Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems

: A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to the corresponding equation. Obtained results are used to an optimal control problem study for a model system which is described by an initial-boundary value problem for a partial differential equation. results on an example of an initial-boundary value problem for a partial differential equation.


Introduction
We consider an optimal control problem for the multi-term fractional equation Here X , Y, U are Banach spaces, L : X → Y is a continuous operator, a linear closed operator M with a dense domain in X acts into Y, linear and continuous operator B acts on control function u from U into Y. Operators N k (t) are linear and continuous for every t ∈ (t 0 , T), k = 1, 2, . . . , n. We mean the Gerasimov-Caputo derivatives under notations D α t and D α k t with 0 ≤ α 1 < α 2 < · · · < α n ≤ m − 1 < α ≤ m ∈ N. 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 kerL = {0}, 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 (Px) (k) (t 0 ) = x k , k = 0, 1, . . . , m − 1, 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 U ∂ is a nonempty closed convex set of a control functions space L q (t 0 , T; U ). The cost functional has the form where q > 1, δ > 0, x d and u d 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.
Denote as x = (x 1 , x 2 , . . . , x n ) a set of n elements. We shall say that operator B : (t 0 , T) × Z n → Z is uniformly Lipschitz continuous in x ∈ Z n , if there exists l > 0, such that the inequality B(t, x) − B(t, y) Z ≤ l n ∑ k=1 x k − y k Z is true for almost all t ∈ (t 0 , T) and for all x, y ∈ Z n .
Caratheodory mapping, which is uniformly Lipschitz continuous in x, at all y 1 , y 2 , . . . , y n ∈ Z and almost everywhere on (t 0 , T) inequality is valid for some a ∈ L q (t 0 , T; R), c > 0. Then problem (5), (6) has a unique strong solution on (t 0 , T).

Degenerate Multi-Term Linear Equation
Let X and Y be Banach spaces. As L(X ; Y ) we denote the space of all linear continuous operators, which act from the space X to Y. Denote by Cl(X ; Y ) the set of all linear closed operators with a dense domain in X and with an image in Y.
Proof. The mapping acts from C m−1 ([t 0 , T]; X ) into the space L q (t 0 , T; Y ) according to the theorem conditions. By the fact imN k ⊂ Y 1 we have (I − Q)N k ≡ 0, QN k ≡ N k , k = 1, 2, . . . , n. Equation (8) where w(t) = (I − P)x(t). Since the operator G is nilpotent and due to Lemma 2, the unique solution of this equation has the form Note that w ∈ C m−1 ([t 0 , T]; X ), D α t w ∈ L q (t 0 , T; X ), and The next step is to prove the unique strong solution existence of the Cauchy problem where v(t) = Px(t), S 1 = L −1 1 M 1 ∈ L(X 1 ) due to Theorem 2. This problem is obtained from (8), (9) after the action of the continuous operator L −1 1 Q. Here the operator L(Y;X ) and it satisfies inequality (7) with c = l, due to Lemma 1. Thus, by Theorem 1 we obtain the required. A unique solution of problem (8), (9) has the form x(t) = v(t) + w(t).
We can take u 0 = 0 in the conditions of Theorem 4. So by that theorem problem (14)-(18) has a unique solution.

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.

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.