# Method for finding solution to
“quasidifferentiable” differential inclusion

Alexander Fominyh  St. Petersburg University, 7–9, Universitetskaya nab., St.
Petersburg, 199034, Russian Federation

###### Abstract.

The paper explores the differential inclusion of a special form. It is
supposed that the support function of the set in the right-hand side of an
inclusion may contain the sum of the maximum and the minimum of the finite
number of continuously differentiable (in phase coordinates) functions. It is
required to find a trajectory that would satisfy differential inclusion with
the boundary conditions prescribed and simultaneously lie on the surface
given. We give substantial examples of problems where such differential
inclusions may occur: models of discontinuous systems, linear control systems
where the control function or/and disturbance of the right-hand side is/are
known to be subject to some nonsmooth (in phase vector) constraints, some real
mechanical models and differential inclusions per se with special geometrical
structure of the right-hand side. The initial problem is reduced to a
variational one. It is proved that the resulting functional to be minimized is
quasidifferentiable. The necessary minimum conditions in terms of
quasidifferential are formulated. The steepest (or the quasidifferential)
descent method in a classical form is then applied to find stationary points
of the functional obtained. Herewith, the functional is constructed in such a
way that one can verify whether the stationary point constructed is indeed a
global minimum point of the problem. The “weak” convergence of the method
proposed is proved for some particular cases. The method constructed is
illustrated by numerical examples.

###### Key words and phrases:

differential inclusion, support function, quasidifferential, dry friction,
sliding mode

###### 1991 Mathematics Subject Classification:

34A60, 49J52, 49K05

The main results of this paper (Sections 4, 5, 6, 7) were obtained in Saint
Petersburg State University and supported by Russian Science Foundation (grant
23-71-01065).

##  1\. Introduction

Differential inclusions are a powerful instrument for modeling dynamical
systems. In this paper the differential inclusions of some special structure
are explored. More specific: the support function of the set in the right-hand
side of a differential inclusion contains the sum of the maximum and the
minimum functions of the finite number of continuously differentiable (in
phase coordinates) functions. So it is required to find a trajectory of such a
differential inclusion which simultaneously would satisfy the boundary
conditions and lie on the surface prescribed.

Such differential inclusions arise from systems of differential equations with
discontinuous right-hand sides (when considering such systems moving in a
sliding mode), from linear control systems when the control function is
subject to some nonsmooth (in phase vector) constraints or/and the nature of
disturbance of the right-hand side is known to contain nondifferentiable
functions of phase coordinates, some real mechanical models when coordinates,
their velocities and accelarations are under some geometrical constraints
leading to the differential inclusions structure considered. A good real
example generating the type of differential inclusion under consideration is
physical model of a system with dry friction and nonsmooth control forcing
this system to move in a sliding mode (see Example 7.3 below).

The majority of papers in literature considering differential inclusions are
devoted to such classical problems as: existence of solutions [1], [2], [3],
[4], dependence of solutions on the parameters [4], [5], attainability and
viability [3], [5], [6]. Let us also give some references [7], [8], [9], [10]
with optimality conditions in problems with differential inclusion. In these
papers differential inclusions of a rather general form are considered. There
are cases of phase constraints as well as nonsmooth and nonconvex ones. On the
other hand, the works listed are more of theoretical significance and some
results seem hard to be employed in practice. Note that the majority of
methods in literature consider only differential inclusions with a free right
endpoint and use some classical approaches as Euler and Runge-Kutta schemes,
various finite differences methods etc. (see, e. g., [11], [12], [13], [14],
[15]). A survey of difference methods for differential inclusions can be found
in [16]. Note one paper [17] where an algorithm to solve boundary value
problems for differential inclusions was constructed.

We reduce the original problem to a variational one; such a reduction is
implemented just like in the author previous works [18], [19]. The functional
obtained is proved to be quasidifferentiable. So the nonsmooth optimization
methods are to be applied. In papers [20], [21] a method for minimizing of a
quasi (sub) differential functional is proposed based on the idea of
considering phase trajectory and its derivative as independent variables (and
taking the natural connection between these variables into account via the
special penalty term). Also note the work [22] where the similar technique is
used to consider systems moving in sliding modes in a more simple case of
smooth functionals.

The “weak” convergence of the modified quasidifferential descent method is
proved in some particlular cases due to the special structure of the
functional to be minimized. The conceptual scheme of the method as well as
ideas of proof are based on the analogous ones of V. F. Demyanov scientific
school on nondifferentiable minimization and taken from book [24] where a
modified subdifferentiable (steepest) descent method for minimizing the
maximum of the finite number of continuously differentiable functions in the
finite dimensional space is proposed justified.

The principial novelity of the method suggested in the paper consists of the
separation of the variables (phase coordinates and its derivatives) and in the
subsequent results such as necessary minimum conditions in new effective form
and construction of the quasidifferential descent method arising from this
idea.

As it was noted above, only a small part of the literature, devoted to
differential inclusions, deals with constructing numerical methods for solving
the corresponding boundary value problem. Moreover, to the best of the author
knowledge literature devoted to differential inclusions all the more do not
deal with the numerical methods for nonsmooth right-hand side of the inclusion
as it is done in the current paper.

##  2\. Basic definitions and notations

In this paper we use the following notations. Denote $\mathcal{N}$ the set of
natural numbers. Let $C_{n}[0,T]$ be the space of $n$-dimensional continuous
on $[0,T]$ vector-functions. Let also $P_{n}[0,T]$ be the space of piecewise
continuous and bounded on $[0,T]$ $n$-dimensional vector-functions. We also
require the space $L^{2}_{n}[0,T]$ of measurable and square-summable on
$[0,T]$ $n$-dimensional vector-functions (strictly speaking the known
corresponding factor-space is considered). If $X$ is some normed space, then
$||\cdot||_{X}$ denotes its norm and $X^{*}$ denotes the space conjugate to
the space given.

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

For some arbitrary set $F\subset R^{n}$ define the support function of the
vector $\psi~{}\in~{}R^{n}$ as $c(F,\psi)=\sup\limits_{f\in F}\langle
f,\psi\rangle$ where $\langle a,b\rangle$ is a scalar product of the vectors
$a,b\in R^{n}$. Denote $S_{n}$ and $B_{n}$ a unit sphere and a unit ball in
$R^{n}$ with the center in the origin respectively, also let $B_{r}(c)$ be a
ball with the radius $r\in R$ and the center $c\in R^{n}$. Let the vectors
$\bf{e_{i}}$, $i=\overline{1,n}$, form the standard basis in $R^{n}$. Let
$0_{n}$ denote a zero element of a functional space of some $n$-dimensional
vector-functions and $\bf{0_{n}}$ denote a zero element of the space $R^{n}$.
If $\varphi_{1}(x)=\max\limits_{i=\overline{1,M}}f_{i}(x)$, where
$f_{i}(x):R^{n}\rightarrow R$, $i=\overline{1,M}$, are some functions, then we
call the function $f_{\overline{i}}(x)$, $\overline{i}\in\\{1,\dots,M\\}$, an
active one at the point $x_{0}\in R^{n}$, if
$\overline{i}\in\mathcal{R}_{1}(x_{0})=\Big{\\{}i\in\\{1,\dots,M\\}\ \big{|}\
f_{i}(x_{0})=\varphi_{1}(x_{0})\Big{\\}}$. If
$\varphi_{2}(x)=\min\limits_{j=\overline{1,K}}g_{j}(x)$, where
$g_{j}(x):R^{n}\rightarrow R$, $j=\overline{1,K}$, are some functions, then we
call the function $g_{\overline{j}}(x)$, $\overline{j}\in\\{1,\dots,K\\}$, an
active one at the point $x_{0}\in R^{n}$, if
$\overline{j}\in\mathcal{R}_{2}(x_{0})=\Big{\\{}j\in\\{1,\dots,K\\}\ \big{|}\
g_{j}(x_{0})=\varphi_{2}(x_{0})\Big{\\}}$.

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

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

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

In this case the function $h_{1}(x,z)$ is called [24] superdifferentiable at
the point $(x,z)$, and the set $\overline{\partial}h_{1}(x,z)$ is called the
superdifferential of the function $h_{1}(x,z)$ at the point $(x,z)$.

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

$h_{1}(x+\alpha d_{1},z+\alpha
d_{2})=h_{1}(x,z)+\alpha\min_{w\in\overline{\partial}h_{1}(x,z)}\langle
w,d\rangle+o(\alpha,x,z,d),$ $\quad\frac{o(\alpha,x,z,d)}{\alpha}\rightarrow
0,\ \alpha\downarrow 0.$

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

$\frac{\partial h_{2}(x,z)}{\partial d}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\Big{(}h_{2}(x+\alpha d_{1},z+\alpha
d_{2})-h_{2}(x,z)\Big{)}=\max_{v\in\underline{\partial}h_{2}(x,z)}\langle
v,d\rangle.$ (2)

In this case the function $h_{2}(x,z)$ is called [24] subdifferentiable at the
point $(x,z)$, and the set $\underline{\partial}h_{2}(x,z)$ is called the
subdifferential of the function $h_{2}(x,z)$ at the point $(x,z)$.

