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The main study of this article is approximate controllability results for neutral stochastic hemivariational inequalities in Hilbert spaces. The mild solution is initially derived by using the theories on the semigroup, Clarke's subdifferential operators and fixed point theorem method. Initially we proved the existence of the mild solution for the neutral stochastic hemivariational inequalities and provided the approximate controllability of the considered system. Finally, a simple application also provided to illustrate the theory obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose the approximate controllability analysis for a second semilinear system under nonlocal conditions using monotone nonlinearity as a tool. Monotonicity is a significant feature in many communications applications that use digital-to-analog converter circuits. While non-monotonicity prevents such applications from working, nonlinearity makes them work. Therefore, it is of great interest to study a problem while assuming the nonlinear function is monotonic. Nonlocal conditions have a finer effect on the solution and are additional specifications for the physical measurements beyond the conventional ones. In order to determine the controllability of the considered system, we formulate the control function. In addition to this, we have broadened the scope of the results for the impulsive system. At last, we discuss two examples. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate controllability of Hilfer fractional stochastic differential system involving non-instantaneous impulses with Rosenblatt process and Poisson jumps. By utilising stochastic analysis, semigroup theory, fractional calculus, and Krasnoselskii's fixed point theorem, we prove our primary outcomes. Firstly, we prove the approximate controllability of the Hilfer fractional system. As a final step, we provide an example to highlight our discussion. [ABSTRACT FROM AUTHOR]

The main objective of this work is to investigate the existence of mild solutions and approximate controllability for some integrodifferential evolution equations with nonlocal conditions. Assuming that the linear part is exactly null and approximately controllable, using resolvent operator theory, we provide our main results. An example is given to illustrate the basic results of this work. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on approximate controllability results for fractional Sobolev type Volterra‐Fredholm integro‐differential inclusions of order 1 < r < 2. By applying the results and ideas belongs to the cosine function of operators, fractional calculus and fixed point approach, the main results are established. Initially, we establish the approximate controllability of the considered fractional system, then continue to examine the system with the concept of nonlocal conditions. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

We study a system composed of a free quantum particle trapped in a box whose walls can change their position. We prove the global approximate controllability of the system: any initial pure state can be driven arbitrarily close to any target pure state in the Hilbert space of the free particle with a predetermined final position of the box. To this purpose we consider weak solutions of the Schrödinger equation and use a stability theorem for the time-dependent Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and approximate controllability for non-autonomous impulsive evolution equations with nonlocal conditions in Banach space. The existence of mild solutions and some conditions for approximate controllability of these non-autonomous impulsive evolution equations are given by using the Krasnoselskii's fixed point theorem, the theory of evolution family and the resolvent operator. In particular, the impulsive functions are supposed to be continuous and the nonlocal item is divided into Lipschitz continuous and completely bounded. An example is given as an application of the results. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.

In this article, we study the existence of mild solutions and the approximate controllability for a class of stochastic elastic systems with structural damping and infinite delay in Hilbert spaces. The estimation of the control function is discussed, where the expression of the control function is constructed by the defined resolvent operator. Under this estimate, the existence of mild solutions for this system is obtained by the Schauder fixed point theorem and the stochastic analysis theory, and sufficient conditions for the approximate controllability are formulated and proved by using the so-called resolvent operator type condition. Finally, an example is given to illustrate the applicability of our conclusion.

This paper focuses on the existence and approximate controllability of second-order non-autonomous impulsive stochastic neutral differential systems in Hilbert spaces. First, by using Schauder's fixed-point theorem, stochastic analysis theory, and evolution operators, a new set of sufficient conditions are formulated, and we prove the existence of mild solutions of second-order non-autonomous impulsive stochastic neutral differential systems. Further, the result is extended to study the approximate controllability results for second-order non-autonomous impulsive stochastic neutral differential systems. Then, a set of sufficient conditions are derived for the approximate controllability of second-order non-autonomous impulsive stochastic neutral differential system by assuming the associated linear system is approximately controllable. The results are obtained with the help of Krasnoselskii's fixed point theorem. Finally, an example is given to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

