partial}I(x(t),z(t),u(t))$ is a convex polyhedron $A(t)\subset
R^{2n+m}$ and analogously the superdifferential
$\overline{\partial}I(x(t),z(t),u(t))$ is a convex polyhedron $B(t)\subset
R^{2n+m}$. For example, if some function is a superposition of the finite
number of maxima and minima of continuously differentiable functions, then its
subdifferential and its superdifferential are convex polyhedra. Herewith, of
course, the sets $A(t)$ and $B(t)$ depend on the point $(x,z,u)$. For
simplicity, we omit this dependence in this paragraph notation. Find the
Hausdorff deviation of the set $-B(t)$ from the set $A(t)$. It is clear that
in this case it is sufficient to go over all the vertices $b_{j}(t)$,
$j=\overline{1,s}$ (here $s$ is a number of vertices of the polyhedron
$-B(t)$): find the Euclidean distance from every of these vertices to the
polyhedron $A(t)$ and then among all the distances obtained choose the largest
one. Let the Euclidean distance sought, corresponding to the vertex
$b_{j}(t)$, $j=\overline{1,s}$, is achieved at the point $a_{j}(t)\in A(t)$
(which is the only one since $A(t)$ is a convex compact). Then the deviation
sought is the value $||b_{\overline{j}}(t)-a_{\overline{j}}(t)||_{R^{2n+m}}$,
$\overline{j}\in\\{1,\dots,s\\}$. (Herewith, this deviation may be achieved at
several vertices of the polyhedron $-B(t)$; in this case $b_{\overline{j}}(t)$
denotes any of them.) Note that the arising here problem of finding the
Euclidean distance from a point to a convex polyhedron can be effectively
solved by various methods (see, e. g. Demyanov & Vasil’ev (1986), Wolfe
(1959)).

In Lemmas 6.1, 6.2 we will write $L_{2}[0,T]$ and $L_{\infty}[0,T]$ instead of
$L_{2}^{1}[0,T]$ and $L_{\infty}^{1}[0,T]$ respectively for the convenience of
notation.

First, give a lemma with a rather simple condition which, on the one hand, is
quite natural for applications and, on the other hand, guarantees that the
function $L(t)$ obtained via piecewise-linear interpolation of the sought
function $p\in L_{\infty}[0,T]$ converges to this function in the space
$L_{2}[0,T]$.

######  Lemma 6.1.

Let the function $p\in L_{\infty}[0,T]$ satisfy the following condition: for
every $\overline{\delta}>0$ the function $p(t)$ 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 number $\overline{\delta}$.

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

###### Proof.

Denote
$M(\overline{\delta}):=\bigcup\limits_{k=1}^{r}\big{(}\overline{t}_{k}(\overline{\delta}),\overline{t}_{k+1}(\overline{\delta})\big{)}$.
We have

$||L-p||^{2}_{L_{2}[0,T]}=\int_{M(\overline{\delta})}\big{(}L(t)-p(t)\big{)}^{2}dt+\int_{[0,T]\setminus
M(\overline{\delta})}\big{(}L(t)-p(t)\big{)}^{2}dt.$

Fix the arbitrary number $\varepsilon>0$. By lemma condition the function
$p(t)$ is bounded, the function $L(t)$ is also bounded by construction for all
(uniform) finite partitions of the interval $[0,T]$. Hence, there exists such
$\overline{\delta}(\varepsilon)$ that the first summand does not exceed the
value ${\varepsilon}/{2}$ for all (uniform) finite partitions of the interval
$[0,T]$. As assumed, the function $p(t)$ is piecewise continuous and bounded
on the set $[0,T]\setminus M(\overline{\delta}(\varepsilon))$, then there
exists (Ryaben’kii (2008)) such a number $\overline{N}(\varepsilon)$ that for
every (uniform) finite partition of the interval $[0,T]$ of the rank
$N>\overline{N}(\varepsilon)$ the second summand (with such
$\overline{\delta}(\varepsilon))$ does not exceed the value
${\varepsilon}/{2}$. This implies the proof of the lemma. ∎

Now give a lemma with a more general but less clear (compared to the previous
lemma) condition which also guarantees that the function $L(t)$ obtained via
piecewise-linear interpolation of the sought function $p\in L_{\infty}[0,T]$
converges to this function in the space $L_{2}[0,T]$.

######  Lemma 6.2.

Let the function $p\in L_{\infty}[0,T]$ satisfy the following conditions: 1)
for every $\overline{\varepsilon}>0$ there exists such a closed set
$T(\overline{\varepsilon})\subset[0,T]$ that the function
$p(t)|_{T(\overline{\varepsilon})}$ is continuous and
$|T^{\prime}(\overline{\varepsilon})|<\overline{\varepsilon}$ where
$T^{\prime}(\overline{\varepsilon}):=[0,T]\setminus
T(\overline{\varepsilon})$; 2) for every $\delta>0$ there exists such a number
$\overline{\overline{N}}(\delta)$ that for each
$N>\overline{\overline{N}}(\delta)$ we have $|M(\delta)|<\delta$ where
$M(\delta):=\bigcup\limits_{k=2}^{N}[t_{k-1},t_{k}]$; here we take the union
of only such intervals $[t_{k-1},t_{k}]$, $k\in\\{2,\dots,N\\}$, in each of
which at least one of the points $t_{k-1}$, $t_{k}$, $k\in\\{2,\dots,N\\}$,
belongs to the set $T^{\prime}(\overline{\varepsilon})$.

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

###### Proof.

Note that the first assumption of the lemma is always satisfied since it is
nothing but formulation of Lusin’s theorem (Kolmogorov & Fomin (1999)).
However, it is given in the lemma formulation since the set
$T^{\prime}(\varepsilon)$ introduced there is used in the second assumption of
the lemma.

Fix some number $\varepsilon>0$.

Let $q(t)$ be a “polygonal extension” of the function
$p(t)|_{T(\overline{\varepsilon})}$ onto the whole interval $[0,T]$ which may
be constructed (Cullum (1969), Lemma 4.1) due to the fact that the set
$T(\overline{\varepsilon})$ is closed (see assumption 1) of the lemma). Then
the function $q(t)$ is continuous on $[0,T]$ and $q(t)=p(t)$ at $t\in
T(\overline{\varepsilon})$. Herewith, one can check (see Cullum (1969), Lemma
4.1) that one may choose $\overline{\varepsilon}$ in such a way that

$\displaystyle{\int_{0}^{T}(q(t)-p(t))^{2}dt<\varepsilon/3}.$ (26)

Consider the expression

$\displaystyle{\int_{0}^{T}\big{(}L(t)-q(t)\big{)}^{2}dt}=\int_{M(\delta)}\big{(}L(t)-q(t)\big{)}^{2}dt+\int_{[0,T]\setminus
M(\delta)}\big{(}L(t)-q(t)\big{)}^{2}dt.$ (27)