From expression (2) one can see that the following formula holds true:

$h_{2}(x+\alpha d_{1},z+\alpha
d_{2})=h_{2}(x,z)+\alpha\max_{v\in\underline{\partial}h_{2}(x,z)}\langle
v,d\rangle+o(\alpha,x,z,d),$ $\quad\frac{o(\alpha,x,z,d)}{\alpha}\rightarrow
0,\ \alpha\downarrow 0.$

If the function $\varsigma(\xi)$ is differentiable at the point $\xi_{0}\in
R^{\ell}$, then its superdifferential (subdifferential) at this point is
represented [24] in the form

$\overline{\partial}\varsigma(\xi_{0})=\\{\varsigma^{\prime}(\xi_{0})\\},\quad\underline{\partial}\varsigma(\xi_{0})=\\{\bf{0_{\ell}}\\}$
(3)
$\Big{(}\underline{\partial}\varsigma(\xi_{0})=\\{\varsigma^{\prime}(\xi_{0})\\},\quad\overline{\partial}\varsigma(\xi_{0})=\\{\bf{0_{\ell}}\\}\Big{)},$

where $\varsigma^{\prime}(\xi_{0})$ is a gradient of the function
$\varsigma(\xi)$ at the point $\xi_{0}$.

Note also that the superdifferential (subdifferential) of the finite sum of
superdifferentiable (subdifferentiable) functions is the sum of the
superdifferentials (subdifferentials) of summands, i. e. if the functions
$\varsigma_{k}(\xi)$, $k=\overline{1,r}$, are superdifferentiable
(subdifferentiable) at the point $\xi_{0}\in R^{\ell}$, then the function
$\varsigma(\xi)=\sum_{k=1}^{r}\varsigma_{k}(\xi)$ superdifferential
(subdifferential) at this point is calculated [24] by the formula

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

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

$\frac{\partial\varphi_{1}(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\Big{(}\varphi_{1}(x+\alpha g_{1},z+\alpha
g_{2})-\varphi_{1}(x,z)\Big{)}=\min_{w\in\overline{\partial}\varphi_{1}(x,z)}w(g).$
(5)

In this case the functional $\varphi_{1}(x,z)$ is called [25]
superdifferentiable at the point $(x,z)$ and the set
$\overline{\partial}{\varphi_{1}(x,z)}$ is called a superdifferential of the
functional $\varphi_{1}(x,z)$ at the point $(x,z)$.

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

$\varphi_{1}(x+\alpha g_{1},z+\alpha
g_{2})=\varphi_{1}(x,z)+\alpha\min_{w\in\overline{\partial}\varphi_{1}(x,z)}w(g)+o(\alpha,x,z,g),$
$\quad\frac{o(\alpha,x,z,g)}{\alpha}\rightarrow 0,\ \alpha\downarrow 0.$

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

$\frac{\partial\varphi_{2}(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\Big{(}\varphi_{2}(x+\alpha g_{1},z+\alpha
g_{2})-\varphi_{2}(x,z)\Big{)}=\max_{v\in\underline{\partial}\varphi_{2}(x,z)}v(g).$
(6)

In this case the functional $\varphi_{2}(x,z)$ is called [25]
subdifferentiable at the point $(x,z)$ and the set
$\underline{\partial}{\varphi_{2}(x,z)}$ is called a subdifferential of the
functional $\varphi_{2}(x,z)$ at the point $(x,z)$.

From expression (6) one can see that the following formula holds true:

$\varphi_{2}(x+\alpha g_{1},z+\alpha
g_{2})=\varphi_{2}(x,z)+\alpha\max_{v\in\underline{\partial}\varphi_{2}(x,z)}v(g)+o(\alpha,x,z,g),$
$\quad\frac{o(\alpha,x,z,g)}{\alpha}\rightarrow 0,\ \alpha\downarrow 0.$

Note that more general definitions (both for the finite-dimensional and the
functional spaces) may be given, when the function (functional) directional
derivative may be represented as a sum of a maximum and a minimum of linear
functionals over some sets (the subdifferential and the superdifferential
respectively), then the function (functional) is called quasidifferentiable
and the pair [subdifferential, superdifferential] is called a
quasidifferential. However we intentionally give the independent definitions
of these objects, since the support function under consideration in this paper
is a sum of a superdifferentiable and a subdifferentiable functions (see
formula (11) below).

##  3\. Statement of the problem and reduction to a variational one

We now give the formal statement of the problem. The motivating examples
leading to such a problem are given in Section 5.

It is required to find such a trajectory $x^{*}\in C_{n}[0,T]$ (with the
derivative $\dot{x}^{*}\in P_{n}[0,T]$) satisfies differential inclusion

$\dot{x}(t)\in F(x(t)),$ (7)

moves along surface

$e(x)={\bf 0_{o}}$ (8)

while $t\in[0,T]$ and meets boundary conditions

$x(0)=x_{0},$ (9) $x_{j}(T)={x_{T}}_{j},\quad j\in J.$ (10)

Assume that there exists such a solution.

In (8) let $e(x)$ be a known continuously differentiable $o$-dimensional
vector-function; in formula (9) the initial point $x_{0}\in R^{n}$ is a given
vector; in formula (10) the final point coordinates ${x_{T}}_{j}$, $j\in J$,
are given numbers corresponding to those ones of the state vector which are
fixed at the right endpoint, here $J\subset\\{1,\dots,n\\}$ is a given index
set.

Now make assumptions regarding the right-hand side support function. Suppose
that differential inclusion in (7) is endowed with the following properties:
1) at each fixed $x\in R^{n}$ the set $F(x)$ on the right-hand side may be
“partitioned” into the sets $F_{1}(x),\dots F_{n}(x)$ (such that
$\dot{x}_{i}\in F_{i}(x)$, $i=\overline{1,n}$) and each of the sets
$F_{1}(x),\dots F_{n}(x)$ is convex and compact; 2) the support functions of
the corresponding sets $F_{i}(x)$, $i=\overline{1,n}$, may be represented as

$c(F_{i}(x),\psi_{i})=c_{1}(F_{i}(x),\psi_{i})+c_{2}(F_{i}(x),\psi_{i}),$ (11)

where

$c_{1}(F_{i}(x),\psi_{i})=\sum_{j=1}^{r}\max\Big{\\{}f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{m(j)}}(x)\psi_{i}\Big{\\}},$
(12)
$c_{2}(F_{i}(x),\psi_{i})=\sum_{j=1}^{s}\min\Big{\\{}g_{i,j_{1}}(x,\psi_{i}),\dots,g_{i,j_{k(j)}}(x,\psi_{i})\Big{\\}},$
(13)

where $f_{i,j_{1}}(x)\psi_{i},\dots,f_{i,j_{m(j)}}(x)\psi_{i}$,
$i=\overline{1,n}$, $j=\overline{1,r}$,
$g_{i,j_{1}}(x,\psi_{i}),\dots,g_{i,j_{k(j)}}(x,\psi_{i})$,
$i=\overline{1,n}$, $j=\overline{1,s}$ (for simplicity of presentation we
suppose that $r$ and $s$ are the same for each $i=\overline{1,n}$), are
continuously differentiable in $x$ functions (at fixed $\psi_{i}\in S_{1}$,
$i=\overline{1,n}$).

A slight difference in the form of formulas of two support functions above is
due to practical problems generating such a structure of support functions
(see Section 5 for corresponding examples).

We will sometimes write $F$ instead of $F(x)$ for brevity. Insofar as $\forall
x\in R^{n}$ the set $F_{i}(x)$, $i=\overline{1,n}$, is a convex compact set in
$R^{n}$, then inclusion (7) may be rewritten [26] as follows:

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

Denote $z(t)=\dot{x}(t)$, $z\in P_{n}[0,T]$, then from (9) one has

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

Let us now realize the following idea. “Forcibly” consider the points $z$ and
$x$ to be “independent” variables. Since, in fact, there is relationship (14)
between these variables (which naturally means that the vector-function $z(t)$
is a derivative of the vector-function $x(t)$), let us take this restriction
into account by using the functional

$\upsilon(x,z)=\frac{1}{2}\int_{0}^{T}\left(x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau\right)^{2}dt.$
(15)

It is seen that relation (14) and condition (9) on the left endpoint is
satisfied iff $\upsilon(x,z)=0$.

