The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics is symmetric and uniformly elliptic and by using the properties of the Mittag-Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.


Introduction
Fractional order calculus is a natural generalization of classical integer order calculus. It deals with integrals and derivatives of an arbitrary real or complex order. Fractional order calculus has become very popular, in recent years, due to its demonstrated applications in many fields of applied sciences and engineering, such as the spread of contaminants in underground water, the charge transport in amorphous semiconductors, and diffusion of pollution in the atmosphere [1][2][3]. Because it generalizes and includes in the limit the integer order calculus, the fractional calculus has the potential to accomplish much more than what integer order calculus achieves [4]. In particular, it has proved to be a powerful tool to describe long-term memory and hereditary properties of various dynamical complex processes [5], diffusion processes, such as those found in batteries [6] and electrochemical and control processes [7], to model and control epidemics [8,9] and mechanical properties of viscoelastic systems and damping materials, such as stress and strain [10].
One can find in the literature several different fractional calculus. Here we use the fractional calculus of Caputo, which was introduced by Michele Caputo in his 1967 paper [11]. Such calculus has appeared, in a natural way, for representing observed phenomena in laboratory experiments and field observations, where the mathematical theory was checked with experimental data. Indeed, the operator introduced by Caputo in 1967, and used by us in the present work, represents an observed linear dissipative mechanism phenomena with a time derivative of order 0.15 entering the stress-strain relation [11]. More recently, a variational analysis with Caputo operators has been developed, which provides further mathematical substance to the use of Caputo fractional operators [12,13].
In the analysis and design of control systems, the stability issue has always an important role [14,15]. For a dynamical system, an equilibrium state is said to be stable if such system remains close to this state for small disturbances, and for an unstable system the question is how to stabilize it, especially by a feedback control law [16]. The stabilization concept for integer order systems and related problems has been considered in several works, see, e.g., [17][18][19][20] and references cited therein. In [17], the relationship between the asymptotic behavior of a system, the spectrum properties of its dynamics, and the existence of a Lyapunov functional is provided. Several techniques are considered to study different kinds of stabilization, for example, the exponential stabilization is studied via a decomposition method [19] while the strong stabilization is developed using the Riccati approach [20].
Similarly as classical dynamical systems, stability analysis is a central task in the study of fractional dynamical systems, which has attracted increasing interest of many researchers [9,21]. For finite dimensional systems, the stability concept for fractional differential systems equipped with the Caputo derivative is investigated in many works [22]. In [23], Matignon studies the asymptotic behavior for linear fractional differential systems with the Caputo derivative, where their dynamics A is a constant coefficient matrix. In this case the stability is guaranteed if the eigenvalues of the dynamics matrix A, λ ∈ σ(A), satisfy |arg(λ)| > πα 2 [23]. Since then, many scholars have carried out further studies on the stability for different classes of fractional linear systems [24,25]. In [24], stability theorems for fractional differential systems, which include linear systems, time-delayed systems, and perturbed systems, are established, while in [25], Ge, Chen and Kou provide results on the Mittag-Leffler stability and propose a Lyapunov direct method, which covers the power law stability and the exponential stability. See also [26], where the Mittag-Leffler and the class-K function stability of fractional differential equations of order α ∈ (1, 2) are investigated. In 2018, the notion of regional stability was introduced for fractional systems in [27], where the authors study the Mittag-Leffler stability and the stabilization of systems with Caputo derivatives, but only on a sub-region of its geometrical domain. More recently, fractional output stabilization problems for distributed systems in the Riemann-Liouville sense were studied [28][29][30], where feedback controls, which ensure exponential, strong, and weak stabilization of the state fractional spatial derivatives, with real and complex orders, are characterized.
An analysis of the literature shows that existing results on stability of fractional systems are essentially limited to finite-dimensional fractional order linear systems, while results on infinite-dimensional spaces are a rarity. In contrast, here we investigate global stability and stabilization of infinite dimensional fractional dynamical linear systems in the Hilbert space L 2 (Ω) with Caputo derivatives of fractional order 0 < α < 1. In particular, we characterize exponential and strong stability for fractional Caputo systems on infinite-dimensional spaces.
The remainder of this paper is organized as follows. In Section 2, some basic knowledge of fractional calculus and some preliminary results, which will be used throughout the paper, are given. In Section 3, we prove results on the global asymptotic and exponential stability of Caputo-time fractional differential linear systems. In contrast with available results in the literature, which are restricted to systems of integer order or to fractional systems in the finite dimensional state space R n , here we study a completely different class of systems: we investigate fractional linear systems where the state space is the Hillbert space L 2 (Ω). We also characterize the stabilization of a controlled Caputo diffusion linear system via a decomposition method. Section 4 presents the main conclusions of the work and some interesting open questions that deserve further investigations.