Consider the first summand in the right-hand side of equality (27). By
construction the function $q(t)$ is bounded, the function $L(t)$ is also
bounded by construction for all (uniform) finite partitions of the interval
$[0,T]$. Then from assumption 2) of the lemma it follows that for
$\varepsilon>0$ there exists such $\delta(\varepsilon)$ that for every
(uniform) partition of the interval $[0,T]$ of the rank
$N>\overline{\overline{N}}(\delta(\varepsilon))$ one has

$\int_{M(\delta)}\big{(}L(t)-q(t)\big{)}^{2}dt<\varepsilon/3.$ (28)

Consider the second summand in the right-hand side of equality (27). Let
$\overline{L}(t)$ be a function obtained via piecewise-linear interpolation
with the nodes $(t_{i},q(t_{i}))$, $i=\overline{1,N}$. Insofar as the function
$q(t)$ is continuous on $[0,T]$, then there exists (Ryaben’kii (2008)) such a
number $\overline{\overline{\overline{N}}}(\varepsilon)$ that for every
(uniform) partition of the interval $[0,T]$ of the rank
$N>\overline{\overline{\overline{N}}}(\varepsilon)$ one has
$\displaystyle{\int_{0}^{T}\big{(}\overline{L}(t)-q(t)\big{)}^{2}dt<\varepsilon/3}$.
But at $t\in[0,T]\setminus M(\delta)$ we have $L(t)=\overline{L}(t)$ by
construction (with the same rank of partitions involved in these functions
construction), insofar as if $t_{i}\in[0,T]\setminus M(\delta)$,
$i\in\\{1,\dots,N\\}$, then $t_{i}\in T(\varepsilon)$, and for such $t_{i}$ we
have $q(t_{i})=p(t_{i})$. For every (uniform) partition of the interval
$[0,T]$ of the rank $N>\overline{\overline{\overline{N}}}(\varepsilon)$ we
then have

$\int_{[0,T]\setminus
M(\delta)}\big{(}L(t)-q(t)\big{)}^{2}dt=\int_{[0,T]\setminus
M(\delta)}\big{(}\overline{L}(t)-q(t)\big{)}^{2}dt\leqslant\int_{0}^{T}\big{(}\overline{L}(t)-q(t)\big{)}^{2}dt<\varepsilon/3.$
(29)

Take
$\overline{N}(\varepsilon)=\max\Big{\\{}\overline{\overline{N}}(\delta(\varepsilon)),\overline{\overline{\overline{N}}}(\varepsilon)\Big{\\}}$.
For every (uniform) partition of the interval $[0,T]$ of the rank
$N>\overline{N}(\varepsilon)$ from (26), (28), (29) we finally have

$||L-p||^{2}_{L_{2}[0,T]}\leqslant\int_{0}^{T}\big{(}L(t)-q(t)\big{)}^{2}dt+\int_{0}^{T}\big{(}q(t)-p(t)\big{)}^{2}dt<\varepsilon/3+\varepsilon/3+\varepsilon/3=\varepsilon.$

The lemma is proved. ∎

######  Remark 3.

The meaning of assumption 2) of Lemma 6.2 is the requirement that $p(t)$ does
not have “too many” discontinuity points on the segment $[0,T]$. It may be
directly verified that if the condition of Lemma 6.1 on the function $p(t)$ is
fulfilled, then the condition required is satisfied. In the picture a simple
example is given of a measurable bounded function with an infinite number of
discontinuity points for which one may construct the function $q(t)$ in such a
way that the set $M(\delta)$ measure is arbitrarily small for a sufficiently
large splitting rank. It is an example of an “appropriate” in the sense of
Lemma 6.2 assumption function.

Let us give an example of the function for which this condition is violated.
Let $p(t)$ be the Dirichlet function on the segment $[0,1]$, i. e. taking the
value $1$ at rational points and taking the value $0$ at irrational points of
this interval. If we take the function $q(t)=0$ $\forall t\in[0,1]$ as a
continuous one and as satisfying Lusin’s theorem applied to the function
$p(t)$, then condition 2) of Lemma 6.2 will be violated with every rank of
(uniform) splitting of the interval $[0,1]$, insofar as with such a splitting
all the splitting points will be rational, i. e. will belong to the set
$T^{\prime}(\overline{\varepsilon})$ $\forall\overline{\varepsilon}>0$, hence
$|M(\delta)|=1$ $\forall\delta>0$ in this case. It is seen that in this
example one has $||L-p||^{2}_{L_{2}[0,1]}=1$ for each function $L(t)$ obtained
via piecewise-linear interpolation of the function $p(t)$ with a uniform
splitting of the segment $[0,1]$, insofar as with such a splitting we always
have $L(t)=1$ $\forall t\in[0,1]$. The Dirichlet function $p(t)$ does not
satisfy condition 2) of Lemma 6.2, insofar as this function has “too many”
discontinuity points on the interval $[0,1]$.

######  Remark 4.

The problem of a rigorous proof of the above method convergence is rather
complicated and remains open; it is beyond the scope of this paper. The
convergence of some modifications of the subdifferential descent method, as a
special case of the quasidifferntial descent method described, was studied in
the finite-dimensional case in papers Demyanov & Vasil’ev (1986), Demyanov &
Malozemov (1990). It is an interesting problem for future investigations to
attempt to spread the ideas of these studies to explore convergence in the
more general problem considered in this paper (the case of a
quasidifferentiable functional in a functional space). Strictly speaking, in
the paper presented only the problem of finding the direction of the steepest
(quasidifferential) descent in the problem posed is completely solved. The
examples below show the adequacy of the method used; nevertheless, as just
noted, its convergence (in whatever sense) requires additional rigorous
justification.

######  Remark 5.

One of the possible problems for the future research is application of the
main concepts of this paper to some more difficult control problems. For
example, consider the problem of minimizing the functional

$J(x,u)=\int^{T}_{0}f_{0}(x(t),\dot{x}(t),u(t),t)dt$

under restrictions (5), (6), (8) and $u\in U$ (see (7)). We don’t give here
the assumptions regarding the integrand of the functional $J(x,u)$ as in this
Remark only the essence of the problem is discussed. For simplicity consider
now only restriction (5) and construct the functional

$J(x,z,u)+\lambda\sum_{i=1}^{n}\int_{0}^{T}\big{|}z_{i}(t)-f_{i}(x(t),u(t),t)\big{|}dt.$

The hypothesis (based on the previous research made in Dolgopolik & Fominyh
(2019)) is that under some natural assumptions this functional is an exact
penalty one. This means that there exists such a value $\lambda^{*}<\infty$
that $\forall\lambda>\lambda^{*}$ the problem of minimizing the functional
constructed is equivalent to the stated problem of minimizing the functional
$J(x,u)$ under restriction (5). We omit the details here. Note only that the
functional constructed may be treated via the method of the paper presented so
it is interesting to try such an approach.

