Controllability of stochastic fractional systems involving state-dependent delay and impulsive effects

In this paper, the controllability concept of a nonlinear fractional stochastic system involving state-dependent delay and impulsive eﬀects is addressed by employing Caputo derivatives and Mittag-Leﬄer (ML) functions. Based on stochastic analysis theory, novel suﬃcient conditions are derived for the considered nonlinear system by utilizing Krasnoselkii’s ﬁxed point theorem. Correspondingly, the applicability of the derived theoretical results is indicated by an example.


Introduction
Fractional order dynamical models based on fractional calculus and real dynamics systems have been significantly improved in a very recent period.The expansive nature of fractional calculus offers a tremendous opportunity to alter the control model.The theory of arbitrary order integrals and derivatives is fractional calculus, which unifies and generalizes the concepts in the fields of science, control theory, and many other areas.The advantages of fractional differential equations (FDEs) have been analyzed by many scientists [16,18,28,31,33].In the branch of qualitative characteristics of dynamical systems, controllability is an essential aspect.Many researchers have expanded the findings of nonlinear and linear systems from integer order to fractional order because of its significance [1,4,6,7].Apart from these works, controllability problems of nonlinear and linear systems involving delay in control were reported in [32,37].Investigation of controllability for a class of switched Hilfer neutral fractional systems with noninstantaneous impulses in finite-dimensional spaces was discussed in [24].Moreover, in [23], the controllability of a fractional neutral dynamic system with noninstantaneous impulsive circumstances has been investigated for an ABC-fractional system with an integral term.
Impulsive differential equations (IDEs) can take adequate consideration of the issues of sudden state jumps in the complete evolution process.Thus, it yields a suitable structure for modeling and illustrating various complex dynamical systems.IDEs have been focused on widely in the literature since these equations appear obviously in electronics, economics, medicine, mechanics, and biology.The monograph by Bainov and Simeonov [5] contains the fundamental understanding of IDEs.In [27], the authors studied sudden transitions such as state modification or in the form of impulses occurring in physics, finance, and aeronautics.The existence, uniqueness, Ulam-Hyers stability, and total controllability results for Hilfer fractional switched impulsive systems in finite-dimensional spaces were discussed in [25].The existence of solutions, stability, and the controllability criteria were analyzed for piecewise impulsive dynamic systems on arbitrary time domain in [22].The total controllability of a novel class of piecewise nonlinear Langevin fractional dynamic equations with noninstantaneous impulses and controllability conditions were investigated in [26].
Stochastic effects have had a major part in the study of fractional differential systems in recent years.Research on the controllability issue of linear, nonlinear, and stochastic systems was considered in [19,20].Mahmudov et al. [30] examined the stochastic controllability of linear, nonlinear systems respectively in Hilbert spaces [29].The results on optimal control of fractional stochastic systems driven by the Wiener process and fractional Brownian motion with noninstantaneous impulsive effects were investigated in [8], and also, in [34], the authors investigated valuable insights into the controllability of fractional stochastic inclusions with fractional Brownian motion effects.Stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses with delay were reported in [21].Approximate controllability of second-order impulsive neutral stochastic integro-differential evolution inclusions with infinite delay was discussed in [35].Further, in [14], necessary criteria for the controllability of the system were given by a particular kind of nonlinear stochastic impulsive system with infinite delay in an abstract space.There was discussion on a generalization of the contraction mapping principle.
Delay differential equation plays a crucial role in the analysis and predictions of life sciences including population dynamics, immunology, and neural networks.However, differential equations examine both the unknown state and its derivatives simultaneously, although for a certain time the instant delay differential system is governed by the past.The delay differential system is independent of the past.The past dependence on a variable is state-dependent delay.The interest in state-dependent delay (SDD) type systems has been enormous in recent years.Existence analysis of fractional system with SDD was reported in [11].Existence analysis of differential inclusions involving stochastic effects, impulsive effects, and SDD was investigated in [13].A fractional integro-differential system involving SDD was studied by Agarwal et al. in [2].The study of fractional impulsive systems involving SDD was discussed by the authors in [9].Moreover, controllability analysis of second-order systems with SDD and impulses was proved in [3].Numerous authors have emphasized how controllability systems are increasingly common and adequate in applications in [12,[38][39][40].
Inspired by the above mentioned results, the paper reports the controllability of a nonlinear system involving SDD, impulsive conditions, and a stochastic term.To the best of the authors knowledge, there are no studies concerning the controllability results of this type of a system, which is the main motivation of this study.Significance of the considered problem is listed below: (i) Stochastic fractional order systems involving state-dependent delay and impulsive effects have been considered.(ii) The considered system solution is derived by employing the Caputo derivative, Laplace transform theory, and the Mittag-Leffler function.(iii) By using certain assumptions, sufficient conditions are derived utilizing Krasnoselkii's fixed point theorem to analyze the controllability result.The paper is organized in the following manner: Preliminaries, lemmas, and basic definitions are explained in Sect. 2. A controllability condition for a nonlinear system with fractional order is analyzed in Sect.3.An example is attainable to exemplify the theoretical result in Sect. 4. Finally, Sect. 5 concludes this paper.

