Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

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.     

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.     

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 a delayed semilinear control system with growing nonlinear term

This paper deals with the approximate controllability of a semilinear control system with delay, where the nonlinear term satisfies the linear growth condition. Sufficient conditions for the approximate controllability of a semilinear control system have been established by assuming that the corresponding linear control system with delay is approximately controllable. To prove our results, the Schauder fixed point theorem is applied and instead of a Co-semigroup associated with the mild solution of the system we use the so-called fundamental solution.     

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 second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.     

Approximate Controllability for Integrodifferential Equations with Multiple Delays

This paper considers the approximate controllability for a class of control systems governed by semilinear delay integrodifferential equations with multiple delays. Sufficient conditions for approximate controllability are established by using the Schauder fixed-point theorem. The results obtained improve some analogous existing results. Several examples are provided to illustrate the application of the approximate controllability result.     

Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices

This paper investigates the relative controllability of delay differential systems with linear impulses and linear parts defined by permutable matrices. We use the impulsive delay Grammian matrix to discuss the relatively controllability of impulsive linear delay controlled systems and we use the Krasnoselskii's fixed point theorem to discuss the relatively controllability of impulsive semilinear delay controlled systems. Finally, two examples are presented to illustrate our theoretical results.     

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 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 ontrollability of second order impulsive functional differential system with infinite delay in Banach spaces

This paper studies the approximate controllability of second order impulsive functional differential system with infinite delay in Banach spaces. Sufficient conditions are formulated and proved for the approximate controllability of such system under the assumption that the associated linear part of system is approximately controllable. The results are obtained by using strongly continuous cosine families of operators and the contraction mapping principle. An example is given to illustrate the obtained theory     

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam's novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

Approximate controllability of fractional order semilinear systems with bounded delay

In this paper, sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order. The existence and uniqueness of mild solution of the system is also proved. The results are obtained by using contraction principle and the Schauder fixed point theorem. Some examples are given to illustrate the theory.     

Controllability of semilinear stochastic systems in Hilbert spaces

We study the weak approximate and complete controllability properties of semilinear stochastic systems assuming controllability of the associated linear systems. The results are obtained by using the Banach fixed point theorem. Applications to stochastic heat equation are given.     

Relative controllability of stochastic nonlinear systems with delay in control

The existence of solutions to systems is a natural premise to carry our study about controllability. Under the basic and readily verified conditions to guarantee the existence of the solutions to a system, in this paper, we prove the relative controllability (approximate controllability ) of the stochastic differential systems with delay in control. Sufficient conditions are given firstly for the relative controllability and relative approximate controllability in finite dimensional spaces, and these results are then generalized to infinite-dimensional Hilbert spaces. Finally, examples are given to illustrate the effectiveness of the proposed methods.     

Approximate controllability of second‐order distributed systems

In this paper, we study the approximate controllability of control systems with state and control in Banach spaces and described by a second‐order semilinear abstract differential equation. We compare the approximate controllability of the system with the approximate controllability of an associated discrete system. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximate controllability and regularity for semilinear retarded control systems

We deal with the approximate controllability for semilinear systems with time delay in a Hilbert space. First, we show the existence and uniqueness of solutions of the given systems with the more general Lipschitz continuity of nonlinear operator f fromR ×V toH. Thereafter, it is shown that the equivalence between the reachable set of the semilinear system and that of its corresponding linear system. Finally, we make a practical application of the conditions to the system with only discrete delay.     

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.