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

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

then put

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

and construct the functional

$\varphi(x,z)=\frac{1}{2}\int_{0}^{T}||h\big{(}x(t),z(t)\big{)}||^{2}_{R^{n}}dt.$
(16)

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

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

holds true, i. e. inclusion (7) takes place iff $\varphi(x,z)=0$.

Introduce the functional

$\chi(z)=\frac{1}{2}\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right)^{2}.$ (17)

We see that if $\upsilon(x,z)=0$, then condition (10) on the right endpoint is
satisfied iff $\chi(z)=0$.

Introduce the functional

$\omega(x)=\frac{1}{2}\int_{0}^{T}||e\big{(}x(t)\big{)}||^{2}_{R^{o}}dt.$ (18)

Obviously, the trajectory $x(t)$ belongs to surface (8) at each $t\in[0,T]$
iff $\omega(x)=0$.

Finally construct the functional

$I(x,z)=\varphi(x,z)+\chi(z)+\omega(x)+\upsilon(x,z).$ (19)

So the original problem has been reduced to minimizing functional (19) on the
space

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

Denote $z^{*}$ a global minimizer of this functional. Then

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

is a solution of the initial problem.

The structure of the functional $\varphi(x,z)$ is natural as the value
$h_{i}(x(t),z(t))$, $i=\overline{1,n}$, at each fixed $t\in[0,T]$ is the
Euclidean distance from the point $z_{i}(t)$ to the set $F_{i}(x(t))$;
functional (16) is half the sum of squares of the deviations in
$L^{2}_{n}[0,T]$ norm of the trajectories $z_{i}(t)$ from the sets $F_{i}(x)$,
$i=\overline{1,n}$, respectively. The meaning of functionals (15), (17), (18)
structures is obvious.

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

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

In this section referring to superdifferential calculus rules (3), (4), we
mean their known analogues in a functional space [27].

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

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

By superdifferential calculus rule (3) one may put

$\underline{\partial}\chi(z)=\left\\{\sum_{j\in
J}\left({x_{0}}_{j}+\int_{0}^{T}z_{j}(t)dt-{x_{T}}_{j}\right){\bf
e_{j}}\right\\},\quad\overline{\partial}\chi(z)=0_{n}$ (20)

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

$\nabla\omega(x,t)=\sum_{j=1}^{o}e_{j}\big{(}x(t)\big{)}\frac{\partial
e_{j}\big{(}x(t)\big{)}}{\partial x}.$

By superdifferential calculus rule (3) one may put

$\underline{\partial}\omega(x,t)=\left\\{\sum_{j=1}^{o}e_{j}\big{(}x(t)\big{)}\frac{\partial
e_{j}\big{(}x(t)\big{)}}{\partial
x}\right\\},\quad\overline{\partial}\omega(x,t)=0_{n}.$ (21)

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

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

By superdifferential calculus rule (3) one may put

$\underline{\partial}\upsilon(x,z,t)=\left\\{\begin{pmatrix}\displaystyle{x(t)-x_{0}-\int_{0}^{t}z(\tau)d\tau}\\\
\displaystyle{-\int_{t}^{T}\left(x(\tau)-x_{0}-\int_{0}^{\tau}z(s)ds\right)d\tau}\end{pmatrix}\right\\},\quad\overline{\partial}\upsilon(x,z,t)=0_{2n}.$
(22)

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 ${L^{2}_{n}[0,T]\times L^{2}_{n}[0,T]}$ (see [28]). Now keeping this
fact in mind, note that strictly speaking, in formulas (20), (21), (22) not
the subdifferentials and superdifferentials theirselves but their images under
the corresponding isometric isomorphisms are given.

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

$\overline{\partial}\Big{(}{\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\Big{)}=h_{i}(x,z)\bigg{(}\psi^{*}_{i}{\bf
e_{i+n}}+\sum_{j=1}^{r}\overline{\partial}\Big{(}-\max\Big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{m(j)}}(x)\psi_{i}^{*}\Big{\\}}\Big{)}\bigg{)}$
(23)

where at $j=\overline{1,r}$ and $j_{q}\in\mathcal{R}^{1}_{ij}(x)$ we have

$\overline{\partial}\Big{(}-\max\Big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{m(j)}}(x)\psi_{i}^{*}\Big{\\}}\Big{)}=\mathrm{co}\left\\{\left[-\psi_{i}^{*}\frac{\partial
f_{i,j_{q}}(x)}{\partial x},\bf{0_{n}}\right]\right\\},$
$\mathcal{R}^{1}_{ij}(x)=\Big{\\{}j_{q}\in\\{j_{1},\dots,j_{m(j)}\\}\ \big{|}\
f_{i,j_{q}}(x)\psi_{i}^{*}=\max\Big{\\{}f_{i,j_{1}}(x)\psi_{i}^{*},\dots,f_{i,j_{m(j)}}(x)\psi_{i}^{*}\Big{\\}}\Big{\\}}$

and

$\underline{\partial}\Big{(}{\textstyle\frac{1}{2}}\,h_{i}^{2}(x,z)\Big{)}=h_{i}(x,z)\sum_{j=1}^{s}\underline{\partial}\Big{(}-\min\Big{\\{}g_{i,j_{1}}(x,\psi_{i}^{*}),\dots,g_{i,j_{k(j)}}(x,\psi_{i}^{*})\Big{\\}}\Big{)}$
(24)

where at $j=\overline{1,s}$ and $j_{p}\in\mathcal{R}^{2}_{ij}(x)$ we have

$\underline{\partial}\Big{(}-\min\Big{\\{}g_{i,j_{1}}(x,\psi_{i}^{*}),\dots,g_{i,j_{k(j)}}(x,\psi_{i}^{*})\Big{\\}}\Big{)}=\
\mathrm{co}\left\\{\left[-\frac{\partial g_{i,j_{p}}(x,\psi_{i}^{*})}{\partial
x},\bf{0_{n}}\right]\right\\},$
$\mathcal{R}^{2}_{ij}(x)=\bigg{\\{}j_{p}\in\\{j_{1},\dots,j_{k(j)}\\}\
\big{|}\
g_{i,j_{p}}(x,\psi_{i}^{*})=\min\Big{\\{}g_{i,j_{1}}(x,\psi_{i}^{*}),\dots,g_{i,j_{k(j)}}(x,\psi_{i}^{*})\Big{\\}}\bigg{\\}}$

and

$\ell_{i}(\psi_{i}^{*}(x,z),x,z)=\max_{\psi_{i}\in
S_{1}}\ell_{i}(\psi_{i},x,z)\quad if\ h_{i}(x,z)>0$
$\psi_{i}^{*}=\psi_{i}^{0}\ ({where}\ \psi_{i}^{0}\ is\ some\ fixed\ element\
of\ the\ set\ S_{1})\quad if\ h_{i}(x,z)=0.$

Note that in the case $h_{i}(x,z)>0$ the element $\psi_{i}^{*}(x,z)$ is unique
from support functions properties [26] and is continuous [29] from maximum
(minimum) function properties.

Formulas (23) and (24) can be easily checked based on the proof given in [22]
and on the quasidifferential calculus rules [30]. Therefore, let us give only
a brief scheme of proof here.

Fix some $i\in\\{1,\dots,n\\}$ and consider two cases.

a) Suppose that $h_{i}(x,z)>0$, i. e. $h_{i}(x,z)=\max_{\psi_{i}\in
S_{1}}\ell_{i}(\psi_{i},x,z)>0$. Our aim is to apply the corresponding theorem
on a directional differentiability from [29]. There the inf-functions are
considered, hence let us apply this theorem to the function
$-\ell_{i}(\psi_{i},x,z)$. Check that the function $h_{i}(x,z)$ satisfies the
conditions:
i) the function $\ell_{i}(\psi_{1},x,z)$ is continuous on $S_{1}\times
R^{n}\times R^{n}$;
ii) there exist a number $\beta$ and a compact set $C\in R$ such that for
every $(\overline{x},\overline{z})$ in the vicinity of the point $(x,z)$ the
level set
${lev}_{\beta}(-\ell(\cdot,\overline{x},\overline{z}))=\\{\psi_{i}\in S_{1}\
|\ -\ell_{i}(\psi_{i},\overline{x},\overline{z})\leq\beta\\}$ is nonempty and
is contained in the set $C$;
iii) for any fixed $\psi_{i}\in S_{1}$ the function
$\ell_{i}(\psi_{i},\cdot,\cdot)$ is directionally differentiable at the point
$(x,z)$;
iv) if $d=(d_{1},d_{2})^{\prime}\in R^{n}\times R^{n}$, $\gamma_{n}\downarrow
0$ and $\psi_{i_{n}}$ is a sequence in $C$, then $\psi_{i_{n}}$ has a limit
point $\overline{\psi}_{i}$ such that

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

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

In order to verify iii) it is sufficient to note that at every fixed
$\psi_{i}\in S_{1}$ the corresponding functions under consideration are the
maxima (minima) of continuously differentiable functions and it is well known
[30] that the $\max$-function ($\min$-function) (of continuously
differentiable functions) is subdifferentiable (superdifferentiable), hence
the explicit expression for their directional derivatives can be obtained.

