1. Introduction
We consider an optimal control problem for the multi-term fractional equation
Here
,
,
are Banach spaces,
is a continuous operator, a linear closed operator
M with a dense domain in
acts into
, linear and continuous operator
B acts on control function
u from
into
. Operators
are linear and continuous for every
,
. We mean the Gerasimov–Caputo derivatives under notations
and
with
Equations, which are not solved with respect to the highest order derivative with respect to time, are often called Sobolev type equations [
1,
2]. Moreover, if (
1) contains the operator
L with a nontrivial kernel
, it often called degenerate evolution equation or degenerate equation [
3]. Here we shall consider this case.
Equations of form (
1) frequently encountered in applications (see references below). The natural initial conditions for degenerate evolution Equation (
1) are
where
P is the projector along the degeneration space of the equation. We require that control functions have to belong the admissible controls set
where
is a nonempty closed convex set of a control functions space
. The cost functional has the form
where
,
,
and
are given functions. We are going to establish solvability conditions of problem (
1)–(
4).
In recent decades, fractional integro-differential calculus has become one of the most important tools for solving mathematical modeling problems [
4,
5,
6,
7,
8]. On the other hand various issues of the control theory, including unique solvability, are of interest to many authors. However, not many papers deal with control problems for fractional differential equations, see [
9,
10,
11,
12] and references therein. The present paper is a continuation of the authors’ works on optimal control problems for the equations with a degenerate operator at the highest-order time-fractional derivative [
13,
14,
15,
16,
17,
18,
19].
In the second section we give the definition of the Gerasimov–Caputo fractional derivative and a result about the existence of a unique strong solution of the Cauchy problem for a semilinear equation which is solved with respect to the highest-order fractional derivative. The third section contains the proof of the unique solvability in the sense of the strong solutions for a class of initial problems of form (
1) and (
2). Here we used the theory of the degenerate evolution equations (see works [
2,
20,
21]). In the fourth section the result on the existence of a unique strong solution for problem (
1), (
2) is applied to the proof of the optimal control existence for (
1)–(
4). The last section of the paper illustrates the obtained abstract results on an example of an initial-boundary value problem for a partial differential equation.
2. Solvability of Nondegenerate Semilinear Equation
Let
be a Banach space. Denote
,
,
for
,
. Let
,
be the usual derivative of the order
,
be the identical operator. The Gerasimov–Caputo derivative of a function
h is (see [
22] (p. 11))
Consider the Cauchy problem
for the nonlinear differential equation
where
, i.e., linear and continuous operator from
to
,
,
, a nonlinear operator
is a Caratheodory mapping, i.e., for arbitrary
it defines a measurable mapping on
and for almost all
it is continuous in
.
A strong solution of problem (
5), (
6) is a function
, such that
, equalities (
5) and (
6) for almost all
are true. Here we use some
.
Denote as a set of n elements. We shall say that operator is uniformly Lipschitz continuous in , if there exists , such that the inequality is true for almost all and for all .
Lemma 1 ([
20]).
Let , . Then Theorem 1 ([
23]).
Suppose that , , , is Caratheodory mapping, which is uniformly Lipschitz continuous in , at all and almost everywhere on inequalityis valid for some , Then problem(
5), (
6)
has a unique strong solution on . 3. Degenerate Multi-Term Linear Equation
Let and be Banach spaces. As we denote the space of all linear continuous operators, which act from the space to . Denote by the set of all linear closed operators with a dense domain in and with an image in Suppose that , , denote by the domain of M, endowed by the graph norm .
Define L-resolvent set of an operator M and its L-spectrum , and denote ,
An operator
M is called
-bounded, if
Under the condition of
-boundedness of operator
M we have the projections
where
(see [
2] (pp. 89–90)). Put
,
,
,
. Denote by
(
) the restriction of the operator
L (
M) on
(
),
.
Theorem 2 ([
2] (pp. 90–91)).
Let an operator M be -bounded. Then- (i)
, , ,
- (ii)
there exist operators , .
Denote , . For operator M is called -bounded, if it is -bounded, , .
Let
,
,
,
. Consider the degenerate multi-term linear equation
Fix a constant
. Strong solution on
of this equation is a function
, such that
and almost everywhere on
equality (
8) holds.
