Optimal feedback control for fractional evolution equations with nonlinear perturbation of the time-fractional derivative term

We study the optimal feedback control for fractional evolution equations with a nonlinear perturbation of the time-fractional derivative term involving Caputo fractional derivatives with arbitrary kernels. Firstly, we derive a mild solution in terms of the semigroup operator generated by resolvents and a kernel from the general Caputo fractional operators and establish the existence and uniqueness of mild solutions for the feedback control systems. Then, the existence of feasible pairs by applying Filippov’s theorem is obtained. In addition, the existence of optimal control pairs for the Lagrange problem has been investigated.


Introduction
Control theory has received considerable attention due to its extensive applications in various areas of science, e.g., ecology, economics, and engineering, particularly in systems with controllability, feedback control, and optimal control [1][2][3][4][5]. Control systems are most often based on the principle of feedback, whereby the signal to be controlled is compared to a desired reference signal and the discrepancy is used to compute a corrective control action.
It is wonderful that the study of fractional control systems has attracted research recently [6][7][8][9][10][11][12]. In [7], Wang et al. considered the optimal feedback control of a nonlinear system, given by fractional evolution equations, that has the form where C D α is Caputo fractional derivative of order α ∈ (0, 1), u 0 ∈ E, and A : D(A ) → E is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators {T(t)} t≥0 in a reflexive Banach space E. The control function v(·) takes values in the Polish space V and f : [0, T] × E × V → E is a given function satisfying suitable assumptions.
Motivated by the previous work, we are concerned with the optimal feedback control of the semilinear fractional evolution equations with a nonlinear perturbation of the timefractional derivative term as follows: ⎧ ⎨ ⎩ C D α;ω 0 (u(t)g(t, u(t))) = A u(t) + f (t, u(t), v(t)), 0 < t ≤ T, u(0) = u 0 , (1) where C D α;ω 0 is the Caputo fractional derivative with arbitrary kernel ω of order α ∈ (0, 1), A : D(A ) ⊆ E → E is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators {T(t), t ≥ 0} in a reflexive Banach space E, and u 0 ∈ E. The control v has a value in a control set V[0, T] and f : [0, T] × E × V → E and g : [0, T] × E → E will be specified in what follows. It should be noted that the nonlinear perturbation term g in (1) contributes to a more complicated derivation of a mild solution, which requires certain assumptions on the semigroup and operator A . Furthermore, when the evolution operator A is defined to be the zero operator on the Banach space E = R, our problem (1) can be modified and rewritten as hybrid fractional differential equations. The fractional derivative of an unknown function is hybrid nonlinear, as a dependent variable is used in this class of equations. Moreover, this problem can be reduced to that considered in [7] where the function g is taken to be zero.
The aim of this paper is to derive a representation of the solution for the problem (1) that depends on fractional derivatives with arbitrary kernels. Furthermore, Krasnoselskii's fixed point theorem is used to investigate the existence results for the nonlinear system (1) under the compactness assumption of the operator semigroup {T(t)} t≥0 . We further investigate the existence of optimal feedback controls for the Lagrange problem. Moreover, our results obtained in this work can be applied for further investigation in many practical problems.
The paper is structured as follows. First, we will outline some definitions and lemmas that will be needed later in Sect. 2. In Sect. 3, we provide a mild solution to the nonlinear system (1) employing the semigroup operator with a function ω that prescribes the generalized Caputo derivative. Next, the Krasnoselskii's fixed point theorem is applied to prove the existence and uniqueness results of mild solutions for the problem (1) in Sect. 4. In Sect. 5, the existence of feasible pairs for the system (1) is also demonstrated. Finally, we will investigate the existence result of the optimal control pairs of the system (1).

Preliminaries
Throughout this paper, E is a reflexive Banach space and f L p is used to denote the Consider C([0, T], E) as the Banach space of continuous functions from [0, T] to E with the usual supremum norm.
We denote by V a Polish space; that is, a separable completely metrizable topological Suppose H and F are two metric spaces.
If is pseudocontinuous at each point t ∈ H, then it is called pseudocontinuous on H.
where is the gamma function.
Definition 2.8 (ω-Caputo fractional derivative, [15,16] The ω-Caputo fractional derivative of a function u of order α is defined by where Furthermore, we also have where E α is the Mittag-Leffler function. Definition 2.11 ([15]) Let u, ω : [a, ∞) → R and ω(t) be a nonnegative increasing function. Then, the Laplace transform of u with respect to ω is given by for all s such that this integral converges.