In order to verify iv) it is sufficient to write out explicitly the right-hand
and the left-hand sides of the inequality required and to check directly that
iv) holds true using the Taylor expansion of the continuously differentiable
functions under $\max$-function ($\min$-function) and the well known [24]
inequalities

$\min_{j=\overline{1,K}}(\nu_{j}+\mu_{j})\geq\min\nu_{j=\overline{1,K}}+\min\mu_{j=\overline{1,K}},$
$\max_{i=\overline{1,M}}(\nu_{i}+\mu_{i})\geq\max_{i=\overline{1,M}}\nu_{i}+\max_{i\in\mathcal{R}}\mu_{i},\quad\mathcal{R}=\Big{\\{}i\in\\{1,\dots,M\\}\
|\ \nu_{i}=\max_{i=\overline{1,M}}\nu_{i}\Big{\\}},$

which are true for the arbitrary numbers $\nu_{i}$, $\mu_{i}$,
$i=\overline{1,K}$ or $\nu_{i}$, $\mu_{i}$, $i=\overline{1,M}$.

So the function $h_{i}(x,z)$ is directionally differentiable at the point
$(x,z)$ but writing out explicitly its directional derivative we make sure
that by definition (see (1)) the function $h_{i}(x,z)$ is quasidifferentiable
at the point $(x,z)$; so the function $h_{i}^{2}(x,z)$ is quasidifferentiable
at the point $(x,z)$ and its subdifferential and superdifferential are
obtained via (24) and (23) respectively by quasidifferential calculus rules
[30].

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

If the interval $[0,T]$ may be divided into a finite number of intervals, in
every of which one (several) of the considered functions is (are) active, then
the function $h_{i}(x,z)$, $i=\overline{1,n}$, quasidifferentiability can be
proved on every of such intervals in a completely analogous fashion. In [20]
it is shown that the functional $\varphi(x,z)$ is quasidifferentiable and that
its quasidifferential is determined by the corresponding integrand
quasidifferential.

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

$\frac{\partial\varphi(x,z)}{\partial g}=\lim_{\alpha\downarrow
0}\frac{1}{\alpha}\Big{(}\varphi(x+\alpha g_{1},z+\alpha
g_{2})-\varphi(x,z)\Big{)}=\max_{v\in\underline{\partial}\varphi(x,z)}v(g)+\min_{w\in\overline{\partial}\varphi(x,z)}w(g)$
(25)

Here the sets $\underline{\partial}\varphi(x,z)$,
$\overline{\partial}\varphi(x,z)$ are of the following form

$\underline{\partial}\varphi(x,z)=\Bigg{\\{}v\in\Big{(}C_{n}[0,T]\times
P_{n}[0,T],||\cdot||_{L^{2}_{n}[0,T]\times L^{2}_{n}[0,T]}\Big{)}^{*}\
\Big{|}$ (26) $v(g)=\int_{0}^{T}\langle v_{1}(t),g_{1}(t)\rangle
dt+\int_{0}^{T}\langle v_{2}(t),g_{2}(t)\rangle dt\quad\forall g_{1}\in
C_{n}[0,T],\,g_{2}\in P_{n}[0,T],$ $v_{1}(t),v_{2}(t)\in
L_{n}^{\infty}[0,T],\quad(v_{1}(t),v_{2}(t))^{\prime}\in{\textstyle\frac{1}{2}}\,\underline{\partial}h^{2}(x(t),z(t))\quad
for\ all\ t\in[0,T]\Bigg{\\}}.$

$\overline{\partial}\varphi(x,z)=\Bigg{\\{}w\in\Big{(}C_{n}[0,T]\times
P_{n}[0,T],||\cdot||_{L^{2}_{n}[0,T]\times L^{2}_{n}[0,T]}\Big{)}^{*}\
\Big{|}$ (27) $w(g)=\int_{0}^{T}\langle w_{1}(t),g_{1}(t)\rangle
dt+\int_{0}^{T}\langle w_{2}(t),g_{2}(t)\rangle dt\quad\forall g_{1}\in
C_{n}[0,T],\,g_{2}\in P_{n}[0,T],$ $w_{1}(t),w_{2}(t)\in
L_{n}^{\infty}[0,T],\quad(w_{1}(t),w_{2}(t))^{\prime}\in{\textstyle\frac{1}{2}}\,\overline{\partial}h^{2}(x(t),z(t))\quad
for\ all\ t\in[0,T]\Bigg{\\}}.$

Using formulas (20), (21), (22), (23), (24), (25), (26), (27) obtained and
superdifferential calculus rule (4) we have the final formula for calculating
the subdifferential and the superdifferential of the functional $I(x,z)$ at
the point $(x,z)$, let us denote

$\underline{\partial}\mathcal{I}(x,z,t):={\textstyle\frac{1}{2}}\underline{\partial}h^{2}(x(t),z(t))+\underline{\partial}\overline{\chi}(z)+\underline{\partial}\overline{\omega}(x,t)+\underline{\partial}\upsilon(x,z,t),\quad\overline{\partial}\mathcal{I}(x,z,t):={\textstyle\frac{1}{2}}\overline{\partial}h^{2}(x(t),z(t)).$

Using the known minimum condition [25] of the functional $I(x,z)$ at the point
$(x^{*},z^{*})$ in terms of quasidifferential, we conclude that the following
theorem is true.

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

$-\overline{\partial}\mathcal{I}(x^{*},z^{*},t)\subset\underline{\partial}\mathcal{I}(x^{*},z^{*},t)$
(28)

at almost each $t\in[0,T]$.

If one has $I(x^{*},z^{*})=0$, then condition (28) is also sufficient.

Theorem 4 already contains a constructive minimum condition, since on its
basis it is possible to construct the quasidifferential descent direction, and
for solving each of the subproblems arising during this construction there
exist some known efficient algorithms for solving them.

##  5\. Problems Generating Nonsmooth Differential Inclusions

Consider the differntial inclusion

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

Let us consider the case when the right-hand side of the differential
inclusion is the convex hull of several sets. For simplicity of presentation,
we will describe the case with only two sets. Consider the mapping

$F(x)=\mathrm{co}\\{F_{1}(x),F_{2}(x)\\},$

where the multivalued mappings $F_{1}(x),F_{2}(x)$ are continuous and at each
fixed $x\in R^{n}$ the sets $F_{1}(x),F_{2}(x)$ are convex and compact. Using
a simple formula [26] for the support function of the union of two convex
compact sets, we have

$c(F(x),\psi)=\max\Big{\\{}c(F_{1}(x),\psi),c(F_{2}(x),\psi)\Big{\\}}.$

Give here one one practical problem which leads to such a system. Let from
some physical considerations the “velocity” $\dot{x}_{1}$ of an object lie in
the range $[\min\\{x_{1},x_{2},x_{3}\\},\max\\{x_{1},x_{2},x_{3}\\}]$ of the
“coordinates” $x_{1}$, $x_{2}$, $x_{3}$. The segment given may be written down
as
$\mathrm{co}\\{x_{1},x_{2},x_{3}\\}=\mathrm{co}\bigcup\limits_{i=1}^{3}\mathcal{F}_{1}^{i}(x)$,
where $\mathcal{F}_{1}^{i}(x)=\\{x_{i}\\}$, $i=\overline{1,3}$. The support
function [26] of this set for $\psi_{1}\in R$ is

$c(F(x),\psi)=\max\Big{\\{}c\big{(}\mathcal{F}_{1}^{1}(x),\psi_{1}\big{)},c\big{(}\mathcal{F}_{1}^{2}(x),\psi_{1}\big{)},c\big{(}\mathcal{F}_{1}^{3}(x),\psi_{1}\big{)}\Big{\\}}=\max\\{x_{1}\psi_{1},x_{2}\psi_{1},x_{3}\psi_{1}\\}.$

Let us consider the case when the right-hand side of the differential
inclusion is the convex hull of several sets. For simplicity of presentation,
we will describe the case with only two sets. Consider the mapping

$F(x)={F_{1}(x)\cap F_{2}(x)},$

where the multivalued mappings $F_{1}(x),F_{2}(x)$ are continuous and at each
fixed $x\in R^{n}$ the sets $F_{1}(x),F_{2}(x)$ are convex and compact. Using
a known formula [26] for the support function of the intersection of two
convex compact sets, we have

