This paper is devoted to the study of the approximate controllability for a class of Sobolev-type fractional differential systems of order sigma is an element of (0, 1) in a separable Hilbert space. The proposed control system is governed by the Hilfer fractional derivative, which provides a unified framework that interpolates between the Riemann-Liouville and Caputo fractional derivatives. By employing resolvent operator theory and semigroup techniques combined with suitable fixed-point arguments, sufficient conditions for the existence of mild solutions and approximate controllability are established under non-local initial conditions. Unlike many existing results, this analysis does not rely on the compactness assumption of the associated semigroup, thereby extending the applicability of the controllability criteria to a broader class of systems with implicit dynamics. The obtained results are further extended to Sobolev-type fractional integro-differential systems. An illustrative example is presented to demonstrate the effectiveness and applicability of the theoretical findings.

This manuscript explores the approximate controllability of Hilfer fractional hemivariational inequalities with non-instantaneous impulses. The study is motivated by the need to establish a realistic and mathematically rigorous framework for fractional control systems by incorporating the non-instantaneous nature of impulses, where impulsive effects act over finite time intervals rather than instantaneously, thereby providing a more accurate representation of real-world phenomena. It captures systems exhibiting non-smooth and non-convex behavior, which significantly enhances the modeling effectiveness. Within the setting of fractional calculus, the model extends classical control frameworks by incorporating memory effects together with non-smooth dynamics. To effectively describe these characteristics, a Banach space of piecewise continuous functions is introduced, which enables a precise and rigorous formulation of such systems. To prove the existence of a mild solution for the proposed system, sufficient conditions are established using a fixed-point theorem for the multivalued maps. The approximate controllability is then proven through a piecewise-defined control function under the assumption that the linear system corresponding to the proposed control problem is approximately controllable. The analysis relies on generalized Clarke directional derivative, generalized Clarke subdifferential, fractional calculus, and semigroup theory. An example is presented in the end as an application to illustrate the effectiveness of the main findings.

This study primarily focuses on establishing the sufficient conditions for the existence and uniqueness of the mild solution along with the approximate and trajectory controllability results for a new class of the non-linear Psi-Caputo fractional neutral-type integro-differential system with finite delay and nonlocal conditions in a Hilbert space. A key advantage of the Psi-Caputo fractional derivative is that it allows to choose a suitable kernel function Psi. First, we derive the existence of the mild solution for the proposed control system by using a fixed point approach. For this purpose, the proposed control system is transformed into an equivalent fixed point problem using the Psi-Riemann-Liouville fractional integral operator. Then, the existence of the mild solution is established by Schauder's fixed point theorem. Then, the uniqueness of the mild solution is studied with the help of the Banach contraction principle. Moreover, the approximate controllability result of the proposed control system is established under the consideration that the corresponding linear system is approximate controllable. Further, the trajectory controllability result is studied by using the Gronwall's inequality. The set of sufficient conditions is derived by using the concepts of fractional calculus, Laplace transform, fixed point techniques, and semigroup theory of bounded linear operators. Finally, an illustrative example is presented to validate and demonstrate the applicability of the theoretical results.

The main objective of this paper is to investigate the sufficient conditions for the existence of mild solution and approximate controllability results for a class of nonlinear fractional neutral-type integro-differential stochastic inclusions with infinite delay in a separable Hilbert space. In the proposed control system, the -Hilfer fractional derivative has been considered, which has a quality to choose a suitable kernel function . First, by making use of the -Riemann-Liouville fractional integral operator, the proposed stochastic control problem is converted into an equivalent fixed point problem, and then, the Bohnenblust-Karlin fixed point theorem is applied to derive the existence of the mild solution. The approximate controllability result for the proposed control system has been established under the consideration that the corresponding linear system is approximate controllable. The theory of multivalued maps, fractional calculus, the concept of stochastic analysis and the fixed point technique have been used to bring off the main results. At the end, we provide a concrete example in order to validate the abstract findings.

The main aim of this paper is to establish the sufficient conditions for the existence and controllability for a class of nonlinear fractional ordered Sobolev-type neutral stochastic integro-differential system with infinite delay in a separable Hilbert space using Psi-Caputo fractional derivative. The Psi-Caputo fractional derivative has a quality to choose the suitable kernel function Psi. To prove the main results, we have relaxed the compactness of the semigroup. Also, the weaker hypotheses (non- Lipschitz conditions rather than the strong Lipschitz-type conditions) are considered in order to prove the main results. The main tools applied in our analysis are fractional calculus, stochastic analysis, the semigroup of bounded linear operators, the measure of noncompactness, and Monch's fixed point theorem. At the end, a concrete example is illustrated to validate our findings. (c) 2024 Published by Elsevier Inc.

