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Our main concern is the existence of a new P C 2 - v -mild solution for Hilfer fractional impulsive evolution equations of order α ∈ (1 , 2) and β ∈ [ 0 , 1 ] as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of P C 2 - v -mild solution for the objective equations by the Laplace transform and probability density function. Secondly, we rely on the Banach contraction mapping principle to discuss a new existence and uniqueness result of P C 2 - v -mild solution when the sine family is compact. Thirdly, a sufficient condition for the approximate controllability result of impulsive evolution equations is formulated and proved under the assumptions that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximately controllable. Finally, two examples are given to illustrate the validity of the obtained results in the application. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximate controllability of semilinear differential inclusion featuring multivalued nonlinearities with nonconvex values. Initially, we explore the approximate controllability result by assuming the underlying state space to be a super-reflexive Banach space. Then, we relax the super-reflexivity condition and prove the approximate controllability in reflexive Banach spaces by employing the relaxation technique. Incorporating multivalued maps with nonconvex values adds complexity to the studied differential inclusions yet enhances their practical applicability, serving as the core motivation for this research endeavor. We conclude by presenting an illustrative example that satisfies the criteria outlined in this paper. [ABSTRACT FROM AUTHOR]

This paper focuses on the approximate controllability of Sobolev-type fuzzy Hilfer fractional neutral integro-differential inclusions involving the Clarke subdifferential. We establish the approximate controllability of this inclusion problem using fuzzy set theory, Sobolev-type methods, Fractional calculus, and Clarke subdifferentials. These techniques help to derive the necessary conditions for approximate controllability of the system. Dhage’s fixed point theorem is applied to prove the results related to approximate controllability. Finally, an example is provided to illustrate the discussion. [ABSTRACT FROM AUTHOR]

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order 1<μ<2$$ 1<\mu <2 $$ using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings. [ABSTRACT FROM AUTHOR]

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We study a class of dynamic control systems described by nonlinear fractional stochastic degenerate evolution equations in Hilbert spaces. We investigate the possibility of replacing a degenerate Cauchy problem by an equivalent problem in the factor space H ⊥ = H \ Ker (L) . The core of this paper is to factorize the degenerate Cauchy problem, to derive a mild solution of stochastic system based on the perturbation theory for linear operators and to study approximate controllability results. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic degenerate evolution equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic degenerate control system under the assumption that the corresponding linear system is approximately controllable. As an application, the abstract results are illustrated by the application in complex media electromagnetic. In addition, the fractional stochastic partial differential equations (fractional SPDEs) are discussed and approximate controllability results are proved by verifying main assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we study approximate controllability of neutral hemivariational inequality with impulses. Initially, we derive the mild solution for the given system, for the main results, we use semigroup theory, fixed point theorem of multivalued map, and properties of generalized Clarke's subdifferentials to prove its existence and approximate controllability results. Finally, an example is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

Stochastic differential equations with Poisson jumps become very popular in modeling the phenomena arising in various fields, for instance in financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. The objective of this paper is to investigate the approximate controllability for a class of control systems represented by second-order stochastic differential equations with time delay and Poisson jumps. The main technique is the theory of fundamental solution constructed through Laplace transformation. By employing the so-called resolvent condition, theory of cosine operators and stochastic analysis, we formulate and prove some sufficient conditions for the approximate controllability of the considered system. In the end an example is given and discussed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The primary focus of this article is to explore the existence and controllability of a new class of Sobolev-type Hilfer fractional stochastic Volterra-Fredholm integrodifferential systems in Hilbert spaces. Our main findings are grounded in ideas from fractional calculus, stochastic analysis, cosine families, and multivalued maps. To begin, we use Bohnenblust-Karlin's fixed point theorem to examine the existence of mild solutions for the given problem. Furthermore, we introduce a novel set of sufficient conditions for the considered nonlinear Hilfer fractional stochastic differential systems, and assuming that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory. [ABSTRACT FROM AUTHOR]

In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.