In this article, we study the approximate controllability for a semilinear second-order stochastic neutral evolution equation with infinite delay in Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. Finally, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability of mild solutions for second-order differential systems. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential systems under the assumption that the associated linear part of the system is approximately controllable. In addition, we extend our system with nonlocal conditions. Our research on approximate controllability was also extended by utilizing impulse systems. To demonstrate the theory of the primary outcomes, an application is shown. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of damped elastic systems with initial conditions without the assumptions that the corresponding linear system is approximately controllable. Firstly, the existence of mild solution is obtained by means of contraction mapping principle and operator semigroup theory. Secondly, using the sequence method, a new set of sufficient conditions for approximate controllability of damped elastic systems are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

It is comprehended that the systems without any limitation on their Zeno action are enthralled in a vast class of hybrid systems. This article is influenced by a new category of nonautonomous second‐order measure differential problems with state‐dependent delay (SDD) and noninstantaneous impulses (NIIs). Some new sufficient postulates are created that guarantee the solvability and approximate controllability. We employ the fixed point strategy and theory of Lebesgue–Stieltjes integral in the space of piecewise regulated functions. The measure of noncompactness is applied to establish the existence of a solution. Moreover, the measured differential equations generalize the ordinary impulsive differential equations. Thus, our findings are more prevalent than that encountered in the literature. At last, an example is comprised that exhibits the significance of the developed theory. [ABSTRACT FROM AUTHOR]

This article is mainly focusing on the approximate controllability of neutral integro‐differential stochastic equations with state‐dependent delay. The main results of this article are proved by applying some ideas about the semigroup theory and resolvent operators. At first, the investigation about the existence of a mild solution and then we discuss the approximate controllability of the considered system. Next, we extend our study to the system with non‐local conditions. Finally, a theoretical application is presented for proving the validity of controllability results obtained. [ABSTRACT FROM AUTHOR]

In our article, we primarily concentrate on the approximate controllability results for Sobolev type fractional stochastic evolution hemivariational inequalities. By applying the facts related to fractional calculus, and fixed‐point technique, the principal results are proved. Initially, we are concentrating the existence and continue to prove the controllability of the fractional evolution system. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

Based on the uniqueness of solution to a coupled system of wave equations associated with incomplete internal observations, we establish the approximate internal synchronization by groups, the induced internal synchronization and the approximate internal synchronization in the pinning sense. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer's resolvent operator theory and Schauder's fixed point theorem. We give an example at the end to ensure the compatibility of the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results. [ABSTRACT FROM AUTHOR]

This paper presents the controllability of a family of linear and semilinear systems with control delay in Hilbert spaces. Firstly, the approximate controllability of the linear control delay system is proved by assuming that the linear system without control delay is completely controllable. Then, Nemytskii operators have been constructed associated to control operators and the nonlinear function. The approximate controllability of the retarded semilinear system is established by using the Rothe type fixed point theorem. The applications of results are explained through examples of parabolic and hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to use resolvent operators to investigate the existence and the controllability of a mild solution to a second-order semilinear integro-differential problem. To construct our criterion, we use a fixed point theorem in conjunction with measures of noncompactness. A practical example is used to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system. [ABSTRACT FROM AUTHOR]

The primary focus of this manuscript is on the approximate controllability of second-order semilinear control systems with impulses. There have been two sets of necessary requirements discussed. Combining the theories of the sine and cosine families, as well as the compactness of the cosine operator along with the fixed point technique (FPT) yields the first set of results. The following discussion avoids the apply of the compactness and fixed point approach of the cosine function and is shown instead using Gronwall’s inequality. The existence and uniqueness of the mild solution are also established. We have also provided a case study of theoretical outcomes that have been confirmed the theory.