This research aims to establish the sufficient conditions for the existence of the mild solution and approximate controllability for a class of P-Caputo fractional neutral-type integro-differential stochastic inclusions with infinite delay in a separable Hilbert space. In the proposed stochastic control system, the P-Caputo fractional derivative is considered, which has the flexibility to choose a suitable kernel function P. Firstly, we derive the existence of the mild solution for the P-Caputo fractional neutral-type delayed integro-differential stochastic system by using the Karlin fixed point approach. For this purpose, the P-Caputo fractional neutral-type delayed integro-differential stochastic inclusions is transferred into an equivalent fixed point problem by implementing the P-Riemann-Liouville integral operator, and then the Karlin fixed point theorem is applied. Further, the approximate controllability results of the proposed stochastic control system are established under the consideration that the corresponding linear system is approximate controllable. The set of sufficient conditions is established by using the concepts of fractional calculus, the general theory of stochastic analysis, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. At the end of the paper, a concrete example is provided to validate the abstract results.

This paper focuses to establish the sufficient conditions for the existence of the integral form mild solution and the approximate controllability for a class of the nonlinear fractional neutral-type integro-differential delayed stochastic system with the instantaneous impulsive effects in a separable Hilbert space by considering the Ψ -Caputo fractional derivative. A key advantage of employing the Ψ -Caputo fractional derivative is its inherent flexibility in choosing a suitable kernel function Ψ , which enhances its compatibility with conventional analytical techniques. Firstly, we derive the existence of the mild solution for the Ψ -Caputo fractional neutral-type delayed integro-differential stochastic system. For this purpose, the proposed stochastic control problem is transferred into an equivalent fixed-point problem using the Ψ -Riemann-Liouville fractional integral operator. Then, the Schauder fixed point theorem is applied. By employing tools from fractional calculus, Laplace transforms, stochastic analysis and fixed point theorem, the set of sufficient conditions is established. The approximate controllability result is established under the consideration that the corresponding linear system is approximately controllable. To illustrate the abstract results, we provide an example at the end of the paper.

The main aim of this research is to study the sufficient conditions for the existence of the mild solution and approximate controllability of a class of Caputo conformable fractional neutral-type stochastic system with the instantaneous impulsive effects and nonlocal conditions in a separable Hilbert space. Since, the conformable fractional derivative retains several classical properties such as the mean value theorem, Rolle’s theorem, the product and quotient rules, and linearity, which distinguish it from traditional fractional derivatives, including the Riemann-Liouville, Caputo, and Hilfer derivatives. Therefore, the conformable derivative is simpler and faster but ignores history, while the Caputo conformable derivative offers a balance capturing some memory with easier calculations. Firstly, the Riemann-Liouville conformable fractional integral operator is used to convert the proposed stochastic control system into an equivalent fixed point problem. Then, the fixed point approach is employed to derive the results. The main tools applied in this study are fractional calculus, semigroups of bounded linear operators, stochastic analysis, and Krasnoselskii’s fixed point theorem. Further, the approximate controllability results of the proposed stochastic control system are established under the consideration that the corresponding linear system is approximate controllable. At the end, we provide an example to illustrate our theoretical findings.

This paper aims to derive a new set of sufficient conditions for the existence and approximate controllability of neutral‐type fractional stochastic integrodifferential inclusions with infinite delay and non‐instantaneous impulse in a separable Hilbert space using the Atangana–Baleanu Caputo fractional derivative. We investigate the existence of a mild solution for the Atangana–Baleanu Caputo fractional neutral‐type delay integrodifferential stochastic system while taking into account the non‐instantaneous impulses. For this purpose, the Atangana–Baleanu Caputo fractional neutral‐type impulsive delay stochastic system is transferred into an equivalent fixed point problem via an integral operator, and then, the Bohnenblust–Karlin fixed point approach is applied. Further, the approximate controllability results of the proposed nonlinear stochastic impulsive control system are established under the consideration that the corresponding linear system is approximately controllable. The set of sufficient conditions is established by using the concepts of stochastic analysis, fractional calculus, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. To illustrate the abstract results, we provide an example at the end of the paper.

The main aim of this paper is to establish a set of sufficient conditions for the existence of integral form mild solution and approximate controllability of a nonlinear multivalued neutral Hilfer fractional differential system in a Hilbert space. In the proposed control problem, we deal with a system that is under the noninstantaneous impulsive effects and infinite delay. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Dhage's fixed-point theorem. Further, the approximate controllability results are established using piecewise defined control function and under the consideration that the corresponding linear system is approximate controllable. The main findings have been established by using the facts concerning the semigroup theory of operators, fixed-point technique, fractional calculus, and the theory of multivalued maps. Finally, an example is given to illustrate the abstract results.