##  7 Numerical Examples

Let us first explain the operation of the quasidifferential descent method
using an illustrative example in which the iterations are given in detail. For
simplicity of presentation this example is not chosen as a control problem but
is a nonsmooth problem of the calculus of variations; however, according to
the minimized functional structure, it fits the formulation of the problem
considered in the paper.

######  Example 7.1.

Consider the functional

$I(x,z)=\int_{0}^{1}|z(t)+|x(t)||dt+\int_{0}^{1}|z(t)|dt+\int_{0}^{T}\Big{(}x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau\Big{)}^{2}dt$

with the initial point $x_{0}=0$ and with the obvious solution $x^{*}(t)=0$,
$z^{*}(t)=0$ for all $t\in[0,1]$. Note that the third summand here means that
$z(t)$ must be the derivative of $x(t)$ (see the functional $I_{4}(x,z)$ in
formula (10) and formula (9)).

Take the functions $x_{(1)}(t)=t-0.5$, $z_{(1)}(t)=0$ as an initial
approximation and discretize the segment $[0,1]$ with a splitting rank, equal
to $2$ (i. e. consider the points $0$, $0.5$, $1$ for further interpolation of
the quasidifferential descent direction). According to the algorithm,
calculate the descent direction separately at these points. At the point $t=0$
we have

$\underline{\partial}I(x_{(1)},z_{(1)})=\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\
-2\end{array}\right),\left(\begin{array}[]{c}2\\\
-2\end{array}\right)\right\\},$
$\overline{\partial}I(x_{(1)},z_{(1)})=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$

Find the deviation of the set $-\overline{\partial}I(x_{(1)},z_{(1)})$ from
the set $\underline{\partial}I(x_{(1)},z_{(1)})$ at the point $t=0$ and obtain
the quasidifferential descent direction
$G(0,(x_{(1)},z_{(1)}))=(0,2)^{\prime}$. At the point $t=0.5$ we have

$\underline{\partial}I(x_{(1)},z_{(1)})=\mathrm{co}\bigg{\\{}\left(\begin{array}[]{c}-0.25\\\
-2\end{array}\right),\left(\begin{array}[]{c}1.75\\\
-2\end{array}\right),\left(\begin{array}[]{c}-0.25\\\ 2\end{array}\right),$
$\left(\begin{array}[]{c}1.75\\\
2\end{array}\right),\left(\begin{array}[]{c}-2.25\\\
0\end{array}\right),\left(\begin{array}[]{c}-0.25\\\
0\end{array}\right)\bigg{\\}},$
$\overline{\partial}I(x_{(1)},z_{(1)})=\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\
-1\end{array}\right),\left(\begin{array}[]{c}0\\\
1\end{array}\right)\right\\},$

hence, one has $G(0.5,(x_{(1)},z_{(1)}))=(0,0)^{\prime}$. At the point $t=1$
we have

$\underline{\partial}I(x_{(1)},z_{(1)})=\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\
2\end{array}\right),\left(\begin{array}[]{c}2\\\
2\end{array}\right)\right\\},$
$\overline{\partial}I(x_{(1)},z_{(1)})=\left(\begin{array}[]{c}0\\\
0\end{array}\right),$

so one has $G(1,(x_{(1)},z_{(1)}))=(0,-2)^{\prime}$. By making the appropriate
interpolation we obtain the quasidifferential descent direction of the
functional $I(x,z)$ at the point $(x_{(1)},z_{(1)})$, namely
$G(x_{(1)},z_{(1)})=(0,-4t+2)^{\prime}$. Construct the next point
$(x_{(2)}(t),z_{(2)}(t))=(0+\gamma_{(1)}\,0,t-0.5+\gamma_{(1)}(-4t+2))^{\prime}$;
having solved the one-dimensional minimization problem $\min_{\gamma\geqslant
0}I(x_{(2)},z_{(2)})$, we have $\gamma_{(1)}=0.25$, hence, $x_{(2)}(t)=0$,
$z_{(2)}(t)=0$ for all $t\in[0,1]$, i. e. in this case the quasidifferential
descent method leads to the exact solution in one step.

Of course, the initial approximation and discretization rank are chosen
artificially here in order to demonstrate the essence of the method developed.
If we take different initial approximation and discretization rank, then in a
general case it will no longer be possible to obtain an exact solution in the
finite number of steps.

Consider now some examples of nonsmooth control problems. In all examples
presented the iterations were carried out till the error on the right endpoint
did not exceed the value $5\times 10^{-3}$ (of the trajectories resulting from
numerical integration of the system (from the initial left endpoint given)
with the control substituted which was obtained via the method of the paper).
Such a choice of accuracy is due to a compromise between the permissible for
practice accuracy of the required value of the considered functional and a not
very large number of iterations. The magnitude $||G{(x,z,u)}||$ (the norm is
considered in the corresponding space) on the solution is also presented and
did not exceed the value $5\times 10^{-3}$ — $8\times 10^{-3}$ (see inclusion
(19) and problem (21)).

The calculations were performed in the package Maple 12.0 Serial Number:
2011-11-11 on a computer with the Intel(R) Core(TM) i5-6200U CPU @ 2.30 GHz
and 8 GB of RAM. The choice of package is justified by the fact that symbolic
computations are natural for some stages of the algorithm presented; it also
made the iterations more demonstrative. On the other hand, a significant part
of the computational time was spent operating with symbolic expressions.

######  Example 7.2.

Consider the system

$\dot{x}_{1}(t)=-|x_{1}(t)|,$ $\dot{x}_{2}(t)=u(t).$

It is required to find such a control $u^{*}\in U$ which brings this system
from the initial point $x(0)=(0,0)^{\prime}$ to the final state
$x(1)=(0,0)^{\prime}$ at the moment $T=1$. Herewith, put $\underline{u}=-1$,
$\overline{u}=1$, i. e. we suppose that $-1\leq u(t)\leq 1$ $\forall
t\in[0,1]$.

The problem given is reduced to an unconstrained minimization of the
functional

$I(x,z,u)=\int_{0}^{1}\big{|}z_{1}(t)+|x_{1}(t)|\big{|}dt+\int_{0}^{1}\big{|}z_{2}(t)-u(t)\big{|}dt+$
$+\frac{1}{2}\Big{(}\int_{0}^{1}z_{1}(t)dt\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{1}z_{2}(t)dt\Big{)}^{2}+$
$+\int_{0}^{1}\max\\{-1-u(t),0\\}dt+\int_{0}^{1}\max\\{{u}(t)-1,0\\}dt+$
$+\frac{1}{2}\int_{0}^{1}\Big{(}x_{1}(t)-\int_{0}^{t}z_{1}(\tau)d\tau\Big{)}^{2}dt+\frac{1}{2}\int_{0}^{1}\Big{(}x_{2}(t)-\int_{0}^{t}z_{2}(\tau)d\tau\Big{)}^{2}dt.$

Take $(x_{(1)},z_{(1)},u_{(1)})=(1,1,1,1,1)^{\prime}$ as an initial point,
then $I(x_{(1)},z_{(1)},u_{(1)})\approx$ $\approx 3.33333$. As the iteration
number increased, the discretization rank gradually increased during the
solution of the auxiliary problem of finding the direction of the
quasidifferential descent described in the algorithm and, in the end, the
discretization step was equal to $10^{-1}$. At the $17$-th iteration the
control $u_{(17)}$ was constructed:

$u\approx 0.56234-1.61222t-0.04788t^{3}+0.78478t^{2},$

with the value of the functional $I(x_{(17)},z_{(17)},u_{(17)})\approx
0.00462$, herewith $x_{1}(T)\approx 0.00357$, $x_{2}(T)\approx 0.00538$. For
the convenience of presentation, the Lagrange interpolation polynomial has
been given which quite accurately approximates (that is, the interpolation
error does not affect the value of the functional and the boundary values) the
resulting control.

