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This manuscript concerns the approximate controllability results in Hilbert space of Hilfer fractional differential inclusions. We show that the Hilfer fractional neutral differential inclusions are approximately controllable using Bohnenblust-Karlin's fixed point theorem. Further, nonlocal conditions are extended to the existence results. Furthermore, an application is shown to demonstrate the theoretical conclusions utility. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the approximate controllability of Hilfer fractional neutral evolution hemivariational inequalities. Using fractional calculus, the theory of operators semigroup and the probability density function, we first construct a new C 1 - β mild solution for Hilfer fractional differential inclusion. Secondly, we prove the approximate controllability of Hilfer fractional evolution hemivariational inequalities to linear and semilinear systems using characteristic solution operators and fundamental features via a fixed point theorem for multi-valued mappings. Finally, two examples are provided to demonstrate our theory. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability of Sobolev‐type Hilfer neutral fractional stochastic differential inclusions in Hilbert spaces is considered. By using the stochastic analysis theory, fractional calculus, multivalued maps, and Karlin's fixed point technique, a new set of necessary and sufficient conditions, are formulated which guarantees the approximate controllability of the nonlinear Hilfer fractional stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, two examples are presented to illustrate the theory of the obtained results. [ABSTRACT FROM AUTHOR]

In this manuscript, our main focus is about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2. With the utilization of the fractional calculations, the findings and facts associated with multivalued maps, the fixed‐point method, and the principal discussions are demonstrated. First, we focus on the approximate controllability of our system. Lastly, we provide an application for the demonstration of theoretical results. [ABSTRACT FROM AUTHOR]

Abstract The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall’s inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated.

This paper is devoted to a class of nonlocal stochastic differential equations involving conformable fractional time derivative operator. Existence of mild solution for the nonlocal conformable stochastic differential equation is proved. Sufficient conditions for controllability and approximate controllability of the nonlocal conformable stochastic differential equation with impulsive conditions are established. An example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this article, we consider the second‐order nonautonomous integro‐differential system with finite delay and noninstantaneous impulse. The aim of this paper is to establish the approximate controllability result for the proposed control problem. We use the evolution operator and Schauder's fixed‐point approach for proving our main result. Also, we provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index 0 < H < 1 / 2 . First, to overcome the irregular or singular properties of fBm with Hurst parameter 0 < H < 1 / 2 , we define a new type of control function. Then, by virtue of the stochastic analysis theory, inequality technique, the semigroup approach, Krasnoselskii's fixed-point theorem and Schaefer's fixed-point theorem, we derive two new sets of sufficient conditions for the existence and approximate controllability of the concerned system. In the end, a concrete example is worked out to demonstrate the applicability of our obtained results. [ABSTRACT FROM AUTHOR]

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of C 0 -semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we deal with fractional neutral evolution systems of hyperbolic type in Banach spaces. We establish the existence and uniqueness of the mild solution and prove the approximate controllability of the systems under different conditions. These results are mainly based on fixed point theorems as well as constructing a Cauchy sequence and a control function. In the end, we give an example to illustrate the validity of the main results. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1 < r < 2. Using principles and ideas from stochastic analysis, the theory of cosine family, fractional calculus, and Banach fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions and then demonstrate the approximate controllability of the fractional stochastic control equation. We then apply our findings to the theory of nonlocal conditions. At last, an application is established for drawing the theory of the main results. [ABSTRACT FROM AUTHOR]

In this work, we offer a set of appropriate conditions for the approximate controllability of second‐order impulsive neutral stochastic integro‐differential evolution inclusions with infinite delay. We validate the major conclusions using concepts from the cosine function, sine function, and the fixed point method. The discussion then shifts to nonlocal second‐order stochastic integro‐differential systems. Finally, we give a theoretical application to help our conversation be more successful. [ABSTRACT FROM AUTHOR]

In our article, we are primarily concentrating on approximate controllability results for nondensely defined fractional neutral stochastic differential systems. By applying the results and ideas belongs to fractional calculus, multivalued maps 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 infinite delay. In the end, we present an example to demonstrate the theory. [ABSTRACT FROM AUTHOR]

This work is to study a control problem for a nonlinear Volterra nonautonomous evolution inclusion subjected to a local initial condition. The evolution inclusion involves a family of (possibly unbounded) operators. It is proved that the solution set for the control problem has Aronszajn-type regularity (also called Rδ-structure). Moreover, the reachability set for the control problem is invariant under nonlinear perturbations. Then, we obtain that the control problem is approximately controllable under some supplementary conditions. We finally present an example to illustrate the applicability of the abstract results. [ABSTRACT FROM AUTHOR]

