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This paper investigates the existence and approximate controllability (ACA) of fractional neutral-type stochastic differential inclusions (NTSDIs) characterized by non-instantaneous impulses within a separable Hilbert space (HS) framework. Employing the Atangana–Baleanu–Caputo (ABC) derivative, we transform the system into an equivalent fixed-point (FP) problem through an integral operator. Subsequently, the Bohnenblust–Karlin FP theorem is leveraged to establish existence results. By assuming ACA of the corresponding linear system, we derive sufficient conditions for the ACA of the nonlinear stochastic impulsive control system. Our analysis relies on concepts from stochastic analysis, fractional calculus, FP theory, semigroup theory, and the theory of multivalued maps (MVMs). The theoretical findings are illustrated through a concrete example. [ABSTRACT FROM AUTHOR]

The main focus of our discussion is the approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral Volterra–Fredholm integro-differential systems. The important outcomes are obtained by applying concepts and ideas from fractional calculus, Dhage's fixed point theorem, and multivalued maps. Finally, an example is given for deriving theoretical results. [ABSTRACT FROM AUTHOR]

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]

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. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multidimensional heat equation governed by a bilinear control. We first investigate the finite-time stability and null controllability properties. Then, we study the global and partial approximate controllability. These results are further applied to establish the exact controllability of the system at hand. Nonnegative controllability results are also discussed. The approaches are based on linear semigroup theory and the controllability of the heat equation with additive controls. An other essential tool that we use here is the classical maximum principle. [ABSTRACT FROM AUTHOR]

We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green's function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of L²-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived. [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. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate controllability of hemivariational inequalities of the Sobolev-type Hilfer fractional neutral stochastic evolution system. Firstly, by using fixed point theorems, fractional calculus, Hilfer fractional derivatives, multivalued maps and stochastic analysis, for stochastic fractional evolution systems, we demonstrate mild solutions. Whereupon we introduce necessary conditions that ensure the approximate controllability of stochastic fractional systems. As a final step, we define our primary outcomes using an example. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove exact and approximate controllability for a multidimensional wave equation with damping. We show, in a constructive way, that a target state can be reached using two multiplicative controls. [ABSTRACT FROM AUTHOR]

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. • The Atangana–Baleanu neutral fractional stochastic hemivariational inequality is considered. • Stochastic analysis, multivalued map theory, and fixed point techniques are used. • The existence of mild solutions and approximate controllability are emphasized. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the approximate controllability of a parabolic equation with memory by boundary control. The proof relies on the explicit solution of the corresponding homogeneous initial boundary value problem and a duality method. [Copyright &y& Elsevier]

Abstract: In this paper we establish sufficient conditions for the approximate controllability of impulsive neutral functional evolution integrodifferential systems in Hilbert spaces. Also we study the exact controllability of the same system. The conditions are obtained by using Schauder’s fixed point theorem when the operator is compact and the Banach fixed point theorem when the operator is not compact. The results are obtained by using the evolution operators. [Copyright &y& Elsevier]

Abstract: In this paper approximate and exact controllability for semilinear stochastic functional integrodifferential systems are established. The results are obtained by using the Banach fixed point theorem. An example is provided to illustrate the theory. [Copyright &y& Elsevier]

We study the boundary controllability properties of a wave equation with structural damping ytt - yxx - ∈ ytxx = 0, y(0, t) = 0, y(1, t) = h(t) where ∈ is a strictly positive parameter depending on the damping strength. We prove that this equation is not spectrally controllable and that the approximate controllability depends on the functional space in which the initial value Cauchy problem is studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder's fixed point theorem. Finally, we provide two concrete examples to validate our results. • Non-autonomous impulsive systems with state-dependent delay in Banach spaces. • Sufficient conditions for approximate controllability. • Extension of the work by Fu (2017) to separable reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

During the proof of Theorem 1 in the paper “controllability of the wave equation with bilinear controls”, we have worked with a constant λ that satisfies the inequality just before (14) (see Ouzahra, 2014 [3, p. 59] ). However, the right term of this inequality depends implicitly on λ , so the existence of λ is not obvious. The aim of this technical note is twofold: firstly, we will give necessary details that clarify the solvability of the above-mentioned inequality. Secondly, we will show that the step of building such a λ is unnecessary, and thus can be removed from the proof. That way, we will optimize the initial proof of Theorem 1 of Ouzahra (2014) [3] . [ABSTRACT FROM AUTHOR]