$c(F(x),\psi)=\overline{\mathrm{co}}\min\Big{\\{}c(F_{1}(x),\psi),c(F_{2}(x),\psi)\Big{\\}}.$
(29)

Give here one practical problem where such differential inclusions may arise.
Let from some physical considerations the “velocity” $\dot{x}_{1}$ of an
object must belong both to the ranges $\mathcal{F}_{1}^{1}(x)=[x_{1},x_{2}]$
and $\mathcal{F}_{1}^{2}(x)=[x_{2}-1,x_{2}+1]$ of the “coordinates” $x_{1}$,
$x_{2}$. The set considered may be written down as
$\mathcal{F}_{1}^{1}(x)\cap\mathcal{F}_{1}^{2}(x)$. The support function of
this set for $\psi_{1}\in S_{1}$ is [26] as follows:

$c(F(x),\psi)=\min\Big{\\{}c\big{(}\mathcal{F}_{1}^{1}(x),\psi_{1}\big{)},c\big{(}\mathcal{F}_{1}^{2}(x),\psi_{1}\big{)}\Big{\\}}=\min\left\\{\frac{x_{1}+x_{2}}{2}\psi_{1}+\frac{|x_{1}-x_{2}|}{2},\,x_{2}\psi_{1}+1\right\\},$

and the first function under the minimum is continuously differentiable in $x$
if $x_{1}\neq x_{2}$.

Note the following fact. As seen in example above in some cases one can
directly check that for $\psi\in S_{n}$ the support function of the
intersection of two sets is the minimum of the support functions of these
sets. However, it is obvious that in general case the right-hand side of
formula (29) does not coincide with

$\min\Big{\\{}c(F_{1}(x),\psi),c(F_{2}(x),\psi)\Big{\\}}$

(there may exist $\psi\in R^{n}$ at which the values of these expressions are
different). On the other hand using the known property of the support
functions, it is easy to show that the point $z$ belongs to the intersection
$F(x)$ iff

$\langle\psi,z\rangle\leq\min\Big{\\{}c(F_{1}(x),\psi),c(F_{2}(x),\psi)\Big{\\}}\quad\forall\psi\in
S_{n}$

This statement can be expanded to a more general case when the right-hand side
is the sum of the finite number of the unions and intersections of the finite
number of convex compacts. For simplicity we write out this condition for the
sum of the union $F^{1}(x)=F_{1}^{1}(x)\cup F_{2}^{1}(x)$ and the intersection
$F^{2}(x)=F_{1}^{2}(x)\cap F_{2}^{2}(x)$: the point $z$ belongs to the set
$F^{1}(x)+F^{2}(x)$ iff

$\langle\psi,z\rangle\leq\max\Big{\\{}c(F_{1}^{1}(x),\psi),c(F_{2}^{1}(x),\psi)\Big{\\}}+\min\Big{\\{}c(F_{1}^{2}(x),\psi),c(F_{2}^{2}(x),\psi)\Big{\\}}\quad\forall\psi\in
S_{n}.$

Hence in practice we may use this condition of belonging of a point to the
set, we actually need not the support function of the intersection, but the
support functions of the sets theirselves at $\psi\in S_{n}$. So instead of
the functional $I(x,z)$ one may construct the similar functional

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

where

$\widetilde{\varphi}(x,z)=\frac{1}{2}\int_{0}^{T}h^{2}\big{(}x(t),z(t)\big{)}dt,\quad$
$\widetilde{\ell}(\psi,x,z)=\langle
z,\psi\rangle-\widetilde{c}(F(x),\psi),\quad\widetilde{h}(x,z)=\max_{\psi\in
S_{n}}\max\\{0,\widetilde{\ell}(\psi,x,z)\\},$

and $\widetilde{c}(F(x),\psi)$ is still the sum of maxima and minima of
corresponding continuously differentiable in $x$ functions.

However, when such a $\widetilde{\varphi}(x,z)$ functional structure is used,
the following drawback arises: since $\widetilde{c}(F(x),\psi)$, is no more a
support function, the crucial property of the uniqueness of the vector
$\psi^{*}$, is no more valid (in the case when the point $z$ does not belong
to the set considered. Roughly speaking this is explained by the fact that
function $\widetilde{h}(x,z)$, is no more the distance to the intersection of
the sets but the maximum of the distances to these sets. While the distance to
the convex compact (as the intersection of convex compacts) is achieved at the
only point of the set, the maximum of the distances to these sets may be
achieved at two points of different sets (if the distances are equal). Let us
illustrate this by the following example. Take $F_{1}=B_{1}((-1,0)^{\prime})$,
$F_{2}=B_{1}((1,0)^{\prime})$. Then one has $c(F_{1},\psi)=-\psi_{1}+1$,
$c(F_{2},\psi)=\psi_{1}+1$. If one takes $z=(0,1)^{\prime}$, then
$\max_{\psi\in
S_{2}}\Big{\\{}\psi_{2}-\min\\{-\psi_{1}+1,\psi_{1}+1\\}\Big{\\}}$ (which is
the distance equal to $\sqrt{2}-1$ of the vector $z$ both to the sets $F_{1}$
and $F_{2}$) is achieved at two elements:
$\psi^{*}=(\sqrt{2}/2,\sqrt{2}/2)^{\prime}$ or
$\psi^{*}=(-\sqrt{2}/2,\sqrt{2}/2)^{\prime}$.

Hence in order for the Theorem 4 to remain true, we must make an additional
assumption. We suppose that in the case $z\notin F(x)$ the vector $\psi^{*}$
is still unique. For instance, in the case in the case when the right side of
an inclusion is represented in the form of intersection of two sets, the
property required will be guaranteed (as explained in the example above) if
$z$ is not a point equidistant from these sets.

An interesting problem for future research is to explore the differential
properties of the functional $\widetilde{I}(x,z)$ in the case of nonuniqueness
of the vector $\psi^{*}$. The hypothesis is that the functional remains
quasidifferentiable in this case as a superposition of the max-function and a
quasidifferentiable function. Also note that in the case of intersection of
two sets in the right-hand side of an inclusion the equality of the functions
under min-function (at the same value of vector $\psi^{*}$) means that the
points at which the minimum distances from the point $z$ to each of these sets
is achieved, coincide, hence this is a distance from the point $z$ to their
intersection which in turn means that the vector $\psi^{*}$ is unique in this
case and the results of the functional $\widetilde{I}(x,z)$
subdifferentiability of this paper remain valid here.

Consider the model of double pendulum, one has

$\ddot{x}_{1}=-\nu^{2}(1+2\mu^{2})x_{1}+\nu^{2}\mu^{2}x_{2},$
$\ddot{x}_{2}=\nu^{2}x_{1}-\nu^{2}x_{2},$

where $\nu=\frac{m_{1}}{m_{2}}$, $\mu=\frac{g}{l}$ and $m_{1}$ and $m_{2}$ are
the masses of the first and the second pendulum respectively, $g$ is the
gravitation acceleration and $l$ is the thread length. In the literature (see,
e. g., [34]) such models are considered with the assumption that the
oscillations are small, therefore one can expect $\sin x_{1}$, $\sin x_{2}$
presented in the exact model to be approximately equal to $x_{1}$, $x_{2}$
respectively. Here it is suggested to consider a more precise model such that,
for instance, the segments $[0.5x_{1},1.5x_{1}]$ and $[0.5x_{2},1.5x_{2}]$ are
presented in the model instead including the $\sin x_{1}$ and $\sin x_{2}$
values (at least at the small values within the segment $[0,0.5]$ of the
coordinates $x_{1}$, $x_{2}$). With such an assumption made one can note that
the support function of the right-hand side is as follows:

$c(F_{1}(x),\psi_{1})=-\nu^{2}(1+2\mu^{2})x_{1}\psi_{1}+0.5\nu^{2}(1+2\mu^{2})|x_{1}||\psi_{1}|+\nu^{2}\mu^{2}x_{2}\psi_{1}+0.5\nu^{2}\mu^{2}|x_{2}||\psi_{1}|,$
$c(F_{2}(x),\psi_{2})=\nu^{2}x_{1}\psi_{2}+0.5\nu^{2}|x_{1}||\psi_{2}|-\nu^{2}x_{2}\psi_{2}+0.5\nu^{2}|x_{2}||\psi_{2}|.$

Consider the control system with disturbance:

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

On the right side there is a control function $u(t)$, as well as a function
$\varpi(t)$ playing the role of some disturbance in the system. Let, based on
some physical considerations, the control as well as the perturbation
$\varpi(t)$ belong to the sets depending on the phase coordinates and suppose
that these sets support functions have the structure discussed in the paper.