The primary aim of this article is to explore the approximate controllability of second-order impulsive hemivariational inequalities with initial conditions in Hilbert space. The mild solution was initially derived using the properties of the cosine and sine family of operators, Clarke's subdifferential, and the fact that the related linear equation has an evolution operator. The results of the approximate controllability of the considered systems are then taken into account using the fixed-point theorem method. An application is provided to support our theoretical findings.

In this paper, we discuss the approximate controllability of control problems governed by evolution hemivariational inequalities in super-reflexive Banach spaces by preassuming the approximate controllability of the associated linear system. We first show that the original problem is connected with a differential inclusion problem involving the Clarke subdifferential operator, and then we prove the approximate controllability of the original problem via the approximate controllability of the differential inclusion problem. The paper offers a unique solution to a challenge introduced by assuming the underlying space X as a super-reflexive Banach space, which presents issues of convexity due to the nonlinear nature of the duality map involved in the expression of the control. Such issues are not present when X is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for approximate controllability of the evolution hemivariational inequality in which X is a super-reflexive Banach space. [ABSTRACT FROM AUTHOR]

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form M(t) Xw(t) with M(t) ≥ 0 and w(t) with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude M(t) = M. In contrast, if the moving control set w(t) is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability problem is discussed for second-order neutral differential inclusions with damping in Banach spaces. The study establishes the existence of mild solutions for the given system by employing sine and cosine operators and applying Bohnenblust-Karlin’s fixed point theorem for multivalued maps. By introducing a new set of sufficient conditions, it has been proven that the given system is approximately controllable. An example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

This article explores the existence and approximate controllability of integral solutions for Hilfer fractional evolution equations in a non-dense domain. Leveraging the well-known generalized Banach contraction theorem, we establish both the existence and uniqueness of the integral solution. Furthermore, we adopt a sequential approach to derive results related to approximate controllability, without relying on the compactness of semigroups or the uniform boundedness of nonlinear functions. To validate our findings, we present and discuss an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we show the approximate controllability for a class of semilinear fractional stochastic systems in abstract space with the Riemann–Liouville fractional derivative. The key of the proof is the existence of the mild solution for the proposed problem. These results are based on new properties of the operator obtained by the subordination principle, compact semigroup and Schauder fixed point theorem. Here we obtain the compactness of the solution operator by using Arzelà–Ascoli theorem. As an application, we establish the approximate controllability of the stochastic Rayleigh–Stokes problem for a generalized second grade fluid. [ABSTRACT FROM AUTHOR]

In this paper, we aim to find a mild solution for a delay control system described by nonlinear fractional evolution differential equations in Banach spaces while being subjected to nonlocal conditions. Further, we explore the sufficient conditions for the approximate controllability of the nonlinear fractional control system, assuming that the associated linear system is approximately controllable. We provide some applications at the end to demonstrate our proposed results. [ABSTRACT FROM AUTHOR]

In this work, we study the approximate controllability problem for a system of nonlocal integro-differential equations with impulsive effects. We start by investigating the existence and uniqueness of solutions for this system. The results are derived using the theory of resolvent operators combined with fixed point theory in a generalized Banach space. Next, we examine approximate controllability without necessarily requiring the nonlinear terms to be uniformly bounded. In particular, we do not impose here the compactness condition for either the resolvent operator or the state-dependent function in the nonlocal condition, as is commonly found in the literature. Finally, we provide an example to demonstrate the abstract results of this work. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to (∗) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system (∗) in terms of the resolvent sequence { S α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