Theorem 2.17 (Krasnoselskii's fixed point theorem) Let B is a nonempty convex, closed, and bounded subset of a Banach space E. Assume that F 1 and F 2 are operators from B to E such that
Now, we outline some facts about the semigroups of linear operators which can be found in [21,22].
The infinitesimal generator of {T(t)} t≥0 of a strongly continuous semigroup (i.e., C 0semigroup) {T(t)} t≥0 is given by We denote the domain of A by D(A ), that is,

Lemma 2.18 ([21, 22]) Let {T(t)} t≥0 be a C 0 -semigroup and let A be its infinitesimal generator. Then
Throughout this work, we assume that the analytic semigroup {T(t)} t≥0 has the following properties: (i) There is a constant M ≥ 1 satisfying (ii) For any 0 < η ≤ 1, there exists a positive constant C η such that

Representation formula of mild solutions based on semigroup theory
Lemma 3.1 Any solution of the problem (1) satisfies the following integral equation: Proof Applying Definition 2.8 and Lemma 2.9 to the problem (1), it can be rewritten in the form of the integral representation as follows: Taking the generalized Laplace transform on both sides of equation (7), we have that for s > 0, It follows that Now, we consider the change of variable It follows that The following one-sided stable probability density in [23] is considered: Using (8), we obtain Then, we have Now, we take the inverse Laplace transform to obtain where φ α (θ ) = 1 α θ -1-1 α ρ α (θ -1 α ) is the probability density function defined on (0, ∞). (1) if satisfies the following integral equation:

Definition 3.2 A function u ∈ C([0, T], E) is called a mild solution of the problem
where the operators Q α;ω (t, τ ) and R α;ω (t, τ ) are defined by

Existence and uniqueness of a mild solution
In order to demonstrate the main results, we outline the following assumptions: The function f is a locally Lipschitz continuous with respect to V, i.e., for all t ∈ [0, T] and u 1 , The following existence of mild solutions for the problem (1) will be proved by using Krasnoselskii's fixed point theorem.
Step 1: We assume that for each k > 0, there exist u k , w k ∈ B k such that

According to (A 3 ) and Lemma 3.3(i), it follows that
Multiplying to both sides by 1 k and taking the limit inferior as k → ∞, we get which is contradiction.
Step 2: F 1 is a contraction on B k . For arbitrary u, w ∈ B k , we have According to (11) of Theorem 4.1, we obtain that F 1 is a contraction.
Step 3: F 2 is a completely continuous operator. Firstly, we claim that F 2 is continuous on B k . Let {u n } ⊂ B k be such that u n → u ∈ B k as n → ∞. For t ∈ [0, T], by Assumptions (A 2 ) and (A 3 ), we have Using the Lebesgue dominated convergence theorem, for any t ∈ [0, T], we obtain as n → ∞. This implies that (F 2 u n )(t) -(F 2 u)(t) C → 0 as n → ∞. Hence F 2 is continuous. Next, we prove the equicontinuity of F 2 (B k ). For any u ∈ B k , we have for 0 ≤ t 1 < t 2 ≤ T, , v(τ ) dτ =: I 1 + I 2 + I 3 .
By Lemma 3.3, we obtain that and hence I 1 → 0 and I 2 → 0 as t 2 → t 1 . For t 1 = 0 and 0 < t 2 ≤ T, it easy to see that I 4 = 0.
Thus, for any ε ∈ (0, t 1 ), we have Therefore I 3 → 0 as t 2 → t 1 and ε → 0 by Lemma 3.3, (iii) and (iv). It follows that Fix t ∈ (0, T], then, for every ε > 0 and δ > 0, we define an operator F ε,δ 2 on B k as By the compactness of T(ε α δ) for ε α δ > 0, it follows that the set N ε,δ (t) = {(F ε,δ 2 u)(t) : u ∈ B k } is relatively compact in E for all ε > 0 and δ > 0. Furthermore, for any u ∈ B k , we Proof For u ∈ B k , we define the operator G on B k by (Gu)(t) = Q α;ω (t, 0) u 0g(0, u 0 ) + g t, u(t) Notice that it is enough to show the uniqueness of a fixed point of G on B k . According to (10), we know that G is an operator from B k into itself. For any u, u * ∈ B k and t ∈ [0, T], according to (A 3 )-(A 5 ), we have This implies that G is a contraction map satisfying (12). Hence the uniqueness of a fixed point of the map G on B k follows from the Banach contraction principle.
In this section, we present the existence of feasible pairs for system (1). To establish our results, we introduce the following hypotheses: We can verify that, for any t ∈ [0, T] and 1 p < α < 1, E j n (t) is bounded. By Lemma 3.3, it is not difficult to verify that E j n (t) is compact in E and also equicontinuous. Due to Ascoli-Arzela theorem, {E j n (t)} is relatively compact in C([0, T], E). Obviously, Q j is a continuous linear operator. Therefore, Q j is a compact operator for j = 1, 2.
We need to investigate the following result in order to solve our optimal feedback control problem.

Existence of optimal feedback control pairs
In this section, we consider the Lagrange problem (P) for the optimal feedback control as follows: find a pair (u 0 , v 0 ) ∈ H[0, T] such that For any (t, u) ∈ [0, T] × E, we denote the set To investigate the existence of optimal control pairs for problem (P), we assume that (C) The map Σ(t, ·) : E → 2 R×E has the Cesari property for a.e. t ∈ [0, T], that is, = Q α;ω (t, 0) u 0g(0, u 0 ) + g t, u(t) For any δ > 0 and sufficiently large l, we have φ l (t), φ 0 l (t) ∈ Σ t, O δ u(t) .