Preliminaries and Notation
In this section, we introduce several definitions and results of fractional calculus that are used in the sequel.

Lemma 1 ([31]
). For any given function g ∈ L 2 (0, T, L 2 (Ω)), we say that function y ∈ C(0, T, L 2 (Ω)) is a mild solution of the system where (S(t)) t≥0 the strongly continuous semigroup generated by operator A, and T α the probability density function defined on (0, ∞) by

Definition 2 ([33]
). The Mittag-Leffler function of one parameter is defined as

Definition 3 ([33]
). The Mittag-Leffler function of two parameters is defined as Remark 2. The Mittag-Leffler function appears naturally in the solution of fractional differential equations and in various applications: see [33] and references therein. The exponential function is a special case of the Mittag-Leffler function [34]: for β = 1 one has E η,1 (z) = E η (z) and E 1,1 (z) = e z .

Main Results
Our main goal is to study the stability and provide stabilization for a class of abstract Caputo-time fractional differential linear systems.

Stability of Time Fractional Differential Systems
Let Ω be an open bounded subset of R n , n = 1, 2, 3, . . ., and let us consider the following abstract time fractional order differential system: where C D α t is the left-sided Caputo fractional derivative of order 0 < α < 1, the second order operator A : D(A) ⊂ L 2 (Ω) −→ L 2 (Ω) is linear, with dense domain and such that the coefficients do not depend on time t, and such that it is also the infinitesimal generator of the C 0 -semi-group (S(t)) t≥0 on the Hilbert state space L 2 (Ω) endowed with its usual inner product < ·, · > and the corresponding norm · . The unique mild solution of system (8) can be written, from Lemma 1, as We begin by proving the following lemma, which will be used thereafter.
Proof. To prove that (S α (t)) t≥0 are bounded, we have to show that By reductio ad absurdum, let us suppose that (10) does not hold, which means that there exists a sequence (t s + τ n ), t s > 0 and τ n −→ +∞, satisfying From relation it follows that the right-hand side goes to 0 as n −→ +∞. Using Fatou's Lemma yields lim inf Therefore, for some s 0 < t s , we may find a subsequence τ n k such that lim k−→+∞ S α (s 0 + τ n k )z = 0.
By virtue of condition (9), one obtains which contradicts (11). The intended conclusion follows from the uniform boundedness principle.

Definition 4.
Let z 0 ∈ L 2 (Ω). System (8) is said to be exponentially stable if there exist two strictly positive constants, M > 0 and ω > 0, such that The next theorem provides necessary and sufficient conditions for exponential stability of the abstract fractional order differential system (8).
Theorem 1. Suppose that the operators (S α (t)) t≥0 fulfill assumption (9) and ∀z ∈ L 2 (Ω) Then, system (8) is exponentially stable if, and only if, for every z ∈ L 2 (Ω) there exists a positive constant δ < ∞ such that Proof. One has Combining assumption (9), Lemma 5, and condition (13), one gets for some N > 0. Therefore, for t sufficiently large, it follows that Then, there exists t 1 > 0 such that ln S α (t) < 0, ∀t ≥ t 1 . Thus, Now, let us show that Let t s > 0 be a fixed number and N = sup S α (t) . Thus, for each t > t s , there exists m ∈ N such that mt s ≤ t ≤ (m + 1)t s . From (12), it follows that Using again (12), it results that Since mt s t ≤ 1 and t s is arbitrary, one obtains Consequently, (14) holds. Hence, we conclude that for all ω ∈ ]0, −ω 0 [, there exists M > 0 such that ∀z ∈ L 2 (Ω) S α (t)z ≤ Me −ωt z , ∀t ≥ 0, which means that system (8) is exponentially stable. The converse is obvious.
In our next theorem, we provide sufficient conditions that guaranty the strong stability of the fractional order differential system (8). The result generalizes the asymptotic result established by Matignon for finite dimensional state spaces, where the dynamics of the system A is considered to be a matrix with constant coefficients in R n [23]. In contrast, here we tackle the stability for a different class of systems. Precisely, we consider fractional systems where the system dynamics A is a linear operator generating a strongly continuous semigroup in the infinite dimensional state space L 2 (Ω). Theorem 2. Let (λ p ) p≥1 and (φ p ) p≥1 be the eigenvalues and the corresponding eigenfunctions of operator A on L 2 (Ω). If A is a symmetric uniformly elliptic operator, then system (8) is strongly stable on Ω.