Take $u_{(17)}$ as an approximation to the control $u^{*}$ sought. In order to
verify the result obtained and to find the “true” trajectory, we substitute
this control into the system given and integrate it via one of the known
numerical methods (here, the Runge-Kutta 4-5-th order method was used). As a
result, we have the corresponding trajectory (which is an approximation to the
one $x^{*}$ sought) with the values $x_{1}(T)=0$, $x_{2}(T)=0.00585$, so we
see that the error does not exceed the value $5\times 10^{-3}$. Herewith,
$||G(x,z,u)||_{L^{2}_{2}[0,T]\times L^{2}_{2}[0,T]\times
L^{1}_{2}[0,T]}=0.00585$.

The computational time was 2 min 48 sec. The pictures illustrate the control
and trajectories dynamics during the algorithm realization.

Figure 1: Example 2, control and trajectories on iterations: 2, 9, 13, 17

######  Example 7.3.

It is required to bring the system

$\dot{x}_{1}(t)=x_{2}(t)x_{3}(t)+u_{1}(t),$
$\dot{x}_{2}(t)=x_{1}(t)x_{3}(t)+u_{2}(t),$
$\dot{x}_{3}(t)=x_{1}(t)x_{2}(t)+u_{3}(t)$

from the initial point $x(0)=(1,0,0)^{\prime}$ to the final state
$x(1)=(0,0,0)^{\prime}$ at the moment $T=1$. Herewith, we suppose that the
total control consumption is subject to the constraint

$\int_{0}^{1}|u_{1}(t)|+|u_{2}(t)|+|u_{3}(t)|dt=1.$

This problem has a practical application to the optimal satellite
stabilization and was considered in work Krylov (1968). With the help of the
new variable
$x_{4}(t)=\displaystyle{\int_{0}^{t}|u_{1}(\tau)|+|u_{2}(\tau)|+|u_{3}(\tau)|d\tau}$
reduce the problem given to problem (5), (6), (8) which is considered in the
paper.

Then we have the system

$\dot{x}_{1}(t)=x_{2}(t)x_{3}(t)+u_{1}(t),$
$\dot{x}_{2}(t)=x_{1}(t)x_{3}(t)+u_{2}(t),$
$\dot{x}_{3}(t)=x_{1}(t)x_{2}(t)+u_{3}(t),$
$\dot{x}_{4}(t)=|u_{1}(t)|+|u_{2}(t)|+|u_{3}(t)|,$

with no restrictions on the control $u^{*}\in P_{3}[0,T]$ which is aimed at
bringing the object from the initial point $x(0)=(1,0,0,0)^{\prime}$ to the
final state $x(1)=(0,0,0,1)^{\prime}$ at the moment $T=1$.

The problem given is reduced to an unconstrained minimization of the
functional

$I(x,z,u)=\int_{0}^{1}\big{|}z_{1}(t)-x_{2}(t)x_{3}(t)-u_{1}(t)|dt+\int_{0}^{1}\big{|}z_{2}(t)-x_{1}(t)x_{3}(t)-u_{2}(t)|dt+$
$+\int_{0}^{1}\big{|}z_{3}(t)-x_{1}(t)x_{2}(t)-u_{3}(t)|dt+\int_{0}^{1}\big{|}z_{4}(t)-|u_{1}(t)|-|u_{2}(t)|-|u_{3}(t)||dt+$
$+\frac{1}{2}\Big{(}1+\int_{0}^{1}z_{1}(t)dt\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{1}z_{2}(t)dt\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{1}z_{3}(t)dt\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{1}z_{4}(t)dt-1\Big{)}^{2}+$
$+\frac{1}{2}\sum_{i=1}^{4}\left(x_{i}(t)-x_{i}(0)-\int_{0}^{t}z_{i}(\tau)d\tau\right)^{2}dt.$

Take $(x_{(1)},z_{(1)},u_{(1)})=(1+t,t,t,t,1,1,1,1,0,0,0)^{\prime}$ as an
initial point, then $I(x_{(1)},z_{(1)},u_{(1)})\approx 5.72678$. As the
iteration number increased, the discretization rank gradually increased during
the solution of the auxiliary problem of finding the direction of
quasidifferential descent described in the algorithm and, in the end, the
discretization step was equal to $10^{-1}$. At the $58$-th iteration the
control $u_{(58)}$ was constructed:

$u_{1}\approx-2.88657t^{5}+8.96619t^{4}-9.30386t^{3}+2.99867t^{2}+0.10679t-1.03399,$
$u_{2}\approx-0.83764t^{5}+0.85068t^{4}+0.43135t^{3}-0.54058t^{2}+0.06933t+0.00945,$
$u_{3}\approx 0.13344t-0.01334,\ t\in[0,0.1),\quad
0.03928t^{3}-0.01848t^{2}-0.00194t,\ t\in[0.1,0.6),$
$-0.94481t^{3}+2.12646t^{2}-1.55331t+0.36987,\ t\in[0.6,1],$

with the value of the functional $I(x_{(58)},z_{(58)},u_{(58)})\approx
0.00551$, herewith, $x_{1}(T)\approx 0.01251$, $x_{2}(T)\approx 0.00431$,
$x_{3}(T)\approx 0.00431$, $x_{4}(T)\approx 1.0069$. For the convenience of
presentation, the Lagrange interpolation polynomial has been given which quite
accurately approximates (that is, the interpolation error does not affect the
value of the functional and the boundary values) the resulting control.

Take $u_{(58)}$ as an approximation to the control $u^{*}$ sought. In order to
verify the result obtained and to find the “true” trajectory, we substitute
this control into the system given and integrate it via one of the known
numerical methods (here, the Runge-Kutta 4-5-th order method was used). As a
result, we have the corresponding trajectory (which is an approximation to the
one $x^{*}$ sought) with the values $x_{1}(T)=0.00514$, $x_{2}(T)=0.00204$,
$x_{3}(T)=0.00051$, $x_{4}(T)=1.00514$, so we see that the error does not
exceed the value $5\times 10^{-3}$. Herewith,
$||G(x,z,u)||_{L^{4}_{2}[0,T]\times L^{4}_{2}[0,T]\times
L^{3}_{2}[0,T]}=0.00757$.