Note that in book [32] an analytic solution to this problem is given using the
apparatus of support functions as well as a conjugate function if one assumes
that the corresponding concave hull of the sets required is realizable. This
solution is obtained in the case when the control as well as the disturbance
functions belong to some convex compact sets. Such analytic theory was
possible due to the known formulas of the linear systems solution (via
fundamental matrix) and support functions for standard convex compact sets of
rather simple structure.

Consider the control system

$\dot{x}(t)=Ax(t)+Bu(t)$ (30)

on the time interval $[-t^{*},T]$ (here $T$ is a given final time moment; see
comments on the time moment $t^{*}$ below).

Let also “discontinuity” surface be given:

$s(x)={\bf 0_{m}}$ (31)

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

Consider the following form of controls:

$u_{i}(x)=-a_{i}|x|\mathrm{sgn}(s_{i}(x)),\ i=\overline{1,m},\quad u_{i}=0,\
i=\overline{m+1,n}$ (32)

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

In book [31] it is shown that if surface (31) is a hyperplane, then under
natural assumptions and with sufficiently big values of the factors $a_{i}$,
$i=\overline{1,m}$, controls (32) ensure system (30) hitting a small vicinity
of this surface (31) from arbitrary initial state in the finite time $t^{*}$
and further staying in this neighborhood with the fulfillment of the condition
$s_{i}(x(t))\rightarrow 0$, $i=\overline{1,m}$, at $t\rightarrow\infty$. Now
we are interested in the behavior of the system on the “discontinuity” surface
(on the time interval $[0,T]$).

We see that the right-hand sides of the first $m$ differential equations in
system (30) with controls (32) are discontinuous on the surfaces $s_{i}(x)=0$,
$i=\overline{1,m}$. So one has to use one of the known definitions of a
discontinuous system solution. Thus, accordingly to Filippov definition [33] a
solution of the discontinuous system considered satisfies differential
inclusion

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

and we have

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

Consider the simpest physical model with dry friction. Let there is a block on
a table subject to Coulomb friction on the contacting surface, pulled by a
force of gravity accelaration $g$. The differential equation [35] for this
system with the velocity $v$ is

$m\frac{dv(t)}{dt}=-\mu F_{N}\mathrm{Sgn}(v(t))+g$ (33)

where $\mathrm{Sgn}$ is a set-valued function given by

$\mathrm{Sgn}(v)=\left\\{\begin{array}[]{lll}&\\{1\\},\ v>0,\\\ &[-1,1],\
v=0,\\\ &\\{-1\\},\ v<0.\end{array}\right.$

The quantity $F_{N}$ is the normal contact force (equal to $mg$ for a block of
mass $m$) and $\mu$ is the coefficient of Coulomb friction. We see that the
right-hand sides in system (33) are discontinuous on the surfaces $s(v)=v=0$.
Here the approach is applied based on the Filippov works [33] on discontinuos
differential equations theory, hence the system satisfies differential
inclusion

$\dot{v}\in\mu F_{N}[-1,1]+g,$

In a more general case a so called friction cone $FC$ is introduced and the
equations of motion under the Coulomb dry friction law are of the form

$M(q)\frac{dv}{dt}\in k(q,v)+FC(q)$

where $M(q)$ is a mass matrix and $k(q,v)$ is a function obtained from the
kinetic and potential energy when applying Lagrange equations with generalized
coordinates $q$ and velocities $v$. One should also impose holonomic or
nonholonomic (unliteral) constraints as well as some complementary conditions
considering: 1) the case when friction vector is on the boundary of the
friction cone in the case of nonzero relative velocity, 2) physical situation
where sliding stops during the contact period. The details may be found in
[36]. The direct discretization is then applied there and the corresponding
finite dimensional problem is solved via methods of linear algebra.

Note that if from physical meaning the velocity itself is a discontinuous
function of time, strictly speaking there is no sense in taking its derivative
(so the acceleration is formally undefined). In order to overcome this
difficulty, a possible generalization of a derivative as a measure and a
relative concept of Radon-Nikodim derivative is used in some literature [37]
and the corresponding so called measure differential inclusions with such a
derivative. The direct discretization is then applied there in order to get
the corresponding finite dimensional problem. It is interesting to note: since
the derivative there is a measure or a generalized function then in many cases
it would be impossible to work with its discrete analogue; the solution
suggested in [37] is to consider the integral equation instead of the initial
differential one and to “discretize” the function itself instead of its
derivative in the left-hand side (and the integral of the function in the
right-hand side) of the corresponding integral equation. This approach
succeeds, for example, when we consider the Heaviside step-function and its
generalized derivative, the delta-function. However, this paper does not deal
with such cases.

##  6\. The $\delta$-Quasidifferential Descent Method

Describe the quasidifferentiable descent method for finding stationary points
of the functional $I(x,z)$.

Fix an arbitrary initial point $(x_{(1)},z_{(1)})\in C_{n}[0,T]\times
P_{n}[0,T]$. Let the point $(x_{(k)},z_{(k)})\in C_{n}[0,T]\times P_{n}[0,T]$
be already constructed. If for each $t\in[0,T]$ minimum condition (28) is
satisfied (in practice, at discrete time moments $t_{i}$, $i=\overline{1,N}$,
with some fixed accuracy $\overline{\varepsilon}$ in sense of Hausdorff norm
in the space $R^{n}$, with some fixed discretization rank $N$), then
$(x_{(k)},z_{(k)})$ is a stationary point of the functional $I(x,z)$ and the
process terminates. Otherwise, put

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

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

$\min_{\gamma\geq 0}I\Big{(}(x_{(k)},z_{(k)})+\gamma
G(x_{(k)},z_{(k)})\Big{)}=I\Big{(}(x_{(k)},z_{(k)})+\gamma_{(k)}G(x_{(k)},z_{(k)})\Big{)}.$
(34)

In practice, the problem above is solved on the interval
$[0,\overline{\gamma}]$ with some fixed $\overline{\gamma}$ value. Then one
has the inequality

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

As seen from the algorithm described, in order to realize the $k$-th
iteration, one has to solve four subproblems. The first subproblem is to
calculate the quasidifferential of the functional $I(x,z)$ at the point
$(x_{(k)},z_{(k)})$. With the help of quasidifferential calculus rules the
solution of this subproblem is obtained in formulas (20), (21), (22), (26) and
(27).
The second subproblem is to find the quasidifferential descent direction
$G(x_{(k)},z_{(k)})$; the following two paragraphs are devoted to solving this
subproblem.
The third subproblem is one-dimensional minimization (34); there are many
effective methods (see, e. g.[38]) for solving this subproblem.
The fourth subproblem is finding the values $\psi_{i}^{*}(x_{(k)},z_{(k)})$,
$i=\overline{1,n}$ at each time moment $t_{i},i=\overline{1,N}$. Solving this
problem is not straightforward since in general case it is of nonlinear
optimization. However, there also many methods (see, e. g.[38]) of nonlinear
programming; moreover for some particular structures of the sets of the righ-
hand sides of a differential inclusion it can be solved analytically.

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

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(35)

Denote $\overline{v}(t)$, $\overline{w}(t)$ its solution. (The vector-
functions $\overline{v}(t)$, $\overline{w}(t)$, of course, depend on the point
$(x_{(k)},z_{(k)})$ but we omit this dependence in the notation for brevity.)
Then the vector-function
$G(x_{(k)},z_{(k)})=-\big{(}\overline{v}+\overline{w}\big{)}$ is a
quasidifferential descent direction of the functional $I(x,z)$ at the point
$(x_{(k)},z_{(k)})$.