Problem formulation
Consider the nonlinear fractional stochastic system involving impulsive and SDD of the form where C 0 D κ t denotes the Caputo derivative with lower bounds 0 of order κ ∈ (0, 1].H denotes a Hilbert space, z(•) ∈ R n is a state variable that takes values in H with the inner product (•, •) and the norm • ; and A ∈ R n×n , B ∈ R n×m are known constant matrices.u ∈ L 2 ([0, T], U) is a control input, where U ∈ H and B is a bounded linear operator on H . z s : (-∞, 0] → H defines the function z s in a Hilbert space H .For some abstract space B, z s (θ ) = z(s + θ ) and continuous function : J × B → (-∞, T].Let ( , F, P) be a complete probability space with filtration {F t } t≥0 generated by an m-dimensional Wiener process and probability measure P on .In ( , F, P), H and K are separable Hilbert spaces.The function z(t) is continuous everywhere except for some t n such that symbolizes the left and right limits at t = t n and z PC = sup t∈J |z(t)| < ∞ in the Banach space.PC(J , L 2 ) is the closed subspace of PC(J , L 2 ( , F, P; H )), which is measurable and F-adapted H -valued process with the norm Stochastic process with filtration {F t } t≥0 in ( , F, P) is a collection of random variables z(t) : → H , t ∈ J .F is measurable, and for t ≥ 0, {w(t)} t≥0 is an m-dimensional Wiener process or a Brownian motion in K , where Qe n equals λ n e n , {β n } n≥1 indicates real-valued Brownian motions, and {e n } n≥1 represents completely orthonormal in K , then An abstract space (B, • B ), i.e., a seminorm linear space of F 0 -measurable function, is defined utilizing the concepts and symbolizations established in [17].
(a) If for every [0, T), z : (-∞, T] → H is a continuous function and Definition 2.1 [33] The Caputo derivative of order κ (0 ≤ m ≤ κ < m + 1) for a function Definition 2.2 [33] The ML function E κ (z) with μ > 0 is defined by and the two-parameter ML function E κ,p (z) with κ, p > 0 is defined by The Laplace transform of ML function E κ,p (z) is For p = 1, we have Consider the following Cauchy fractional problem: To obtain the solution of (4), take the Laplace transform Now, taking the inverse Laplace transform, we have Then Definition 2.4 z(t) ∈ J → H is known as a stochastic process if system (1)-( 3) satisfies (1) z(t) is F t -adapted measurable ∀t ∈ J ; (2) z(t) ∈ H satisfies the following: Lemma 2.5 [10,36] Assume that N is a convex, nonempty, and closed subset belonging to the Banach space X.Suppose that L and M are operators such that (iii) M is a contraction mapping, then there ∃T ∈ N such that T = L z + M z.