The computational time was 15 min 32 sec. The pictures illustrate the control
and trajectories dynamics during the algorithm realization.

Figure 2: Example 3, control on iterations: 2, 13, 29, 45, 58

Figure 3: Example 3, trajectories on iterations: 2, 13, 29, 45, 58

######  Example 7.4.

Consider the system

$\dot{x}_{1}(t)=x_{2}(t)$ $\dot{x}_{2}(t)=u(t)-Px_{2}(t)|x_{2}(t)|-Qx_{2}(t).$

It is required to find such a control $u^{*}\in U$ which brings this system
from the initial point $x(0)=(0,0)^{\prime}$ to the final state
$x(48)=(200,0)^{\prime}$ at the moment $T=48$. Herewith, put
$\underline{u}=-2/3$, $\overline{u}=2/3$, i. e. we suppose that $-2/3\leq
u(t)\leq 2/3$ $\forall t\in[0,48]$. The parameters of the problem are
$P=0.78\times 10^{-4}$ and $Q=0.28\times 10^{-3}$. This problem has a
practical application to the optimal train motion and was considered in work
Outrata (1983). Not that the right-hand side of the system contains a
nondifferentiable term. On the other hand, essential assumption of the paper
is that only continuous differentiable functions in the right-hand side of the
system are allowed (see Outrata (1983), Par. 2, Assumption. (ii)). So the
paper Outrata (1983) method presumably can not be implemented on those
iterations when the nondifferentiability in the system right-hand side occurs
(as for example, on the first iteration in our case since the initial
approximation point $z_{(1)}=(0,0,0)^{\prime}$ is taken (see the next
paragraph)). (However, the paper Outrata (1983) method eventually succeeded in
this example, since $x_{2}^{*}(t)>0$ for all $t\in(0,48)$.)

In fact, in paper Outrata (1983) a more complicated (if we “ignore” the fact
that the right-hand side of the system is formally nondifferentiable) problem
is considered with some phase constraints and the functional

$\int_{0}^{48}x_{2}(t)\max\\{u(t),0\\}dt$

to be minimized. The paper gives neither the optimal control nor the resulting
cost value obtained; the rough estimate of the “characteristic” segments
(where $u(t)>0$) given in Fig. 1, Par. 5.1 of Outrata (1983) (we duplicate
this picture as well) shows that the functional optimal value exceeds the
magnitude $13.3$.

We consider a simplified problem; however, let us try to obtain this value of
the functional via the new variable
$x_{3}(t)=\displaystyle{\int_{0}^{t}x_{2}(\tau)\max\\{u(\tau),0\\}d\tau}$ and
explore the system with the additional equation

$\dot{x}_{3}(t)=x_{2}(t)\max\\{u(t),0\\}$

and with the boundary conditions $x(0)=(0,0,0)^{\prime}$ and
$x(48)=(200,0,13.3)^{\prime}$ (and without any phase constraints).

The problem given is reduced to an unconstrained minimization of the
functional

$I(z,u)=\int_{0}^{48}\big{|}z_{2}(t)-u(t)+Pz_{1}(t)|z_{1}(t)|+Qz_{1}(t)\big{|}dt+\int_{0}^{48}\big{|}z_{3}(t)-z_{1}(t)\max\\{u(t),0\\}\big{|}dt$
$+\frac{1}{2}\Big{(}\int_{0}^{48}z_{1}(t)dt-200\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{48}z_{2}(t)dt\Big{)}^{2}+\frac{1}{2}\Big{(}\int_{0}^{48}z_{3}(t)dt-13.3\Big{)}^{2}$
$+\int_{0}^{48}\max\\{-2/3-u(t),0\\}dt+\int_{0}^{48}\max\\{{u}(t)-2/3,0\\}dt+$
$+\frac{1}{2}\int_{0}^{48}\Big{(}z_{1}(t)-\int_{0}^{t}z_{2}(\tau)d\tau\Big{)}^{2}dt.$

The functional is slightly simplified beforehand using the fact that
$x_{2}(t)=z_{1}(t)$, also put
$\displaystyle{x_{1}(t)=\int_{0}^{t}z_{1}(\tau)d\tau}$,
$\displaystyle{x_{3}(t)=\int_{0}^{t}z_{3}(\tau)d\tau}$, $t\in[0,48]$,
throughout iterations.

Take $(z_{(1)},u_{(1)})=(0,0,0,0)^{\prime}$ as an initial point, then
$I(z_{(1)},u_{(1)})=2\times 10^{4}$. As the iteration number increased, the
discretization rank gradually increased during the solution of the auxiliary
problem of finding the direction of the quasidifferential descent described in
the algorithm and, in the end, the discretization step was equal to $2\times
10^{-1}$ on the intervals $[0,9]$ and $[40,48]$ and equal to $3.875$ on the
interval $[9,40]$ (it was reasonable to use a nonuniform (and such a rough in
the middle interval) splitting here due to the qualitative different
“behavior” of the trajectories on these intervals). At the $216$-th iteration
the control $u_{(216)}$ was constructed:

$u\approx 0.666667,\ t\in[0,4.383),$
$0.09896701901t-0.01727775583t^{2}+0.5648118155,\ t\in[4.383,9),$
$-3.398320586\times 10^{-10}\,t^{6}+6.378460924\times
10^{-8}\,t^{5}-0.000003933576898t^{4}+$
$+0.00010132955t^{3}-0.00075674151t^{2}-0.01180182196t+0.1719284027,\
t\in[9,40),$ $-0.03302426450t^{3}+4.273406929t^{2}-184.2002092t+2644.154001,\
t\in[40,42),$ $-0.666667,\ t\in[42,48],$

with the value of the functional $I(z_{(216)},u_{(216)})\approx 0.00124$,
herewith $x_{1}(T)\approx 200.001$, $x_{2}(T)\approx 0.00521$,
$x_{3}(T)\approx 13.30413$. For the convenience of presentation, the Lagrange
interpolation polynomial has been given which quite accurately approximates
(that is, the interpolation error does not affect the value of the functional
and the boundary values) the resulting control.

Note that during the method implementation the control was obtained bringing
the trajectory from an initial position given to the final point satisfying
the right endpoint constraint with the satisfactory accuracy (and delivering a
rather small value to the functional minimized as well); however, this control
itself violated its own restrictions by the value $2\times 10^{-2}$ — $4\times
10^{-2}$ at the beginning and the end of the time interval (see the control
dynamics in the picture). Since the exact satisfaction of the restrictions on
control is important for applications, on several last iterations it was
decided to improve the accuracy via multiplying the $6$-th and the $7$-th
summands of the functional by a large factor (of order $10^{2}$) and
minimizing this modified functional during the last iterations.