The goal of this research is to look at some of the sufficient conditions for approximate controllability in nonlinear resolvent integro-differential evolution control systems. We have considered that nonlinear term is satisfying Lipschitz continuity. To show the key results, we employ Gronwall’s inequality, semigroup theory, and the resolvent operators. The main results have been discussed under two sets of assumptions. Application of common fixed point theorems such as Banach, Schauder, Sadovskii, etc. is avoided as discussed earlier by several researchers in the available literature. Finally, one case study based on the proposed problem is discussed in order to verify the theoretical findings. Approximate controllability, Gronwall’s inequality, Integro-differential system, Resolvent operators [ABSTRACT FROM AUTHOR]

This article investigates the issue of optimal control and approximate controllability results for fractional integrodifferential evolution equations with infinite delay of r∈(1,2) in Banach space. In the beginning, we analyze approximate controllability results for fractional integrodifferential evolution equations using the fractional calculations, cosine families, and Banach fixed point theorem. After, we developed the continuous dependence of the fractional integrodifferential evolution equations by using the Henry–Gronwall inequality. Furthermore, we tested the existence of optimal controls for the Lagrange problem. Lastly, an application is presented to illustrate the theory of the main results. [ABSTRACT FROM AUTHOR]

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. We focus on a specific category of nonlocal integrodifferential equations. We apply the theory of the resolvent operator in the sense of Grimmer, as well as the fixed point strategy and the theory of the Lebesgue-Stieljes integral, in the context of the space of regulated functions. In light of this, the prevalence of our findings is greater than that which is found in the literature. At last, an example is comprised that exhibits the significance of developed theory. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and approximate controllability outcomes for Hilfer fractional neutral evolution equations. To begin, we explore existence outcomes using fractional computations and Banach contraction fixed point theorem. In addition, we illustrate that a neutral system with a time delay exists. Further, we prove the considered fractional time-delay system is approximately controllable using the sequence approach. Finally, an illustration of our main findings is offered. [ABSTRACT FROM AUTHOR]

This paper presents approximate controllability results for impulsive stochastic integro-differential systems with statedependent delay in a Hilbert space. The use of the resolvent operator in the sense of Grimmer, as well as stochastic analysis techniques, yields a new set of results. Finally, an example is given to show how the theory that has been worked out can be put into practice. [ABSTRACT FROM AUTHOR]

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that b > 0 and α - τ c 2 b > 0 , we show that if 0 < s < 1 and Ω ⊂ R N ( N ≥ 1 ) is a bounded domain with a Lipschitz continuous boundary ∂ Ω , then there is no control function g such that the following system τ u ttt + α u tt + c 2 (- Δ) s u + b (- Δ) s u t = 0 in Ω × (0 , T) , u = g χ O in (R N \ Ω) × (0 , T) , u (· , 0) = u 0 , u t (· , 0) = u 1 , u tt (· , 0) = u 2 in Ω , is exactly or null controllable in time T > 0 . However, we prove that for 0 < s < 1 , the system is approximately controllable for every g ∈ H 1 ((0 , T) ; L 2 (O)) , where O ⊂ R N \ Ω ¯ is an arbitrary non-empty open set. [ABSTRACT FROM AUTHOR]

We prove in this paper the global approximate controllability of the 1-D Boussinesq equation-subjected to internal control and free boundary conditions-on a bounded domain. The key ingredients of the proof relies Coron's return method for the exact global controllability of the nonlinear control system y t t + (y 2) x x = u (t) y t t + (y 2) x x = u (t) , combined with some priori estimates for nonlinear weak-hyperbolic systems defined respectively in Gevrey class of functions, and in Sobolev spaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the approximate controllability of fractional evolution inclusions with hemivariational inequalities of order 1 < r < 2 $1< r

In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed.

This article is primarily targeting the approximate controllability results of Atangana–Baleanu neutral fractional stochastic hemivariational inequality. The primary conclusions were validated using principles and ideas from stochastic analysis, fractional calculus, multivalued map theory, and fixed point techniques. We begin by emphasizing the existence of mild solutions, and then demonstrate the fractional control system’s approximate controllability. Our findings are then applied to the notion of nonlocal circumstances. At last, an example is included to show the applicability of our results.

This paper is devoted to studying the approximate controllability for second-order impulsive differential inclusions with infinite delay. For proving the main results, we use the results related to the cosine and sine function of operators, Martelli's fixed point theorem, and the results when combined with the properties of differential inclusions. Firstly, we prove the approximate controllability for second-order impulsive differential inclusions with initial conditions. Then, we extend the discussion to the second-order impulsive system with nonlocal conditions. Finally, we provide an example for the illustration of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

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

This paper includes investigation of approximate controllability results for a multi-term time-fractional stochastic differential system in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard type iterations. The basic tools of fractional calculus, generalization of semigroup, and stochastic analysis techniques have been utilized. Finally, an application is discussed to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