A solution of the generalized Showalter-Sidorov problem
to Equation (
8) is a solution of the equation, such that conditions (
9) are true. Note here that
, hence the smoothness of
is not less the smoothness of
, since
due to Theorem 2.
Lemma 2 ([
19]).
Let be a nilpotent operator of a power , a function , and for . Then the equation has a unique strong solution. Moreover, it has the form Theorem 3. Let , , an operator M be -bounded, mappings be measurable, essentially bounded on , for almost all , , , for all there exist , . Then problem (
8), (
9)
has a unique strong solution. Proof. The mapping
acts from
into the space
according to the theorem conditions. By the fact
we have
,
,
. Equation (
8) after the action of the operator
has the form
where
. Since the operator
G is nilpotent and due to Lemma 2, the unique solution of this equation has the form
Note that
,
, and
The next step is to prove the unique strong solution existence of the Cauchy problem
where
,
due to Theorem 2. This problem is obtained from (
8), (
9) after the action of the continuous operator
. Here the operator
satisfies the conditions of Theorem 1. Indeed, let
then the operator
B is uniformly Lipschitz continuous in
with the constant
and it satisfies inequality (
7) with
,
due to Lemma 1. Thus, by Theorem 1 we obtain the required. A unique solution of problem (
8), (
9) has the form
. □
4. Optimal Control Problem
Let
,
,
. Consider an optimal control problem for a degenerate multi-term linear equation
where
is a subset of
of admissible controls,
J is the cost functional,
, such that
,
are given,
.
Introduce the spaces at
Lemma 3 ([
13,
14]).
and are Banach spaces with the norms , respectively. Introduce the continuous operator ,
Admissible pairs set
of problem (
10)–(
13) is a set of such pairs
, that
,
is a strong solution of (
10), (
11). To solve problem (
10)–(
13) means to find the set of pairs
, which minimize the cost functional, i.e.,
Theorem 4. Let , , an operator M be -bounded, mappings be measurable, essentially bounded on , for almost all , . Assume that is a non-empty closed convex subset in , there exists such , that , at . Then problem (
10)–(
13)
has a unique solution . Proof. By Theorem 3 the set
of admissible pairs is nonempty. We use Theorem 2.4 from the monograph [
24] for the proof of an optimal control existence. Take spaces
,
,
,
, the operator
and the vector
of the form
The continuity of the linear operator
from
to
follows from the inequalities
Here we applied Lemma 1.
For a pair
we have
if
. Thus, functional (
13) is coercive. □
5. Example
Consider a control problem for the model equation
,
,
,
. In order to reduce problem (
14)–(
16) to (
8), (
9) we choose spaces and operators:
,
,
,
,
,
,
,
.
The admissible controls set and the cost functional are given by the following
where
,
,
,
.
Theorem 5. Let , at some , , , are measurable and essentially bounded, . Then there exists a unique solution of problem(
14)–(
18)
on Proof. We have
, and Equation (
14) is degenerate. The operator
M is
-bounded, since for sufficiently large
we have
and
This implies that
and we can reduce problem (
9) to the problem
,
, of form (
16). Note that conditions (
19) mean that
. Moreover,
, since
,
.
We can take
in the conditions of Theorem 4. So by that theorem problem (
14)–(
18) has a unique solution. □
6. Conclusions
We studied the unique solvability of initial value problems for a class of degenerate evolution fractional multi-term equations. The obtained results are applied to study of some optimal control problems for systems, which state is described by such initial value problem. Abstract results can be used for investigation of optimal control problems for multi-term time-fractional partial differential equations, it is illustrated on an example. The results of the work in future will be extended to problems with start control and with mixed control to degenerate evolution fractional multi-term equations, to stochastic degenerate fractional evolution equations with white noise, and some others.
Author Contributions
Conceptualization, M.P.; methodology, M.P.; validation, G.B.; formal analysis, G.B.; investigation, G.B.; writing–original draft preparation, G.B.; writing–review and editing, M.P.; supervision, M.P.; project administration, M.P. All authors have read and agreed to the published version of the manuscript.
Funding
The work is supported by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011.
Conflicts of Interest
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
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