Take $u_{(216)}$ as an approximation to the control $u^{*}$ sought. In order
to verify the result obtained and to find the “true” trajectory, we substitute
this control into the system given and integrate it via one of the known
numerical methods (here, the Runge-Kutta 4-5-th order method was used). As a
result, we have the corresponding trajectory (which is an approximation to the
one $x^{*}$ sought) with the values $x_{1}(T)\approx 200.00214$,
$x_{2}(T)\approx 0.00297$, $x_{3}(T)\approx 13.3054$, so we see that the error
does not exceed the value $5\times 10^{-3}$. Herewith,
$||G(z,u)||_{L^{3}_{2}[0,T]\times L^{1}_{2}[0,T]}=0.00299$.

The computational time was 102 min 43 sec. The pictures illustrate the control
and trajectories dynamics during the algorithm realization. For convenience
also the resulting control and trajectories are depicted.

Figure 4: Example 4, control and trajectories on iterations: 20, 50, 100, 150,
175, 200, 216

Figure 5: Example 4, the resulting control and trajectories

##  8 Discussion

As already noted, the main advantage of the method proposed is of theoretical
nature: it is original as it is qualitatively different from existing methods
based on the direct discretization of the initial problem. Besides, the method
preserves attractive geometrical interpretation of quasidifferentials (see
Example 7.1 and Ch. V, Par. 3 in book Demyanov & Rubinov (1977) for more
examples with geometrical illustration in the finite-dimensional case).

It also has some practical advantages. The following four paragraphs give
examples of some specific problems demonstrating these advantages. In order to
simplify the presentation and just to get essence we give examples of some
problems of calculus of variations and an example of one simplest control
problem only in a smooth case.

Consider the problem of minimizing the functional

$J(x,z)=\int_{0}^{1}z^{4}(t)/48+z^{2}(t)+x^{2}(t)-6x(t)\,dt$

under the constraints

$x(0)=1,\quad x(1)=0,\quad\int_{0}^{1}x(t)\,dt=2/3.$

In this example the steepest descent method appeared to be very effective. On
the contrary: in order to construct approximations by the Ritz-Galerkin
method, a lot of calculations are required and it is also necessary to solve
essentially non-linear systems with parameters.

In a problem of minimizing the the functional

$J(z)=\int_{0}^{10}z^{2}(t)-x^{2}(t)\,dt$

under the constraints

$x(0)=0,\quad x(10)=0$

both the Euler equation and the Ritz-Galerkin method give a trajectory
delivering neither a strong nor a weak minimum. The steepest descent method
“points” to the fact that there is no solution in this problem: the one-
dimensional minimization problem has no bounded solution.

Minimize the functional

$J(z)=\int_{0}^{2}z^{3}(t)\,dt$

under the constraints

$x(0)=0,\quad x(2)=4.$

This example illustrates that the steepest descent method “points” to the fact
that, on a solution obtained, the functional reaches a weak minimum rather
than a strong one, while both the Euler equation and the Ritz-Galerkin method
give only a trajectory delivering a weak minimum.

All the details have been omitted for brevity. One can find a detailed
description of these problems and more interesting examples as well as
justification of the statements posed in the original papers Demyanov &
Tamasyan (2011), Demyanov & Tamasyan (2010). Note also that a method used in
these papers is slightly different from the one presented but it preserves its
many properties, so the comparative analysis is correct.

Consider the system

$\dot{x}(t)=-x(t)+u(t)$

on the time interval $[0,2]$. It is required to find a control $u^{*}\in
P_{1}[0,T]$ such that the corresponding trajectory satisfies the boundary
conditions

$x(0)=x(2)=0.$

Apply direct discretization to this system via the formula

$x(i+1)-x(i)=\frac{1}{N}(-x(i)+u(i))$

where $i=\overline{0,19}$ and the discretization rank $N=20$. If we use the
initial condition $x_{0}=x(0)=0$ and calculate $x_{20}=x(2)$ as an explicit
function of the variables $u_{i}$, $i=\overline{0,19}$, we will obtain

$x_{20}(u)\approx
0.01886768013\,u_{0}+0.01986071592\,u_{1}+0.02090601676\,u_{2}+\,0.0220063334\,u_{3}+$
$+0.02316456151\,u_{4}+0.02438374896\,u_{5}+0.02566710416\,u_{6}+0.02701800438\,u_{7}+$
$+0.02844000461\,u_{8}+0.02993684696\,u_{9}+0.03151247049\,u_{10}+0.03317102156\,u_{11}+$
$+0.03491686480\,u_{12}+0.03675459453\,u_{13}+0.03868904688\,u_{14}+0.04072531250\,u_{15}+$
$+0.04286875000\,u_{16}+0.045125\,u_{17}+0.0475\,u_{18}+0.05\,u_{19}.$

So in order to get the required finite position, one has to solve the equation
$x_{20}(u)=0$ with respect to the variables $u_{i}$, $i=\overline{0,19}$. In
other words, it is required to minimize the functional $|x_{20}(u)|$. Take the
initial point $u_{(0)}$ with the following coordinates: $u_{2i}=10$,
$i=\overline{0,9}$, $u_{2i+1}=-10$, $i=\overline{0,8}$,
$u_{19}\approx-6.7101842175$ (see the picture). Note that $x_{20}(u_{(0)})=0$,
so the point $u_{(0)}$ delivers a global minimum to the functional
$|x_{20}(u)|$ (i. e. the point $u_{(0)}$ solves the discretized problem).
However, if we substitute this control $u_{(0)}$ into the original system, we
will get the corresponding trajectory $x_{(0)}(t)$ with the finite value
$x(2)\approx-0.1189349683$.

Now try to solve this problem via the method of the paper, i. e. minimize the
functional

$I(x,z,u)=\frac{1}{2}\int_{0}^{2}\left(z(t)+x(t)-u(t)\right)^{2}dt+\frac{1}{2}\int_{0}^{2}\left(x(t)-\int_{0}^{t}z(\tau)d\tau\right)^{2}dt+\frac{1}{2}\left(\int_{0}^{2}z(t)dt\right)^{2}$

(for simplicity of presentation we have taken a square-function instead of the
abs-function as the integrand in the first summand). Take the same point
$(x_{(0)},z_{(0)},u_{(0)})^{\prime}$ (where $z_{(0)}=\dot{x}_{(0)}$) as an
initial approximation. One can check that on the first iteration we will get
such a control that the corresponding trajectory takes the finite value
$|x(2)|<0.11894$ (i. e. “better” than that one obtained via discretization
method).

