Approximate Controllability of Semilinear Fractional Stochastic Dynamic Systems with Nonlocal Conditions in Hilbert Spaces

A class of dynamic control systems described by semilinear fractional stochastic differential equations of order 1 < q < 2 with nonlocal conditions in Hilbert spaces is considered. Using solution operator theory, fractional calculations, fixed-point technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for nonlocal approximate controllability of semilinear fractional stochastic dynamic systems is formulated and proved by assuming the associated linear system is approximately controllable. As a remark, the conditions for the exact controllability results are obtained. Finally, an example is provided to illustrate the obtained theory.     

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

Sobolev-Type Fractional Stochastic Integrodifferential Equations with Nonlocal Conditions in Hilbert Space

We prove the existence of mild solution to Sobolev-type fractional stochastic integrodifferential equation with nonlocal conditions in Hilbert space. Also, we study the controllability of Sobolev-type fractional stochastic integrodifferential equations with impulsive conditions. We use uniformly continuous semigroups and fixed point technique for the main results. An example is provided to illustrate the theory.     

Existence of solutions for nonlinear fractional stochastic differential equations

The fractional stochastic differential equations have wide applications in various fields of science and engineering. This paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in Hilbert spaces. Using fractional calculations, fixed point technique, stochastic analysis theory and methods adopted directly from deterministic fractional equations, new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay. Further, we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions. Examples are provided to illustrate the obtained theory.     

Approximate Controllability of Fractional Impulsive Partial Neutral Stochastic Differential Inclusions with State-Dependent Delay and Fractional Sectorial Operators

In this article, the approximate controllability of fractional impulsive partial neutral stochastic differential inclusions with state-dependent delay and fractional sectorial operators in Hilbert spaces is studied. By using the stochastic analysis, the fractional sectorial operators and a fixed point theorem for multi-valued maps combined with approximation techniques, we discuss a new set of su?cient conditions for the approximate controllability of the systems under the mixed Lipschitz and Carathéodory conditions. An example is provided to illustrate the obtained theory.     

Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.     

Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions

In this paper, we investigate the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus, and multivalued analysis. An interesting example is provided to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Approximate Controllability for a Class of Abstract Second-Order Functional Evolution Equations

Results providing sufficient conditions for the approximate controllability of a class of second-order abstract functional evolution equations governed by the generator of a strongly continuous cosine family of linear operators, together with nonlocal initial conditions, are developed. This work extends results included in recent work by Naito, Park, Zhou, and others. The abstract theory is then applied to a parabolic partial integrodifferential equation.     

Existence and Approximate Controllability of Solutions for an Impulsive Evolution Equation with Nonlocal Conditions in Banach Space

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.     

Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay

In this paper we study a kind of second-order impulsive stochastic differential equations with state-dependent delay in a real separable Hilbert space. Some sufficient conditions for the approximate controllability of this system are formulated and proved under the assumption that the corresponding deterministic linear system is approximately controllable. The results concerning the existence and approximate controllability of mild solutions have been addressed by using strongly continuous cosine families of operators and the contraction mapping principle. At last, an example is given to illustrate the theory.     

Controllability of stochastic semilinear functional differential equations in Hilbert spaces

In this paper approximate and exact controllability for semilinear stochastic functional differential equations in Hilbert spaces is studied. Sufficient conditions are established for each of these types of controllability. The results are obtained by using the Banach fixed point theorem. Applications to stochastic heat equation are given.     

Approximate controllability of evolution systems with nonlocal conditions

We study the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linearized equation we obtain sufficient conditions for the approximate controllability of the semilinear evolution equation. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to show the application of our result.     

Approximate controllability of retarded semilinear stochastic system with non local conditions

This paper deals with the approximate controllability of retarded semilinear stochastic system with nonlocal conditions in Hilbert Spaces under the assumption that the corresponding linear system is approximately controllable. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. With this control function, the sufficient conditions for the approximate controllability of the proposed problem in Hilbert Space are established. The results are obtained by using Banach fixed point theorem. Finally, two examples are provided to illustrate the application of the obtained results.     

Approximate controllability of Hilfer fractional differential equations

In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1<α<2 and type 0 ≤ β ≤ 1 is considered. The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations. Finally, an example is presented to illustrate the obtained results.     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.     

Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.     

Some results on the controllability of forward stochastic heat equations with control on the drift

In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations.     

Approximate controllability of impulsive fractional evolution equations of order 1<α<2$$ 1&amp;lt;\alpha &amp;lt;2 $$ with state-dependent delay in Banach spaces

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state-dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear-regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results.