Stabilization of Time Fractional Differential Systems
Let Ω be an open bounded subset of R n , n = 1, 2, 3, . . .. We consider the following Caputo-time fractional differential linear system: with the same assumptions on A as in Section 3.1 and where B is a bounded linear operator from U into L 2 (Ω), where U is the space of controls, assumed to be a Hilbert space. By Lemma 1, the unique mild solution z(·) of system (16) is defined by where S α (t) and K α (t) are given, respectively, by (3) and (4) .

Definition 6.
System (16) is said to be exponentially (respectively strongly) stabilizable if there exists a bounded operator K ∈ L(L 2 (Ω), U) such that the system is exponentially (respectively strongly) stable on Ω.

Remark 4.
It is clear that the exponential stabilization of system (16) implies the strong stabilization of (16). Note that the concept is general: when α = 1, we obtain the classical definitions of stability and stabilization.
Let (S k (t)) t≥0 be the strongly continuous semi-group generated by A + BK, where K ∈ L(L 2 (Ω), U) is the feedback operator. The unique mild solution of system (16) can be written as where Ψ α (θ) is defined by (5).
Proof. The proof is similar to the proof of Theorem 1.

Theorem 4.
Let (λ k p ) p≥1 and (φ k p ) p≥1 be the eigenvalues and the corresponding eigenfunctions of operator A + BK on L 2 (Ω). If A + BK is a symmetric uniformly elliptic operator, then system (16) is strongly stabilizable on Ω.
Proof. The proof is similar to the proof of Theorem 2.

Example 2.
Let us consider, on Ω =]0, 1[, the following fractional differential system of order α = 0.2: with the linear bounded operator B = I and where we take K = −B * = −I. The operator with spectrum given by the eigenvalues λ k p = − 1 2 − 1 100 (pπ) 2 , p ≥ 1, and the corresponding eigenfunctions Furthermore, the solution of system (19) can be written as It is clear that A + BK is a symmetric and uniformly elliptic operator. Hence, from Theorem 4, we deduce that system (19) is strongly stabilizable on Ω, i.e., the system is strongly stabilizable by the feedback control u(t) = −B * z(t). Figure 2 shows, for z(x, 0) = x(x − 1), that the state z(x, t) of system (19) is unstable at t = 0. Moreover, we see that the state evolves close to 0 at t = 10. Numerically, the state is stabilized by u(t) = −B * z(t) with an error equal to 1.75 × 10 −04 .

Decomposition Method
Now, we study the stabilization of system (16) using the decomposition method, which consists in decomposing the state space and the system using the spectral properties of operator A.
Let ξ > 0 be fixed and assume that there are at most finitely many nonnegative eigenvalues of A and each with finite dimensional eigenspace. In other words, assume there exists l ∈ N such that where H u = PH = span{φ 1 , φ 2 , . . . , φ l } and H s = (I − P)H = span{φ l+1 , φ l+2 , . . .} with P ∈ L(H) the projection operator [40]. Hence, system (16) can be decomposed into the following two sub-systems: and C D α t z s (t) = A s z s (t) + (I − P)Bu(t), z 0s = (I − P)z 0 , where A s and A u are the restrictions of A on H s and H u , respectively, and are such that σ(A s ) = σ s (A), σ(A u ) = σ u (A), and A u is a bounded operator on H u . Our next result asserts that stabilization of system (16) is equivalent to the one of system (22).

Theorem 5.
Let the spectrum σ(A) of A satisfy the above spectrum decomposition assumptions (20) for some ξ > 0 and A s be a symmetric uniformly elliptic operator. If system (22) is strongly stabilizable by the control u(t) = D u z u (t) (24) with D u ∈ L(H, U) such that z u (t) then system (16) is strongly stabilizable using the feedback control v(t) = D u z u (t).

Conclusions and Future Work
We investigated the stability problem of infinite dimensional time fractional differential linear systems under Caputo derivatives of order α ∈ (0, 1), where the state space is the Hillbert space L 2 (Ω). We proved necessary and sufficient conditions for exponential stability and obtained a characterization for the asymptotic stability, which is guaranteed if the system dynamics is symmetric and uniformly elliptic.
Moreover, some stabilization criteria were also proved. Finally, we investigated the strong stabilization of the system via a decomposition method, where an explicit feedback control is obtained. Illustrative examples were given, showing the effectiveness of the theoretical results. As future work, we intend to extend our work to the class of infinite dimensional time fractional differential nonlinear systems. Various other questions are still open and deserve further investigations, such as, studying boundary stability and gradient stability for time fractional differential linear systems or considering the more recent notion of Λ-fractional derivative [41], and thus obtaining a geometrical interpretation.
Author Contributions: Each author equally contributed to this paper, read and approved the final manuscript.