Approximate Controllability Results Research Articles

Approximate controllability results of $$\psi$$-Hilfer fractional neutral hemivariational inequalities with infinite delay via almost sectorial operators

Approximate controllability results of $$\psi$$-Hilfer fractional neutral hemivariational inequalities with infinite delay via almost sectorial operators

Approximate Controllability Result for Backward Stochastic Evoluation Inclusions in Hilbert Spaces

In this paper, we study semilinear Backward Stochastic Evolution Inclusion systems in Hilbert spaces. First, we prove the existence of mild solution of the semilinear Backward Stochastic Evolution Inclusion systems using a multivalued fixed point theorem. Then, we obtain the approximate controllability result for semilinear Backward Stochastic Evolution Inclusion systems through the linear systems corresponding to these semilinear Backward Stochastic Evolution Inclusion systems under appropriate conditions. In particular, our study extends the results of the concept of approximate controllability to Backward Stochastic Evolution Inclusion systems.

A New Approach on the Approximate Controllability Results for Hilfer Fractional Stochastic Hemivariational Inequalities of Order $$1<\mu <2$$

A New Approach on the Approximate Controllability Results for Hilfer Fractional Stochastic Hemivariational Inequalities of Order $$1<\mu <2$$

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

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.

Approximate controllability of third order dispersion systems

The current paper focuses on the approximate controllability of Volterra-Fredholm type integrodifferential third order dispersion system. The Banach fixed point theorem and semigroup operator are used to obtain the existence and uniqueness of mild solutions to the considered system. In order to examine the approximate controllability results, a different approach is used, namely, the sequential method.

Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping

Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping

An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order 1 < r < 2 using sectorial operators

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, γ ), 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.

Approximate controllability analysis of impulsive neutral functional hemivariational inequalities

This article mainly focuses on the existence and approximate controllability of impulsive neutral functional hemivariational inequalities. The existence of mild solutions and approximate controllability results are formulated and proved by using the semigroup of operators theory, a fixed point theorem of multivalued maps, and properties of generalized Clarke subdifferential. In the end, some examples are provided to illustrate the applicability of the main results.

An analysis on the approximate controllability of neutral functional hemivariational inequalities with impulses

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.

A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

Approximate controllability of second-order impulsive neutral stochastic differential equations with state-dependent delay and Poisson jumps

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.

Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions☆

The study of approximate controllability results ensures the essential conditions required for a solution. Keeping the importance of the study, we initiate the existence and approximate controllability an Atangana-Baleanu-Caputo (ABC)-fractional order neutral delay integrodifferential stochastic systems with nonlocal conditions. For this purpose, the suggested ABC-fractional order neutral integrodifferential stochastic system is transferred into an equivalent fixed point problem via an integral operator. The operator is then analyzed for boundedness, continuity and equicontinuity. Then Arzela-Ascolli theorem ensures the compact multivalued function of the operator and Martelli’s fixed point theorem is utilized for the existence of the mild solution. Based on the fixed point theorem and Dunford-Pettis theorem joined with ζ -resolvent operators, we develop the approximate controllability results. Finally, an example is given to justify the theoretical results. The achieved results reveal that the proposed method is systematic and suitable for dealing with the stochastic fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

Global Controllability of the Boussinesq System with Navier-Slip-with-Friction and Robin Boundary Conditions

In this paper, we deal with the global exact controllability to the trajectories of the Boussinesq system posed in 2D or 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier-slip-with-friction boundary condition, and a Robin boundary condition is imposed to the temperature. We assume that one can act on the velocity and the temperature on a small part of the boundary. For the proof, we first transform the boundary control problem into a distributed control problem. Then, we prove a global approximate controllability result by adapting the strategy of Coron, Marbach, and Sueur [J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 1625–1673]; this relies on the controllability properties of the inviscid Boussinesq system and the analysis of appropriate asymptotic boundary layer expansions. Finally, we conclude with a local controllability result; as in many other cases, this can be established as a consequence of the null controllability of a linearized system through a fixed-point argument. Our contribution can be viewed as an extension of the results in [J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 1625–1673], where thermal effects were not considered. Thus, we prove that the ideas behind the controllability properties of the Euler system and the well-prepared dissipation technique can be adapted to the present situation. Furthermore, we cover all the classical boundary conditions for the temperature, that is, those of the Robin, Neumann, and Dirichlet kinds.

Approximate controllability of neutral fractional stochastic differential systems with control on the boundary

In this paper, the approximate boundary controllability results for neutral fractional stochastic differential system with fractional Brownian motion in Hilbert space is discussed. The existence mild solution of neutral fractional stochastic differential system with fractional Brownian motion is proved by using fractional calculus, compact semigroup, Schauder fixed-point theorem and stochastic analysis. The sufficient conditions for approximate boundary controllability of this system is established under the corresponding linear system is approximately controllable. Finally, an example is given to show the application of our results.

Atangana–Baleanu Semilinear Fractional Differential Inclusions With Infinite Delay: Existence and Approximate Controllability

Abstract The main focus of this paper is centered around approximate controllability results of Atangana–Baleanu fractional differential systems with infinite delay. Using principles and ideas from the theory of multivalued maps, fractional calculus, and Bohnenblust–Karlin 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 Atangana–Baleanu fractional control system. We then apply our findings to the theory of the neutral system.

Results on the approximate controllability of Atangana-Baleanu fractional stochastic delay integrodifferential systems

The approximate controllability results of Atangana-Baleanu fractional stochastic integrodifferential systems with infinite delay are the topic of this paper. The essential discoveries are developed by employing techniques and theories from stochastic analysis, multivalued map theory, fractional calculus, and fixed point procedure. Before establishing the fractional stochastic control system’s approximate controllability, we illustrate the existence of mild solutions. Ultimately, an example is offered to clarify our main points.

Approximate controllability of the semilinear population dynamics system with diffusion

This paper aims to study the approximate controllability of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the approximate controllability results of the system are obtained using ‐semigroup approach and some other simple conditions on the nonlinear term and operators involved in the model.

A note on existence and approximate controllability outcomes of Atangana–Baleanu neutral fractional stochastic hemivariational inequality

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.

Approximate controllability of delay nonautonomous integro‐differential system with impulses

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.

A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay

In this paper, we investigate the approximate controllability results of Atangana-Baleanu fractional neutral stochastic systems with infinite delay. Using principles and ideas from stochastic analysis, the theory of multivalued maps, fractional calculus, and Bohnenblust-Karlin fixed point theorem, a new set of sufficient conditions are formulated and proved for the approximate controllability of the fractional stochastic control system. We then apply our findings to the theory of nonlocal conditions. Finally, an example is given to illustrate the theory.