This manuscript aims to establish a new set of sufficient conditions for the existence and approximate controllability of a neutral‐type fractional integro‐differential system employed with noninstantaneous impulse and finite delay in Hilbert space. In order to bring off the main results, the author used fractional calculus, semigroup theory, and Schauder's fixed‐point theorem. At the end, we provide an example to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper is interested in studying the one-phase Stefan problem. For this purpose, we use the nonlinear sentinel method, which relies typically on the approximate controllability and the Fanchel-Rockafellar duality of the minimization problem, to prove the existence and uniqueness of a solution to this problem. In particular, our research focuses on the application of the nonlinear sentinel method to the single-phase Stefan problem. This approach aids in identifying an unspecified boundary section within the domain undergoing a liquid-solid phase transition. We track the evolution of the temperature profile in the liquid-solid material and the corresponding movement of its interface over time. Eventually, the local convergence used for the iterative numerical scheme is demonstrated. [ABSTRACT FROM AUTHOR]

The article deals with the approximate controllability of Atangana–Baleanu semilinear control systems. The outcomes are derived by applying Gronwall's inequality and Cauchy sequence, and avoid the use of the fixed point theorem. We have also included an example for the validation of theoretical results. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to the approximate controllability of semilinear infinite-delayed stochastic differential inclusions involving impulses. The presence of a mild solution for the aforementioned system is shown by employing the theory of stochastic analysis and the fixed point theorem due to Covitz and Nadler. Then, we construct a group of sufficient conditions and utilize Grownwall's inequality to establish the approximate controllability results. Finally, two examples are constructed to exemplify the feasibility of derived outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type (P; η r; β), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1 < α < 2 with delay in Banach spaces. Firstly, we study the existence of the P C -mild solution for our objective system via some characteristic solution operators related to the Mainardi's Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results. [ABSTRACT FROM AUTHOR]

This article deals with the existence and uniqueness of solutions, as well as the approximate controllability of fractional neutral differential equations (ACFNDEs) with deformable derivatives. The findings are achieved using Banach's, Krasnoselskii's, and Schauder's fixed-point theorems and semigroup theory. Three numerical examples are used to illustrate the application of the theories discussed in the conclusion. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of the mild solutions and the approximate controllability of some integrodifferential evolution systems with multi-valued nonlocal conditions. We use the resolvent operators theory to obtain sufficient conditions for the existence and controllability results. We present a general Kalman's controllability criterion to study the approximate controllability in linear cases. Consequently, we prove the approximate controllability in the nonlinear cases. Some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This artifact introduces the concept of total approximate controllability in order to seek approximate controllability of the impulsive systems at break points in addition to the final point. The chosen system of inspection is a nonlinear fractional differential system affected by non-instantaneous impulses. The system is governed by Riemann-Liouville derivatives of higher-order with fixed lower limits. The appropriate integral-type initial conditions are determined differently depending on the impulsive functions. Firstly, mild solution of the concerned system is constructed, followed by a sufficient set of assumptions required to manifest the existence, uniqueness, and controllability results. Next, the definition of total approximate controllability is interposed. Further, the total approximate controllability of the concerned fractional system is established using Riemann-Liouville fractional resolvent, appropriately defined interval-wise Nemytskii operators, and an iterative technique. Lastly, an illustration is put forward to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

This manuscript is about infinite-delayed semilinear fractional differential systems. The considered system is of weighted type so that the initial condition may be non null at time zero. The principle contribution is the approximate controllability of the considered system. The state space is not required to be reflexive and the non linear function is not supposed to be Lipschitz. More than that, the nonlinear function in this work can involve spatial derivatives which enlarges the area of application of the obtained result. [ABSTRACT FROM AUTHOR]

The article is dedicated towards the study of fractional order nonlinear differential systems with non‐instantaneous impulses involving Riemann–Liouville derivatives with fixed lower limits and appropriate integral type initial conditions in Banach spaces. First, a mild solution of the system is constructed, and subsequently, its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using the concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out corrections in examples presented in previous articles. [ABSTRACT FROM AUTHOR]

The main aim of this article is to focus on approximate controllability for Sobolev‐type fractional control problems in Hilbert spaces without uniqueness. By using the fixed point theorem for multivalued maps with nonconvex values, the main results of our article are proved. Moreover, we obtain some results on the continuity of the solution map and the topological structure of the solution set of the considered Sobolev‐type fractional differential system. Finally, we provide a theoretical application to assist in the effectiveness of our discussion. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order r ∈ (1 , 2). By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result. [ABSTRACT FROM AUTHOR]