In fact, applying the method to this example one can get a solution with any
given accuracy. This is due the fact that the Gateaux gradient in this case is
as follows:

$\nabla
I(x,z,u,t)=\begin{pmatrix}\big{(}z(t)+x(t)-u(t)\big{)}+\displaystyle{\left(x(t)-\int_{0}^{t}z(\tau)d\tau\right)}\\\
\big{(}z(t)+x(t)-u(t)\big{)}-\displaystyle{\left(\int_{t}^{2}\Big{(}x(\tau)-\int_{0}^{\tau}z(s)ds\Big{)}d\tau\right)}+\int_{0}^{2}z(t)\,dt\\\
-\big{(}z(t)+x(t)-u(t)\big{)}\end{pmatrix}.$

Hence, the stopping criteria (that is $||\nabla
I(x^{*},z^{*},u^{*})||_{L_{2}^{1}[0,2]\times L_{2}^{1}[0,2]\times
L_{2}^{1}[0,2]}=0$ in this case) may be only fulfilled (with some accuracy)
when the third component of the gradient vanishes. This fact implies that the
second summand in the first component vanishes. This fact finally implies that
the third summand in the second component vanishes. Thus, we see that there
are no local minima of the functional considered, so the method of the paper
will lead to the desired solution (with any given accuracy). Roughly speaking,
the method proposed “analyzes” the “behavior” of the whole trajectory, rather
than only its points considered at some discrete moments of time.

Although in this example the initial control is chosen in a special way, one
can easily check that there is a “huge” number of other controls with the same
properties (i. e. delivering a global minimum to the functional $|x_{T}(u)|$
but giving an error to the right endpoint value). Both increasing the time
moment $T$ and adding some kind of nonsmoothness in the right-hand side of the
system given (for example taking the function $-|x(t)|$ instead of $-x(t)$
now) will only increase this “huge” number. Of course, while increasing
discretization rank, one can decrease the error on the right endpoint. On the
other hand, even the discretization rank taken seems to be not very high for
control problems solved via discretization or a “control parametrization”
technique (see, e. g. Teo & Goh & Wong (1991)). Besides, for any
discretization rank taken the initial control may be chosen in such a way that
it will deliver the global minimum to the functional minimized (i. e.
$x_{T}(u_{(0)})=0$) but give an arbitrary large error to the right endpoint
value.

As noted, only the smooth case in the examples given is considered for
simplicity. A nonsmooth case may lead to even more difficulties: for example,
in the control problem presented any kind of nondifferentiability in the
right-hand side will significantly increase the number of “local” minima with
direct discretization used.

List also some secondary advantages of the method proposed:

1) although large discretization rank gives a good approximation to the
original problem, the choice of the appropriate discretization rank is not
straightforward;

2) in many cases the quasidifferential descent method rapidly demonstrates the
structure of the desired solution, although then the convergence to this
solution may be very slow;

3) any integral restriction (for example, $||u(t)||^{2}_{L_{2}^{m}[0,T]}\leq
C$, $C\in R$ given) will generate a complicated constraint with a large number
of variables (equal to the discretization rank) after direct discretization is
applied to the problem; on the contrary: the integral restriction is very
natural for the variational statement of the problem solved and it is easy to
add a corresponding summand to the functional $I(x,z,u)$ in order to take this
restriction into account.

The main disadvantage of the method presented in this paper reduces to the
computational effort: the number of iterations may be very large and the
accuracy obtained is often not very high (although mostly appropriate for
applications). Despite the comparative disadvantages listed, in many problems
direct discretization is more preferable than the method proposed, both in
terms of the number of iterations implemented and of the accuracy achieved.
The current and future research work is devoted to implementing some
modifications and simplicities aimed at increasing the method computational
efficiency.

##  9 Conclusion

The paper is devoted to developing a direct “continuous” method for a
nonsmooth control problem. The problem of bringing a system with a
nondifferentiable (but only quasidifferentiable) right-hand side from one
point to another is considered. The admissible controls are those from the
space of piecewise continuous vector-functions which belong to some
parallelepiped at each moment of time. The problem of finding the steepest
(the quasidifferential) descent direction was solved and the quasidifferential
descent method was applied to some illustrative examples. The method is
original and is qualitatively different from the existing methods as most of
them are based on direct discretization of the original problem. The main and
new idea implemented is to consider phase trajectory and its derivative as
independent variables and to take the natural relation between these variables
into account via penalty function of a special form. This idea gives
possibility to calculate the quasidifferential of the minimized functional and
eventually to obtain the steepest descent direction. As computational effort
of the method proposed is rather high, the goal of the future research is to
improve computational efficiency. Another goal is to applicate the main ideas
presented to other control problems.

## Appendix

With the help of quasidifferential calculus rules (Demyanov & Rubinov (1977),
Ch. III, Par. 2) at each $i\in\\{1,\dots,n\\}$, $j\in\\{1,\dots,m\\}$ and at
each $t\in[0,T]$ calculate the quasidifferentials below.

$\mathcal{D}\,\big{|}z_{i}(t)-f_{i}(x(t),u(t),t)\big{|}=\left[\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ 1\\\ 0\\\ \vdots\\\
0\end{array}\right)-\overline{\partial}f_{i}(x,u,t),\
-\underline{\partial}f_{i}(x,u,t)\right],$

if $z_{i}-f_{i}(x,u,t)>0$. Here $1$ is on the $(n+i)$-th place.

$\mathcal{D}\,\big{|}z_{i}(t)-f_{i}(x(t),u(t),t)\big{|}=\left[\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ -1\\\ 0\\\ \vdots\\\
0\end{array}\right)+\underline{\partial}f_{i}(x,u,t),\
\overline{\partial}f_{i}(x,u,t)\right],$

if $z_{i}-f_{i}(x,u,t)<0$. Here $-1$ is on the $(n+i)$-th place.

$\mathcal{D}\,\big{|}z_{i}(t)-f_{i}(x(t),u(t),t)\big{|}=$
$=\left[\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\ \vdots\\\ 0\\\ 1\\\
0\\\ \vdots\\\
0\end{array}\right)-2\,\overline{\partial}f_{i}(x,u,t),\,\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ -1\\\ 0\\\ \vdots\\\
0\end{array}\right)+2\,\underline{\partial}f_{i}(x,u,t)\right\\},\
-\underline{\partial}f_{i}(x,u,t)+\overline{\partial}f_{i}(x,u,t)\right],$

if $z_{i}-f_{i}(x,u,t)=0$. Here $1$ and $-1$ are on the $(n+i)$-th place.

$\mathcal{D}\max\big{\\{}u_{j}(t)-\overline{u}_{j},0\big{\\}}=\left[\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ 1\\\ 0\\\ \vdots\\\ 0\end{array}\right),\ 0_{m}\right],$

if $u_{j}-\overline{u}_{j}>0$. Here $1$ is on the $j$-th place.

$\mathcal{D}\max\big{\\{}u_{j}(t)-\overline{u}_{j},0\big{\\}}=\left[0_{m},\
0_{m}\right],$

if $u_{j}-\overline{u}_{j}<0$.

$\mathcal{D}\max\big{\\{}u_{j}(t)-\overline{u}_{j},0\big{\\}}=\left[\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ 1\\\ 0\\\ \vdots\\\ 0\end{array}\right),\,0_{m}\right\\},\
0_{m}\right],$

if $u_{j}-\overline{u}_{j}=0$. Here $1$ is on the $j$-th place.