By a procedure of successive projections, the authors decompose a coupled system of wave equations into a sequence of sub-systems. Then, they can clarify the indirect controls and the total number of controls. Moreover, the authors give a uniqueness theorem of solution to the system of wave equations under Kalman's rank condition. [ABSTRACT FROM AUTHOR]

In this paper, we give some qualitative results on the behavior of a nonsimple elastic plate with memory corresponding to anti-plane shear deformations. First we describe briefly the equations of the considered plate and then we study the well-posedness of the resulting problem. Secondly, we perform the spectral analysis, in particular, we establish conditions on the physical constants of the plate to guarantee the simplicity and the monotonicity of the roots of the characteristic equation. The spectral results are used to prove the exponential stability of the solutions at an optimal decay rate given by the physical constants. Then we present an approximate controllability result of the considered control problem. Finally, we give some numerical experiments based on the spectral method developed with implementation in MATLAB for one and two-dimensional problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence of mild solutions and approximate controllability for measure driven evolution system with nonlocal conditions. First, we obtain the existence of mild solutions and approximate controllability for the concerned problem by constructing new Green's function and control function involving Gramian controllability operator. This research relies on Schauder's fixed point theorem and the theory of operator semigroup. At the end, two examples are also given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]

This work studies the approximate controllability of a class of second-order retarded semilinear differential equations with nonlocal conditions and with delays in control. First, we deduce the existence of mild solutions using cosine family and fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz. Controllability of the system is shown using an approximate and iterative technique. The results are illustrated using an example. [ABSTRACT FROM AUTHOR]

This manuscript is mainly focusing on the approximate controllability and the null controllability for fractional neutral stochastic partial differential equations with delay driven by Rosenblatt process. We discuss the sufficient conditions for approximate controllability and null controllability for Hilfer fractional neutral stochastic partial differential equations driven by Rosenblatt process. Finally, we provide two examples to verify the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, noninstantaneous impulsive conformable fractional stochastic delay integro-differential system driven by Rosenblatt process is studied. Sufficient conditions for approximate controllability and null controllability for the considered problem are established. Finally, an example is introduced to explain the obtained results. [ABSTRACT FROM AUTHOR]

Abstract In this article, we are concerned with the existence of mild solutions and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators and nonlocal conditions. The existence results are obtained by first defining Green’s function and approximate controllability by specifying a suitable control function. These results are established with the help of Schauder’s fixed point theorem and theory of almost sectorial operators in a Banach space. An example is also presented for the demonstration of obtained results.

In this manuscript, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and memory in Hilbert space. By using semigroup theory, stochastic analysis techniques and fixed point approach, we derive a new set of sufficient conditions for the approximate controllability of nonlinear stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate our results.

Abstract We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main results, we use fractional calculus, stochastic analysis, fractional power of operators and Sadovskii’s fixed point theorem. At the end, an example is also given to show the applicability of our obtained theory.

We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution remains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control.

This paper mainly focuses on the approximate controllability for second‐order impulsive neutral differential inclusions with infinite delay. By using the results related to cosine and sine function of operators, Dhage's fixed point theorem, and their outcome when combined with the properties of differential inclusions, our primary discussions are proved. Finally, we provided an example for the illustration of the theory obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the topological properties of mild solution sets for a control system described by Hilfer fractional delay evolution equations with nonlocal initial conditions. We firstly obtain the nonemptiness, the compactness and R δ -property of the mild solution set by applying Schauder's fixed point theorem, a fixed point theorem of nonconvex valued maps and the fractional calculus. Then we apply this obtained result to show that the presented control problem has a reachable invariant set under nonlinear perturbations. Furthermore, we also apply the obtained results to characterize the approximate controllability of the presented control problem. Finally, we present an example to illustrate the application of abstract results. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictive assumptions. The sufficient conditions for approximate controllability are derived under the natural assumptions on the nonlinear terms, which is an odd homogeneous and the nonlinear inverse considered as a multivalued mapping is bounded. An example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder's fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h (⋅) ↦ ∫ 0 ⋅ U (⋅ , s) h (s) d s : L 1 ([ 0 , T ] ; Y) → C ([ 0 , T ] ; Y) , where Y is a Banach space and U (⋅ , ⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results. [ABSTRACT FROM AUTHOR]

In the paper, the problems of controllability and approximate controllability are studied for the heat equation wt = 1/ρ (kwx)x + γw, x > 0, t ∈ (0, T), controlled by the Dirichlet boundary condition. Control is considered in L∞(0, T). It is proved that each initial state of this system is approximately controllable to any its end state in a given time T > 0. [ABSTRACT FROM AUTHOR]

In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number of Fourier modes. In order to obtain the irreducibility, we first investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a finite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator, we establish the irreducibility of Kuramoto-Sivashinsky equation. [ABSTRACT FROM AUTHOR]