Main result
Now, we prove the controllability results for system (1)-(3) by using the following hypothesis.
(H 1 ) The function g : J × B → H is continuous.There exists L g such that (H 2 ) The function ς : J × B → H is continuous.There exists L ς such that (H 3 ) η g : [0, ∞) → (0, ∞) is a nondecreasing continuous function, and there exists an (H 4 ) η ς : [0, ∞) → (0, ∞) is a nondecreasing continuous function, and there exists an (H 5 ) The map I n : B → H is continuous and (H 6 ) A bounded and continuous function J ϕ : X ( -) → (0, ∞) exists and t → ϕ t is a welldefined function from The linear operator W is defined by there exists a bounded invertible operator For brevity, Define the control function where Theorem 3.1 The nonlinear system (1)-( 3) is controllable on J if and (H 1 )-(H 7 ) are satisfied.
Proof The operator is defined as follows: Now, we can find a fixed point of , which implies that system (1)-( 3) is controllable, by dividing the proof into several steps using Lemma 2.5.Define the set B r = {z ∈ B : z ∞ ≤ r}.Clearly, B r is a convex, closed, and bounded set in B for each r, then by Lemma 2.3 Step 1: B r ⊂ B r .If we assume B r ⊂ B r is false, then there ∃z ∈ B r ∀r > 0 such that r ≤ E z(t) 2 , t ∈ J , we get and hence This contradicts the assumption.Thus, for some r > 0, B r ⊂ B r . where Step 2: 1 (z) is contractive.Suppose z 1 , z 2 ∈ B r , where Therefore L 0 ≤ 1, 1 (z) is contractive.
Step 5: Let 0 ≤ ≤ t for any z ∈ B r .Now, define an operator on B r by Since T(t) is a compact operator, V (t) = { 2 z(t), z ∈ B r } is relatively compact in H for every > 0. Also, for every z ∈ B r , we have Therefore, E 2 (z)(t) -( 2 (z)(t)) 2 → 0 as → 0, and for each t ∈ J , there are relatively compact sets arbitrarily close to the set V(t).Hence V (t) = { 2 z(t), z ∈ B r } is relatively compact in H .So from the Arzela-Ascoli theorem, 2 is completely continuous.

Corollary 3.2
In the absence of a stochastic system, system (1)-( 3) reduces to the following form: where A , B, g are defined the same as before.Then the solution of system (5)-( 7) can be written as follows: and it satisfies hypotheses (H 1 )-(H 4 ), (H 6 ), and (H 7 ), then for any t ∈ J the control can be chosen as Then the solution of system (5)-( 7) satisfies z(t) = z 1 , and hence the system is controllable on J .

Corollary 3.3
In the absence of an impulsive condition, system (1)-( 3) reduces to the following form: where A , B, g, ς are defined the same as before.Then the solution of system (8)-( 9) can be written as follows: and it satisfies hypotheses (H 1 ), (H 3 ), and (H 5 )-(H 7 ).Then, for any t ∈ J , the control can be chosen as follows: Then the solution of system (8)-( 9) satisfies z(t) = z 1 , and hence the system is controllable on J .

Corollary 3.4 The solution of the linear system
can be expressed as The linear system (4)-( 6) is controllable if and only if the controllability Grammian matrix Remark 3.5 The research on the results for optimal control of fractional stochastic systems driven by the Wiener process and fractional Brownian motion with noninstantaneous impulsive effects was investigated in [8] and in [34].The authors investigated valuable insights into the controllability of fractional stochastic inclusions with fractional Brownian motion effects.The analysis of a system with a fractional derivative that can be steered from one state to another by an admissible control input was discussed by differing the type of stochastic, impulsive, and delay effects and the applications and implications of their findings.To the best of the authors' knowledge, there is no work concerning the controllability of stochastic fractional systems with state-dependent delay and impulsive effects, which is the main motivation of the work.which is positive definite.Further, ς and g satisfy the hypotheses of Theorem 3.1.Also, the corresponding linear system is controllable.Hence the nonlinear impulsive fractional stochastic system (10)-( 12) is controllable on J = [0, T].

Conclusion
Controllability results for nonlinear impulsive fractional stochastic systems involving SDD have been derived based on stochastic theory and fractional calculus under certain assumptions in a Hilbert space.On the basis of the control input, sufficient conditions for the controllability criteria have been obtained using Krasnoselskii's fixed point theorem.
An example is included to validate the obtained criteria.FDEs system with delay arises in many applications; moreover, the proposed approach could be applied to other kinds of multi-order fractional dynamical systems involving various delay effects, which will be the focus of future research.