The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $ t $ is obtained as a linear combination of the control evaluated at time $ t $ and of the state evaluated at finitely many previous instants of time $ t-\Lambda_1,\dots,t-\Lambda_N $. Based on the realization theory developed by Y. Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $ L^q $ spaces, $ q \in [1, +\infty) $. We also provide a necessary condition for $ L^1 $ exact controllability, which can be seen as the closure of the $ L^1 $ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $ d\max\{\Lambda_1,\dots,\Lambda_N\} $, where $ d $ is the dimension of the state space. [ABSTRACT FROM AUTHOR]

Control systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed‐point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results. [ABSTRACT FROM AUTHOR]

Abstract We consider the approximate controllability for a class of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps in a real separable Hilbert space. Under the sufficient conditions, we obtain approximate controllability results by virtue of the theory of a strongly continuous cosine family of bounded linear operators combined with stochastic inequality technique and the Sadovskii fixed point theorem. Finally, we illustrate the main results by an example.

In this article, we presented some sufficient conditions for the approximate controllability of a class of fractional differential systems with fixed delay in Banach space. The existence of a mild solution is discussed with the help of the fixed point theorem. Controllability results are obtained using generalized Gronwall's inequality, Cauchy sequence, and the basics of functional analysis. As a last note, we have included an example for the illustration of the obtained theory. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the existence and approximate controllability for fractional stochastic Volterra–Fredholm integro-differential systems having order 1 < r < 2. The primary outcomes are obtained by applying concepts and ideas from fractional calculus, multivalued maps, the theory of cosine family, Martelli and Dhage, and Leray–Schauder fixed point techniques. We begin by emphasizing the existence, and then demonstrate the approximate controllability of the considered system. Additionally, we determine the approximate controllability outcomes for the system with infinite delay. At last, an application is established for drawing the theoretical conclusions of primary outcomes. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of mild solutions and approximate controllability for a type of semilinear system with finite time delay, instantaneous, and non-instantaneous impulses. First, we investigate the existence result for the considered system by applying Kuratowski measure of noncompactness and $ \rho $-set contractive fixed point theorem without using Lipschitz conditions, which are stronger than our assumptions for impulses and the nonlinear term. Then, on this basis, the approximate controllability for the considered system is obtained. Finally, an example is worked out to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of Hilfer fractional stochastic evolution inclusion with nonlocal conditions is investigated. By using fractional calculus, semigroups theory, stochastic analysis and the fixed point theorem for multi-valued maps, a new sufficient condition for the existence of mild solution of the Hilfer fractional stochastic system in the space of weighted continuous functions is derived. Furthermore, a novel sufficient condition for the approximate controllability of the Hilfer fractional stochastic evolution inclusion with nonlocal conditions is established. Finally, an example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In the paper, the problems of controllability and approximate controllability are studied for the control system wt = 1/ρ(kwx)x + γw, (√k/ρwx)|x=0 = u, x > 0, t ∈ (0,T), where u is a control, u ∈ L∞(0,T). It is proved that each initial state of the control system is approximately controllable to any target state in a given time T>0. To obtain this result, the transformation operator generated by the equation data ρ, k, γ is applied. The results are illustrated by examples. [ABSTRACT FROM AUTHOR]

This study focused on the approximate controllability results for the Hilfer fractional delay evolution equations of the Sobolev type without uniqueness. Initially, the Lipschitz condition is derived from the hypothesis, which is represented by a measure of noncompactness, in particular, nonlinearity. We also examined the continuity of the solution map of the Sobolev type of Hilfer fractional delay evolution equation and the topological structure of the solution set. Furthermore, we prove the approximate controllability of the fractional evolution equation of the Sobolev type with delay. Finally, we provided an example to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer's resolvent operator theory and Schauder's fixed point theorem. We give an example at the end to ensure the compatibility of the results. [ABSTRACT FROM AUTHOR]