In this manuscript, we explore the sufficient conditions for the existence and controllability of a new class of the nonlinear fractional Sobolev-type neutral integro-differential stochastic system with instantaneous impulses and finite delay in a separable Hilbert space. We investigate the existence of the mild solution for the proposed stochastic control problem while taking into account the instantaneous impulsive effects. For this purpose, the Riemann-Liouville fractional integral operator is used to convert the proposed stochastic control system into an equivalent fixed point problem. Then, we establish the controllability results by employing the theory of fractional calculus, semigroups of bounded linear operators, stochastic analysis, Mönch’s condition, and the measure of noncompactness. In our approach, we relax the compactness assumption on the semigroup operator. Instead of using strong Lipschitz-type conditions, we derive the sufficient conditions under non-Lipschitz assumptions. At the end, an example is given to demonstrate and support the theoretical results.

. In this paper, we aim to establish a set of sufficient conditions for the existence and controllability of an integral form mild solution of a fractional neutral-type integro-differential stochastic system with finite delay and noninstantaneous impulses in a separable Hilbert space. The results are derived using fractional calculus, the principle of stochastic analysis, semigroup theory, measure of noncompactness and Monch's condition. Sufficient conditions are established under the non-Lipschitz conditions instead of the strong Lipschitztype conditions. Finally, an example is given to support the validation of the obtained abstract results.

In this paper, we aim to establish a set of sufficient conditions for the existence of an integral form mild solution and approximate controllability for a class of Sobolev-type Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. In the proposed control problem, we deal with a system that is under non-instantaneous impulsive effect. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Schauder’s fixed point theorem. The approximate controllability results for the proposed control problem have been established under the consideration that the corresponding linear system is approximate controllable. By utilising stochastic analysis, the theory of resolvent operator, fractional calculus, and the fixed point technique, sufficient conditions have been established. At the end, an example is given to illustrate the abstract results.

This manuscript explores the approximate controllability of first order nonlinear hemivariational inequalities with non instantaneous impulses in a real Hilbert space. The proposed problem formulates a practical scenario by considering systems with non smooth and non convex behavior, while also incorporating the non instantaneous nature of impulses. To effectively model these dynamics, space of piecewise continuous functions is defined, and then enables a precise representation of such systems. The problem is formulated using hemivariational inequalities, and then is helpful for offering a robust and comprehensive framework to address the complexities associated with non convex and non smooth dynamics. To prove the existence of a mild solution for the proposed system, sufficient conditions are established by employing fixed point theorem for the multivalued maps. The approximate controllability is then proven through a piecewise defined control function under the assumption that the linear system corresponding to the proposed control problem is approximate controllable. The key tools utilized in our analysis are properties of generalized Clarke subdifferential, generalized Clarke directional derivative, theory of resolvent operator, and semigroup theory. An example is presented in the end as an application to illustrate the main findings.

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolevtype in a Hilbert Space as well as to generalize the results that existed in the literature on this topic.We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system.With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function.The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique.Finally, we provide two examples as applications to illustrate our main results.

This paper aims to establish a new set of sufficient conditions for the existence and exact controllability of nonlocal fractional integro-differential neutral stochastic system involving finite delay in a separable Hilbert space. Firstly, we establish sufficient conditions for the mild solution of the system and then established the controllability results by means of fractional calculus, semigroup theory and Mainardii’s Wright type function, principles of stochastic systems, measure of noncompactness. Finally, we provide an example to illustrate our main findings.

This manuscript aims to study a new class of non-instantaneous impulsive Hilfer stochastic differential system of fractional-order 1/2≤β≤ 1, driven by fractional Brownian motion and Poisson jumps in a Hilbert space. A new set of sufficient conditions is derived to ensure the existence of the mild solution and exact null controllability of the proposed control system utilizing the stochastic analysis theory, fractional calculus, fixed point technique, and the theory of a resolvent operator. Further, we extend the proposed fractional-order system to the nonlocal differential system in which we consider the nonlocal initial conditions, and briefly discuss the exact null controllability result. At the end, we provide two examples to demonstrate the applicability of the abstract results.

This article deals with the existence and approximate controllability results for a class of nonautonomous second-order evolution systems with infinite delay. To establish sufficient conditions for the proposed control problem the theory of evolution operator with Schauder’s fixed point theorem is used. Further, we extend the approximate controllability results to the integro-impulsive differential system. Finally, to emphasize our theoretical concepts, an example is provided.

This paper is concerned with the existence and controllability results for a class of Hilfer fractional differential equations of Sobolev-type with non-instantaneous impulse in Banach space. In order to bring off the main results, the author used the theory of propagation family {P(?)}??0 (generated by the operator pair (A,R)), measure of non-compactness, and the fixed point methods. The primary goal of this study is to determine the controllability of a dynamical system without assuming that R?1 is a bounded operator, and no relationship between the domain of the operators A and R. At the end, we provide an example to illustrate the main results.