In [20] it is shown that

$\max_{w\in\overline{\partial}I(x_{(k)},z_{(k)})}\min_{v\in\underline{\partial}I(x_{(k)},z_{(k)})}\int_{0}^{T}\big{(}v(t)+w(t)\big{)}^{2}dt=$
$=\int_{0}^{T}\max_{w(t)\in\overline{\partial}\mathcal{I}(\xi_{(k)}(t),t)}\min_{v(t)\in\underline{\partial}\mathcal{I}(\xi_{(k)}(t),t)}\big{(}v(t)+w(t)\big{)}^{2}dt.$
(36)

The equality (36) justifies that in order to solve problem (35) it is
sufficient to solve the problem

$\max_{w(t)\in\overline{\partial}\mathcal{I}(\xi_{(k)}(t),t)}\min_{v(t)\in\underline{\partial}\mathcal{I}(\xi_{(k)}(t),t)}\big{(}v(t)+w(t)\big{)}^{2}$
(37)

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 (37) 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 in our case; 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 $G((x_{(k)},z_{(k)}),t_{i})$ where
$t_{i}\in[0,T]$, $i=\overline{1,N}$, are discretization points (see notation
of Lemma 6.1 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
$L^{2}_{2n}[0,T]$ (as the discretization rank $N$ tends to infinity) to the
vector-function $G(x_{(k)},z_{(k)})$ 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 $\mathcal{I}(x,z)$ at each time moment
of a (uniform) partition of the interval $[0,T]$. In this paragraph we
describe in detail a solution of this subproblem for some fixed value
$t\in[0,T]$. It is known [30] that in our case the subdifferential
$\underline{\partial}\mathcal{I}(x,z,t)$ is a convex polyhedron $A(t)\subset
R^{2n}$ and analogously the superdifferential
$\overline{\partial}\mathcal{I}(x,z,t)$ is a convex polyhedron $B(t)\subset
R^{2n}$. Herewith, of course, the sets $A(t)$ and $B(t)$ depend on the point
$(x,z)$. 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
$b_{j}(t)$, $j=\overline{1,V}$ (here $V$ 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
$b_{j}(t)$, $j=\overline{1,V}$, 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 $||b_{\overline{j}}(t)-a_{\overline{j}}(t)||_{R^{2n}}$,
$\overline{j}\in\\{1,\dots,V\\}$. (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 problem of finding the Euclidean
distance from a point to a convex polyhedron can be effectively solved by
various methods (see, e. g. [39]).

Give a lemma [20] with a rather simple condition which, on the one hand, is
quite natural for applications and, on the other hand, guarantees that the
function $\mathcal{L}(t)$ obtained via piecewise-linear interpolation of the
sought function $\mathcal{G}\in L^{\infty}_{1}[0,T]$ converges to this
function in the space $L^{2}_{1}[0,T]$.

Let the function $\mathcal{G}\in L^{\infty}_{1}[0,T]$ satisfy the following
condition: for every $\overline{\delta}>0$ the function $\mathcal{G}(t)$ 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 $\mathcal{G}(t_{i})$,
$i=\overline{1,N}$, at these points. Let $\mathcal{L}(t)$ be the function
obtained with the help of piecewise linear interpolation with the nodes
$(t_{i},\mathcal{G}(t_{i}))$, $i=\overline{1,N}$. Then for each
$\overline{\varepsilon}>0$ there exists such a number
$\overline{N}(\overline{\varepsilon})$ that for every
$N>\overline{N}(\overline{\varepsilon})$ one has
$||\mathcal{L}-\mathcal{G}||^{2}_{L^{2}_{1}[0,T]}\leq\overline{\varepsilon}$.

Note that when implementing the algorithm using the rules of quasidifferential
calculus, those functions that are active only with some error $\delta$ (or
$\delta$-active) are considered. Let’s introduce the concept of
$\delta$-active function. Fix some value $\delta>0$. We call the function
$f_{\overline{i}}$, $\overline{i}\in\\{1,\dots,M\\}$, a $\delta$-active one at
the point $x_{0}\in R$ if
$\varphi_{1}(x_{0})=\max\limits_{i=\overline{1,M}}f_{i}(x)-f_{\overline{i}}\leq\delta$.
We call the function $g_{\overline{j}}$, $\overline{j}\in\\{1,\dots,K\\}$, a
$\delta$-active one at the point $x_{0}\in R$ if
$\varphi_{2}(x_{0})=\min\limits_{j=\overline{1,K}}g_{j}(x)-g_{\overline{j}}\geq-\delta$.
Taking this error into account justifies the method name, and also is crucial
when proving its convergence in some sense in one special case (see Theorem 6
below).

It is clear that in most practical examples, the subproblems of the algorithm
are also solved approximately. Herewith, the accuracy parameters there depend
on specific methods for solving these problems and are also selected in
advance.

Consider now a particular case when
$c(F_{i}(x),\psi_{i})=c_{1}(F_{i}(x),\psi_{i})$, $i=\overline{1,n}$. Herewith,
denote the functional minimized as $I_{1}(x,z)$.

Put

$\Psi_{1}(x,z)=\min_{||g||_{L_{n}^{2}[0,T]\times
L_{n}^{2}[0,T]}=1}\frac{\partial I_{1}(x,z)}{\partial g}=\frac{\partial
I_{1}(x,z)}{\partial G}.$ (38)

Then necessary minimum condition [40] of the functional $I_{1}(x,z)$ at the
point $(x^{*},z^{*})$ is $\Psi_{1}(x^{*},z^{*})\geq 0$.

Give now some results on the convergence of the quasidifferential descent
method as applied to the functional $I_{1}(x,z)$. For this we have to make a
few additional assumptions.

Fix the initial point $(x_{(1)},z_{(1)})$. Suppose that the set

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

is bounded in $L_{n}^{2}[0,T]\times L_{n}^{2}[0,T]$-norm (due to the
arbitrariness of the initial point, in fact, one must assume that the set
$lev_{I_{1}(x_{(1)},z_{(1)})}I_{1}(\cdot,\cdot)$ is bounded for every initial
point taken).

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

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

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

$||\nabla\hat{I}_{1}(\overline{x},\overline{z})-\nabla\hat{I}_{1}(\overline{\overline{x}},\overline{\overline{z}})||_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}\leq
L||({\overline{x}},{\overline{z}})^{\prime}-(\overline{\overline{x}},\overline{\overline{z}})^{\prime}||_{L^{2}_{n}[0,T]\times
L^{2}_{n}[0,T]}.$ (39)

Under the assumptions made one has the inequality

$\underline{\lim}_{k\rightarrow\infty}\Psi_{2}(x_{(k)},z_{(k)})\geq 0$ (40)

for the sequence built according to the rule above. (See [23].)

In [23] the explanations of the assumptions above with illustrating examples
are made as well.

##  7\. Numerical Examples

Let us return to some examples noted in Section 5. In all the examples
considered the values $\delta=10^{-3}$, $\varepsilon=10^{-2}$ and
$\overline{\gamma}=1$ of the parameters were taken in the
$\delta$-quasidifferential descent method.

Consider the differential inclusion

$\dot{x}_{1}\in[x_{1},x_{2}]\cap[x_{2}-1,x_{2}+1],$ $\dot{x}_{2}=x_{1}+1$

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

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

Consider the more general problem of minimizing the functional

$\mathcal{J}(x)=\int_{0}^{1}|x_{1}(t)|dt.$

One of the obvious solutions is $x^{*}_{1}(t)=0$, $x_{2}^{*}(t)=t$ for all
$t\in[0,1]$.

Such an illustrative example is taken intentionally in order for the cost
functional to be substantially nonsmooth (but only subdifferentiable) on the
optimal trajectory. In order to solve this optimization problem with
restrictions consider the correpsonding unconstrained problem of minimizing
the functional

$J(x,z):=\mathcal{J}(x)+\lambda I(x,z)$

with sufficiently big penalty factor $\lambda$ value. Also note that
$J(x^{*},z^{*})=0$.

We have

$c(F_{1}(x),\psi_{1})=\min\left\\{\frac{x_{1}+x_{2}}{2}\psi_{1}+\frac{|x_{1}-x_{2}|}{2},\,x_{2}\psi_{1}+1\right\\},\quad
c(F_{2}(x),\psi_{2})=\psi_{2}(x_{1}+1)$

for $\psi_{1},\psi_{2}\in S_{1}$.

Consider the support function $c(F_{1}(x),\psi_{1})$ for $\psi_{1}\in S_{1}$
in details. As proved in Section 4, when both functions are active, this
function is superdifferentiable provided that $x_{1}\neq x_{2}$. But it is
interesting to note that in the case $x_{1}=x_{2}$ the first function is the
only one to be active, and since abs-function can be represented as the max-
function of two continuous differentiable functions (the function under abs
itself and the opposite one), then as again proved in Section 4 this function
is subdifferentiable in the case $x_{1}=x_{2}$.

Take $\lambda^{*}=10$ and $(x_{(1)},z_{(1)})=(-1,-2,0,0)^{\prime}$ as the
first approximation. We intentionally take such a point in order to make both
functions under minimum in the first support function to be active (see the
formula above and note that, as one can check,
$\psi_{1}^{*}(x_{(1)},z_{(1)})=1$), hence the support function would be
substantionally nonsmooth at this point. At the end of the process the
discretization step was equal to $10^{-1}$. Figure 1 illustrates the
trajectories obtained. From Figure 1 we see that the differential inclusion is
practically satisfied (we see that the resulting curve $\dot{x}_{1}(t)$
practically coincides with the “lower” boundary $x_{1}(t)$ of the set
$F_{1}(x(t))$ obtained and the resulting curve $\dot{x}_{2}(t)$ practically
coincides with the curve $x_{1}(t)+1$ (which is formally the set $F_{2}(x(t))$
obtained)). The boundary values error doesn’t exceed the magnitude $5\times
10^{-3}$. To obtain such an accuracy $37$ iterations have been required. The
functional value on the trajectory obtained is of order $10^{-3}$.

