1. Dependence of the Solution to the Integral Inclusion from Perturbation
Let
be the n-dimensional Euclidean space. The set of all nonempty compact (convex compact) subsets in
we will designate as
;
is the continuous matrix function, wherewith
being the set of all square
matrices of real elements (
);
the continuous function;
the setvalued mapping.
Assume that if a vector is multiplied by a matrix, then the vector is a row vector, if a matrix is multiplied by a vector, then the vector is a column vector.
Let us consider a problem for inclusion
(1)
The function
satisfying (1) we will call the solution to problem (1) (see [1] ).
Let
, if
is the continuous matrix function.
Theorem 1. Let
be the continuous matrix function,
the continuous function,
the multivalued mapping,
is measurable on t, and there exists a summable function
such that
for
. Moreover, let
and
be such that
for
. Then there exists such a solution
to problem (1) that
for
, where
.
2. On Subdifferential of the Integral Functional
Let
is the normal convex integrant (see [2] ).
Let consider a subdifferential of the integral functional
in
.
Theorem 2. If
be the continuous matrix function,
the continuous function,
is the
normal convex integrant and function
is
summable for
,
, where
, then
is nonempty and
belongs to
if and only if, there exist
,
, such, that
, where
is the transpose of the matrix
.
Theorem 3. If
be measurable bounded matrix function,
be measurable bounded function,
is
the normal convex integrant and function
is
summable for
,
, where
, then
is nonempty and functional
belongs to
if and only if,
there exist
,
, such, that
.
3. On Subdifferential of the Terminal Functional
Let
continuous matrix function,
continuous function,
proper convex function in
. Consider a subdifferential of the terminal functional
in
, where
.
Theorem 4. If
-proper convex function in
and continuous in the
point
, then
4. Convex Extremal Problem for Integral Inclusions
Let
be the continuous matrix function,
the continuous function. Hereafter we will assume that
is the normal convex integrant,
the convex function. Let
,
is the multivalued mapping.
The problem of minimization of the functional
(2)
is considered under the following constraints
(3)
where
,
.
Introducing the notation
we have that problem (2) and (3) is equivalent to the minimization of the functional
among all functions
.
Let the mapping
be measurable on
, the set
be closed and convex for almost all
and
be compact for all
. From here it follows that
is a convex normal integrant on
.
Let us consider the following functional
where
. Let
. The problem (2) and (3) is called stable, if
is finite and function h is subdifferentiable at zero (see [3] ).
Lemma 1. Let
; the mapping
be measurable on
; the mapping
be closed and convex for almost all
, i.e.
be closed and convex for almost all
; there exist such a summable function
that
for
; there exist a solution
to the problem
such that
belongs to
coupled with some
tube, i.e.
;
the normal convex integrant;
the convex function and
is finite; the function
be summarized for
,
, where
, and function
be continuous at the point
. Then the function h is subdifferentiable at zero, i.e. problem (2) and (3) is stable.
Let
. Assume
,
where
.
Theorem 5. Let
; the mapping
be measurable on
; the mapping
be closed and convex for almost all
; f be the normal convex integrant on
;
the convex function on
;
the continuous matrix function;
the continuous function. For the function
to minimize the functional (2) among all the solutions to the problem (3), it is sufficient that there exist
,
and
such that
1)
,
2)
,
3)
4)
and if for
the condition of lemma 1 is satisfied, then conditions 1) - 4) become necessary.
5. Nonconvex Extremal Problem for Integral Inclusions
Let
be the continuous matrix function;
the continuous function, i.e.
. Hereafter we will assume that
is the normal integrant and
is the function. Let
,
be the multivalued mapping.
We consider the following problem of minimization of the functional
(4)
under the following constraints
(5)
where
,
.
Let
and consider the minimization of the functional
among all the functions
.
Theorem 6. If
is the solution to the problem (4) and (5),
and
are measurable on t,
, there exist
,
,
,
and
such that
at
and
for
,
,
. Then there exist a number
such that
minimizes the functional
in D for
, where
,
,
.
Theorem 7. Let the condition of the theorem 6 be satisfied and the function
among all solutions to the problem (5) minimizes the functional (4). Then there exists
and
such that
1)
2)
where
is Clarke subdifferential of the function g at the point
(see [4] ).
6. A Higher Order Necessary Condition in the Extremal Problem for the Volterra Type Inclusion
Consider the problem (4) and (5), where
. Assume
We consider the following problem of minimization of the function
among all functions
.
Let
be the solution to the problem (4) and (5). Let
.
Theorem 8. Let
be the multivalued mapping, the mapping
be measurable on
; f be the normal integrant on
;
the function in
;
the continuous matrix function;
the continuous function, and there exists a summable function
such that
for
; there exist
,
and number
such that
for
,
for
(see [5] ). If the function
among all solutions of the problem (5) minimizes the functional (4), then there exists a number
such that
minimizes the functional
in
for
.
Let
be the normal integrant on
;
the function.
Let assume
Theorem 9. Let
the multivalued mapping, the mapping
be measurable on
; f be the normal integrant on
;
the function in
;
the continuous matrix function;
the continuous function and there exist a summable function
such that
for
; there exist the normal integrant
, the functions
,
,
and number
such that
for
, where
,
for
and let
solutions of the problem (4)-(5). Then there exist a number
such that
minimizes the functional
in
for
, where
solutions to the problem (5), which satisfy the main theorem 1 for
.
It’s possible to get the local variant of theorems 8 and 9 analogical to theorem 6.
Let assume
for
.
Corollary 1. If the condition of theorem 8 is satisfied, then there exist a number
such that
for
and
.
Let assume
Theorem 10. If the condition of theorem 8 is satisfied, then there exist a number
such that
for
and
.