$\mathcal{D}\max\big{\\{}\underline{u}_{j}-u_{j}(t),0\big{\\}}=\left[\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ -1\\\ 0\\\ \vdots\\\ 0\end{array}\right),\ 0_{m}\right],$

if $\underline{u}_{j}-{u}_{j}(t)>0$. Here $-1$ is on the $j$-th place.

$\mathcal{D}\max\big{\\{}\underline{u}_{j}-u_{j}(t),0\big{\\}}=\left[0_{m},\
0_{m}\right],$

if $\underline{u}_{j}-{u}_{j}(t)<0$.

$\mathcal{D}\max\big{\\{}\underline{u}_{j}-u_{j}(t),0\big{\\}}=\left[\mathrm{co}\left\\{\left(\begin{array}[]{c}0\\\
\vdots\\\ 0\\\ -1\\\ 0\\\ \vdots\\\ 0\end{array}\right),\,0_{m}\right\\},\
0_{m}\right],$

if $\underline{u}_{j}-{u}_{j}(t)=0$. Here $-1$ is on the $j$-th place.

In the previous paragraph formulas the subdifferentials
$\underline{\partial}f_{i}(x,u,t)$ and the superdifferentials
$\overline{\partial}f_{i}(x,u,t)$, $i=\overline{1,n}$, are calculated via
quasidifferential calculus apparatus as well. Book Demyanov & Rubinov (1977)
(see Ch. III, Par. 2 there) contains a detailed description of these rules for
a rich class of functions. Let us give just some of these rules which were
used in the formulas of the previous paragraph. Let $\xi\in R^{l}$. If the
function $\varphi(\xi)$ is quasidifferentiable at the point $\xi_{0}\in R^{l}$
and $\lambda$ is some number, then we have

$\lambda\,\mathcal{D}\varphi(\xi_{0})=[\lambda\,\underline{\partial}\varphi(\xi_{0}),\lambda\,\overline{\partial}\varphi(\xi_{0})],\quad\text{if}\
\lambda\geqslant 0,$
$\lambda\,\mathcal{D}\varphi(\xi_{0})=[\lambda\,\overline{\partial}\varphi(\xi_{0}),\lambda\,\underline{\partial}\varphi(\xi_{0})],\quad\text{if}\
\lambda<0.$

If the functions $\varphi_{k}(\xi)$, $k=\overline{1,r}$, are
quasidifferentiable at the point $\xi_{0}\in R^{l}$, then the quasidifferntial
of the function $\varphi(\xi)=\max_{k=\overline{1,r}}\varphi_{k}(\xi)$ at this
point is calculated by the formula

$\mathcal{D}\varphi(\xi_{0})=[\underline{\partial}\varphi(\xi_{0}),\overline{\partial}\varphi(\xi_{0})],$
$\underline{\partial}\varphi(\xi_{0})=\mathrm{co}\Big{\\{}\underline{\partial}\varphi_{k}(\xi_{0})-\sum_{i\in
P(\xi_{0}),\ i\neq k}\overline{\partial}\varphi_{i}(\xi_{0}),\ k\in
P(\xi_{0})\Big{\\}},$ $\overline{\partial}\varphi(\xi_{0})=\sum_{i\in
P(\xi_{0})}\overline{\partial}\varphi_{i}(\xi_{0}),$
$P(\xi_{0})=\\{k\in\\{1,\dots,r\\}\ |\
\varphi_{k}(\xi_{0})=\varphi(\xi_{0})\\}.$

If the functions $\varphi_{k}(\xi)$, $k=\overline{1,r}$, are
quasidifferentiable at the point $\xi_{0}\in R^{l}$, then the quasidifferntial
of the function $\varphi(\xi)=\min_{k=\overline{1,r}}\varphi_{k}(\xi)$ at this
point is calculated by the following formula

$\mathcal{D}\varphi(\xi_{0})=[\underline{\partial}\varphi(\xi_{0}),\overline{\partial}\varphi(\xi_{0})],$
$\underline{\partial}\varphi(\xi_{0})=\sum_{j\in
Q(\xi_{0})}\underline{\partial}\varphi_{j}(\xi_{0}),$
$\overline{\partial}\varphi(\xi_{0})=\mathrm{co}\Big{\\{}\overline{\partial}\varphi_{k}(\xi_{0})-\sum_{j\in
Q(\xi_{0}),\ j\neq k}\underline{\partial}\varphi_{j}(\xi_{0}),\ k\in
Q(\xi_{0})\Big{\\}},$ $Q(\xi_{0})=\\{k\in\\{1,\dots,r\\}\ |\
\varphi_{k}(\xi_{0})=\varphi(\xi_{0})\\}.$

Note also that if the function $\varphi(\xi)$ is subdifferentiable at the
point $\xi_{0}\in R^{l}$, then its quasidifferential at this point may be
represented in the form

$\mathcal{D}\varphi(\xi_{0})=[\underline{\partial}\varphi(\xi_{0}),0_{l}],$

and if the function $\varphi(\xi)$ is superdifferentiable at the point
$\xi_{0}\in R^{l}$, then its quasidifferential at this point may be
represented in the form

$\mathcal{D}\varphi(\xi_{0})=[0_{l},\overline{\partial}\varphi(\xi_{0})].$

These two formulas can be taken as definitions of a subdifferentiable and a
superdifferentiable function respectively. If the function $\varphi(\xi)$ is
differentiable at the point $\xi_{0}\in R^{l}$, then its quasidifferential may
be represented in the forms

$\mathcal{D}\varphi(\xi_{0})=[\varphi^{\prime}(\xi_{0}),0_{l}]\ \mathrm{or}\
\mathcal{D}\varphi(\xi_{0})=[0_{l},\varphi^{\prime}(\xi_{0})],$

where $\varphi^{\prime}(\xi_{0})$ is a gradient of the function $\varphi(\xi)$
at the point $\xi_{0}$. The latter fact indicates that there is not the only
way to construct quasidifferential. We also note that the subdifferential (the
superdifferential) of the finite sum of quasidifferentiable functions is a sum
of subdifferentials (superdifferentials) of the summands, i. e. if the
functions $\varphi_{k}(\xi)$, $k=\overline{1,r}$, are quasidifferentiable at
the point $\xi_{0}\in R^{l}$, then the quasidifferential of the function
$\varphi(\xi)=\sum_{k=1}^{r}\varphi_{k}(\xi)$ at this point is calculated by
the formula

$\mathcal{D}\varphi(\xi_{0})=\left[\sum_{k=1}^{r}\underline{\partial}\varphi_{k}(\xi_{0}),\sum_{k=1}^{r}\overline{\partial}\varphi_{k}(\xi_{0})\right].$

Via the rules given and formulas (12) and (13) we find the quasidifferentials
$\mathcal{D}I_{1}(x,z,u)$ and $\mathcal{D}I_{3}(u)$.

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