The primary aim of this article is to explore the approximate controllability of second-order impulsive hemivariational inequalities with initial conditions in Hilbert space. The mild solution was initially derived using the properties of the cosine and sine family of operators, Clarke's subdifferential, and the fact that the related linear equation has an evolution operator. The results of the approximate controllability of the considered systems are then taken into account using the fixed-point theorem method. An application is provided to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is investigated in the sense of integral solution in Hilbert spaces. Some sufficient and necessary conditions are obtained. Firstly, the existence and uniqueness of integral solutions of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions are considered by GE-evolution operator theory and Sadovskii's fixed point theorem, the existence and uniqueness theorem of solutions is given. Secondly, the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is studied in the sense of integral solution. The criterion for approximate controllability is provided. The obtained results have important theoretical and practical value for the study of controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1 < q < 2 . Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the approximate controllability for a class of non-autonomous delayed integrodifferential inclusions with unbounded delay. Our technique starts with the search for the optimal control for a linear quadratic regulator problem. The existence of such an optimal control aids to establish sufficient conditions insuring our inclusion problem's approximate controllability. The findings we acquired represent a generalization and extension of previous results on this topic. Finally, we present an example to illustrate the abstract theory. [ABSTRACT FROM AUTHOR]

This paper aims to investigate a new class of Atangana-Baleanu fractional stochastic differential systems under the influence of the Rosenblatt process and non-instantaneous impulses. We analyze the existence and uniqueness of piecewise continuous mild solutions by using the fixed point method, fractional calculus, and resolvent family. Further, we define a new piecewise control function and investigate the approximate controllability results for the proposed problem. Finally, the main results are validated with the aid of an example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability for Hilfer fractional stochastic hemivariational inequalities of order 1 < μ < 2 in Hilbert spaces. Initially, we define the concept of a mild solution for our problem in terms of fractional calculus, cosine families, stochastic analysis, and generalized Clarke subdifferential. Then, the existence and approximate controllability for Hilfer fractional stochastic evolution hemivariational inequalities are formulated and proven under appropriate conditions using fixed point theorems for multivalued maps. Finally, an example is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the existence of mild solutions and approximate controllability for a class of fractional semilinear systems with instantaneous and non-instantaneous impulses. By applying the Kuratowski measure of noncompactness and $ \rho $-set contractive fixed-point theorem, the results for the considered system were obtained. In the end, an example was studied to support the main results.

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present the existence of mild solutions and approximate controllability for a class of non-autonomous measure driven evolution systems with nonlocal conditions in Banach spaces. Firstly, we obtain the existence of mild solutions for the concerned problem by the semigroup theory and Schauder's theorem. Secondly, some sufficient conditions of approximate controllability are proved. At the end, an example is also given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the existence of mild solutions and approximate controllability for a class of fractional semilinear systems with instantaneous and non-instantaneous impulses. By applying the Kuratowski measure of noncompactness and ρ -set contractive fixed-point theorem, the results for the considered system were obtained. In the end, an example was studied to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the sufficient conditions for approximate controllability and null controllability of neutral delay stochastic Hilfer fractional integrodifferential system with Rosenblatt process are established. The required results are obtained based on a fixed point technique, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type 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. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev-type hemivariational differential systems of order r ∈ (1, 2) with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed-point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of solutions to Hilfer fractional neutral delay integro‐differential equations subject to nonlocal conditions are discussed in this study. The main results of this study are based in part on the fixed point techniques of Banach contraction and Krasnoselskii's fixed point theorem from calculus theory. First, we determine whether or not the fractional system has a mild solution. The uniqueness of the mild solution is further illustrated by expanding our results. The optimal control problems are governed by a new class of neutral delay integro‐differential equations in Banach spaces, and we also develop sufficient conditions for the approximate controllability of a nonlinear fractional system. An example is given at the end to strengthen the compatibility of the results. [ABSTRACT FROM AUTHOR]

In this paper we show that bilinear wave equations (on a torus of arbitrary dimension) are globally approximately controllable in any positive time, when the initial state of the system has a finite number of active modes. This is in contrast to the global approximate controllability of bilinear Schrödinger equations where the minimal time may be arbitrarily large. The control technique we adopt is explicit and based on a small-time limit of conjugated dynamics to displace the system along non-directly accessible directions (a.k.a. Lie brackets of the generators). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]