Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of order q ∈ ]0, 1[. +e result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result.


Introduction
Time-fractional systems have been proved, with the development of science and technology, to be one of the most effective tools in modeling many phenomena arising in physics, engineering, and real world problems [1][2][3][4][5][6].
erefore, the research studies of fractional-order calculus attract lots of attention for these kinds of systems with several fractional derivatives (for more details, see [7][8][9][10] and the references therein). Zhou and Jiao [11] introduced a concept of a mild solution based on Laplace transform and probability density functions; several authors presented a tremendous amount of valuable results on controllability and observability, stability analysis, and so on [12][13][14][15].
Similar to the integer-order control systems [16][17][18][19][20][21], the regional controllability problem of fractional systems is a class of control problems presented in many applications in real world. Regional controllability of linear and some nonlinear fractional systems is referred to in literature [22][23][24] and the references therein. However, regional controllability of Riemann-Liouville fractional semilinear evolution equations with analytic semigroup problem is still open. en, this paper focuses on the existence of a bounded control steering the system into a bounded desired state defined only in a subregion of the whole evolution domain. Based on Banach fixed-point theorem and some properties of fractional operators, the main result is deduced. e rest of this paper is organized as follows. In Section 2, some definitions and preparation results are introduced. In Section 3, the regional controllability of the considered system, using theory of analytical semigroup, is established under some conditions. At last, two examples are given to illustrate our given algorithm.

Preliminaries and Problem Formulation
In this section, we introduce some basic definitions of fractional operators present in the considered system which will be specified later and some properties which are used further in this paper.
Definition 1 (see [7]). e left sided Riemann-Liouville fractional integral (resp. derivative) of a function y at a point t of order q ∈ ]0, 1[ can be written as Let us consider X and Y to be two Banach spaces; we have the following two propositions.

4)
Now, we present the considered system. For that, let Ω be a bounded subset of R n with a smooth boundary zΩ . Let us consider T > 0 and denote We consider the following semilinear fractional system involving Riemann-Liouville derivative of order q ∈ ]0, 1[: where − A is the infinitesimal generator of an analytic semigroup of uniformly bounded operator G(t) { } t≥0 on the Hilbert space X � L 2 (Ω). Without loss of generality, let 0 ∈ ρ(A) where ρ(A) is the resolvent set of A. en we define the fractional power A α for 0 < α < 1, which is a closed linear operator. Its domain is D(A α ) � X α , which is a Banach space equipped with the norm ‖.‖ X α � ‖A α (.)‖ X , N: L 2 (0, T; X α ) ⟶ L 2 (0, T; X) is a nonlinear operator, and B is the control operator which is linear (bounded or unbounded) from R p into X where p is the number of actuators, u is given in U ≔ L 2 (0, T, R p ), and the initial state y 0 is in X α .
We use the following definition of mild solution for the previous problem.
Definition 2 (see [26]). For t ∈ ]0, T] and any given u ∈ U, we say that a function y u ∈ C(0, T; X) is a mild solution of system (6) if it satisfies the following formula: where in which ϕ q is a probability density function defined in ]0, ∞[. Moreover A α and K q have the following properties.

en, we have
, and by the previous proposition, we have the result.
For the rest of this paper, we denote

Regional Controllability
In this section, we formulate and prove conditions for the regional controllability of semilinear Riemann-Liouville fractional control systems. To do this, let ω be a subregion of Ω, and we define the restriction operator in ω by and we denote by χ * ω its adjoint. We have the following definition.

Definition 3.
e system (6) is said to be exactly (respectively, approximately) ω-controllable if for all y d ∈ L 2 (ω) (respectively, for all ε > 0 and for all y d ∈ L 2 (ω)), there exists a control u ∈ U such that χ ω y u (T) � y d (respectively, For the rest of this paper, we can write the mild solution as follows: and we define the restriction of the controllability operator in ω by H q Consider now the following associate linear system of equation (6): which we assume to be approximately ω− controllable.

Proof
(1) Let us consider Using the limit of L N (., .) near (0, 0), we can see that for ρ > 0, there exists l > 0 such that We have m > 0 and the mapping f: B(0, m) ⟶ B(0, ρ) such that f(u) � y u is a Lipschitz mapping with constant β/1 − A 3 . In fact, and hence m > 0.
To show the Lipschitz condition of the function f, we use equation (15), and Corollary 1, we have for all u, v ∈ B(0, m) erefore, by hypothesis (H 1 ), we obtain and hence f is a Lipschitz mapping with constant β/1 − A 3 .
(2) Let us consider z d and y d in B(0, μ); we have On the other hand, en, erefore, F satisfies the Lipschitz condition.
□ Remark 1. In the case where B is unbounded, we suppose that B is an admissible control operator for H q (see [28]), and consequently we can demonstrate the same result with a suitable β. We give the following proposition.

Proposition 4. e sequence
Proof. Let us consider n, k ∈ N * ; we have From inequality (31), we obtain en, (u n ) n is a Cauchy sequence on B(0, m), and we conclude that (u n ) n converges to u * in B(0, m).
Accordingly, we implement the algorithm as follows.

Numerical Simulations
In this section, we present two numerical simulations illustrating our theoretical result where the first one is given by using zonal actuator and the second example is given by using a pointwise actuator.

Zonal Actuator.
Let us consider the following one-dimensional fractional system with q � 0.3.
In Figure 1, we remark that with the given zonal sensor, we obtain successful results which validate the used method and the previous algorithm; indeed, the desired and estimated final states are very close in the subregion ω � [0.2, 0.5] with the error ϵ � 3 × 10 − 6 which is very small. Figure 2 presents the evolution of control function which has a transfer cost equal to ‖u * ‖ 2 � 0.21.

Pointwise Actuator.
We consider the following system: we have the following result. In Figure 3, like the previous part, the given algorithm leads to good results; we remark that the desired state and the reached one are close in w � [0.35, 0.55] using the given pointwise actuator. In this case, the reconstruction error is very small, and it is of order ε � 10 − 5 . Figure 4 shows the evolution of control function depending on the time t with the transfer cost ‖u * ‖ 2 � 4 × 10 − 3 .

Conclusion
In this paper, we have established the regional controllability for a class of Riemann-Liouville fractional semilinear control systems. e idea of applying control theory for this kind of systems is very interesting and constitutes a new issue in the applications. e presented method in this paper covers a large class of this kind of systems. We have also given an algorithm which has been implemented numerically and has very satisfactory results. In addition, the problem of regional controllability remains open for other types of fractional systems that will be the subject of future research.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.