Figure 1. Solution of Example 7.1

Consider the differential inclusion

$\dot{x}\in F(x),\quad F(x)=B_{2}\cap E_{2}(x),\quad t\in[0,1],$

where $E_{2}(x)$ is an ellipse depending on the phase coordinate $x_{1}$:

$\displaystyle{E_{2}(x)=\Big{\\{}(\overline{x},\overline{y})\in R^{2}\
\big{|}\ \frac{(\overline{x}-2)^{2}}{x_{1}^{2}+3}+\overline{y}^{2}\leq
1\Big{\\}}}.$

The boundary conditions are

$x(0)=(0,0)^{\prime},\quad(0.75,0.5)^{\prime}.$

We have

$c(B_{2},\psi)=\sqrt{\psi_{1}^{2}+\psi_{2}^{2}}=1,\quad
c(E_{2}(x),\psi)=\sqrt{(x_{1}^{2}+3)\psi_{1}^{2}+\psi_{2}^{2}}+2\psi_{1}.$

Note that here we use the functional $\widetilde{I}(x,z)$ (see Remark 5).

Take an initial point $x_{(1)}=(t,t)^{\prime}$ and $z_{(1)}=(1,1)^{\prime}$.
At the end of the process the discretization step was equal to $10^{-1}$.
Figure 2 illustrated the trajectories obtained. In Figure 2 the points
$z^{*}(t)$ and the allowed set $B_{2}\cap{E}_{2}(x(t))$ of these points
location are depicted at some $t$-values from the segment $[0,1]$, it is seen
that the differential inclusion considered is satisfied for these values
$t\in[0,1]$ (it is easy to chek that it is correct for all the others time
moments as well $t\in[0,1]$). The boundary values error doesn’t exceed the
magnitude $5\times 10^{-3}$. To obtain such an accuracy $15$ iterations have
been required. The functional $I(x^{*},z^{*})$ value on the trajectory
obtained is of order $5\times 10^{-3}$.

Note that as can be easily checked the distances from this point to the sets
considered is not equal for each time moment $t\in[0,1]$, hence the
requirement of the vector $\psi^{*}$ uniqueness (in order for the Theorem 4 to
be valid) is fullfilled here (see Remark 5). The same is true for the point
$z_{(k)}$, $k=1,2,\dots$ on all the other iterations of the algorithm
implemented.

Figure 2. Solution of Example 7.2

The test problem used here is a modified problem from [41] with a system of
three masses connected by springs with a forcing term and friction as shown in
Figure 3. This object combines the problems of two types: modeling dry
friction generates a discontinuous system moving in a sliding mode, thus can
be considered as a differential inclusion; the sliding mode on the surface
$s(x)=x_{4}=0$ is provided by an additional nonsmooth control function
$u_{1}(x)=-a_{1}|x|\mathrm{sgn}(s(x))$ where the gain factor $a_{1}=1.05$,
giving a support function of the set in the right-hand side of this inclusion
in the form discussed in the paper. The aim of external forcing term $u(t)$ is
to bring the system (moving in a sliding mode) to the point prescribed at the
final moment of time.

The actual differential equations to be solved are

$\ddot{x}_{1}=(-x_{1})+(x_{2}-x_{1})-\dot{x}_{1}-1.05|x|\,\mathrm{sgn}(\dot{x}_{1}),$
$\ddot{x}_{2}=(x_{1}-x_{2})+(x_{3}-x_{2})-\dot{x}_{2}-0.3\,\mathrm{sgn}(\dot{x}_{2}),$
$\ddot{x}_{3}=(x_{2}-x_{3})-\dot{x}_{3}-0.3\,\mathrm{sgn}(\dot{x}_{3})+u,$

or, in a normal form

$\dot{x}_{1}=x_{4},\ \dot{x}_{2}=x_{5},\ \dot{x}_{3}=x_{6}$
$\dot{x}_{4}=(-x_{1})+(x_{2}-x_{1})-x_{4}-1.05|x|\,\mathrm{sgn}(x_{4}),$
$\dot{x}_{5}=(x_{1}-x_{2})+(x_{3}-x_{2})-x_{5}-0.3\,\mathrm{sgn}(x_{5}),$
$\dot{x}_{6}=(x_{2}-x_{3})-x_{6}-0.3\,\mathrm{sgn}(x_{6})+u,$

with the initial conditions

$x_{1}(0)=-1,\ x_{2}(0)=1,\ x_{3}(0)=-1,$ $x_{4}(0)=-1,\ x_{5}(0)=1,\
x_{6}(0)=1.$

Figure 3. Model of Example 7.3

Due to Filippov definition [33] the corresponding differential inclusion on
the discontinuity surface $x_{4}=0$ is

$\dot{x}_{1}=x_{4},\ \dot{x}_{2}=x_{5},\ \dot{x}_{3}=x_{6}$
$\dot{x}_{4}\in(-x_{1})+(x_{2}-x_{1})-x_{4}+1.05|x|\,[-1,1]$
$\dot{x}_{5}=(x_{1}-x_{2})+(x_{3}-x_{2})-x_{5}-0.3\,\mathrm{sgn}(x_{5}),$
$\dot{x}_{6}=(x_{2}-x_{3})-x_{6}-0.3\,\mathrm{sgn}(x_{6})+u,$

with the same boundary conditions.

Suppose that $t^{*}=-0.2$ and that on the interval $[-0.2,0]$ no external
force is applied, therefore $u(t)=0$, $t\in[-0.2,0]$. Such a time interval is
enough to bring the system to the vicinity of the surface $x_{4}(t)=0$, so we
approximately put $x_{4}(0)=0$; herewith the other “initial” conditions
obtained from the movement during the time interval $[-t^{*},0]$ are as
follows $x_{5}(0)\approx 0.026$, $x_{6}(0)\approx 1.119$,
$x_{1}(0)\approx-1.053$, $x_{2}(0)\approx 1.099$, $x_{3}(0)\approx-0.787$.
Take these values as the initial ones for the differential inclusion and apply
the paper method in order to find the control $u(t)$ and the correpsonding
solution of differential inclusion above on the next time interval $[0,1]$.
The aim of the control $u(t)$ is to bring the third block to the origin at the
final time moment $T=1$, i. e. we impose the restriction $x_{3}(1)=0$ on the
right endpoint. The “discontinuous” surface $x_{4}=0$ of course remains the
same.

Note that in this example the control $u(t)$ is the function sought as well.
The functional structure (put
$h_{6}(x,z,u)=z_{6}-((x_{2}-x_{3})-x_{6}-0.3\,\mathrm{sgn}(x_{6})+u)$) and the
method can be easily modified in a obvious way in order to seek for this
variable as well. We also naturally simpify the functions $h_{i}(x,z)$ and
write them down without the $\psi_{i}$-variable, $i=1,2,3,5$, since they
respond to exact differential equations, not inclusions. Take
$(x_{(1)},z_{(1)},u_{(1)})=(0,0,0,0,0,0,0,0,0,0,0,0,0)^{\prime}$ as the first
approximation. In practice we can reduce by $6$ the number of variables (the
coordinates $x_{i}$ and their derivatives $z_{i}$, $i=\overline{1,3}$), since
the first three equations are obviously resolved. At the end of the process
the discretization step was equal to $10^{-1}$. Figure 4 illustrates the
trajectories obtained. From Figure 4 we see that the relation with the
differential inclusion is approximately satisfied. The trajectory practically
lies on the surface required as well; therefore the variable $x_{1}(t)$
remains approximately equal to $-1.053$ for all $t\in[0,1]$. The boundary
values error doesn’t exceed the magnitude $5\times 10^{-3}$. To obtain such an
accuracy $113$ iterations have been required. The functional value on the
trajectory obtained is approximately $10^{-3}$. The control obtained is
$u_{(113)}=0.914t^{2}-4.05t+4.45t^{3}$ (we give an approximation via an
interpolation polynomial for brevity).

Note that the right-hand side is formally nondifferentiable in phase
coordinates function, however it is differentiable if $y_{1}\neq 0$ and
$y_{2}\neq 0$. While algorithm realization $y_{1}$ and $y_{2}$ vanishes only
at finite number of isolated time moments; hence one can split the time
interval in the segments where $y_{1}$ and $y_{2}$ remains its signum and
Gateaux differentiability of the functional in these variables remains valid.

Figure 4. Solution of Example 7.3

The detailed discussion on crucial role of an idea to consider $z$ and
$\dot{x}$ as “independent” variables in nonsmooth problems of control and
variational calculus as well as its general advantages and disadvantages with
illustrating examples is given in [20].

## References

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