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 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 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.     

Approximate controllability of semilinear functional equations in Hilbert spaces

In this paper approximate and complete controllability for semilinear functional differential systems is studied in Hilbert spaces. Sufficient conditions are established for each of these types of controllability. The results address the limitation that linear systems in infinite-dimensional spaces with compact semigroup cannot be completely controllable. The conditions are obtained by using the Schauder fixed point theorem when the semigroup is compact and the Banach fixed point theorem when the semigroup is not compact.     

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 of Neutral Stochastic Functional Differential Systems with Infinite Delay

In this article, we consider a class of control systems governed by the neutral stochastic functional differential equations with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.     

Controllability of semilinear stochastic functional integrodifferential systems in Hilbert spaces

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.     

Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay

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.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

On the approximate controllability for fractional evolution hemivariational inequalities

In this paper, we focus on the approximate controllability of control systems described by a large class of fractional evolution hemivariational inequalities. Firstly, we introduce the concept of mild solutions and present the existence of mild solutions for this kind of problems. Next, we show the approximate controllability of the corresponding linear control system. Finally, the approximate controllability of the fractional evolution hemivariational inequalities is formulated and proved under some appropriate conditions. An example demonstrates the applicability of our results. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.     

Approximate controllability of non-linear evolution systems with time-varying delays

^** Email: balachandran_k{at}lycos.com In this paper, the approximate controllability of non-linearevolution time-varying delay systems with preassigned responsesis studied. These controllability results are for non-linearsystems that are not associated with linear systems and no compactnessassumption is imposed.     

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.     

On the Approximate Controllability for Hilfer Fractional Evolution Hemivariational Inequalities

The paper is concerned with the approximate controllability of some Hilfer fractional evolution hemivariational inequalities. Using two classes of operators and their fundamental properties, we derive sufficient conditions for approximate controllability of linear and semilinear controlled systems via a fixed point theorem for multivalued maps. Finally, an example is given to illustrate our theory.     

Approximate controllability for a class of hemivariational inequalities

In this paper, we deal with the approximate controllability for control systems described by a class of hemivariational inequalities. Firstly, we introduce the concept of mild solutions for hemivariational inequalities. Then the approximate controllability is formulated and proved by utilizing a fixed-point theorem of multivalued maps and properties of generalized Clarke subdifferential.     

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.     

On controllability of semilinear stochastic systems in Hilbert spaces

The Nussbaum fixed-point theorem together with conditions forapproximate controllability of linear systems are used to obtainsufficient conditions for approximate controllability of associatedsemilinear stochastic systems in Hilbert spaces.     

APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL BILINEAR SYSTEMS

1.IntroductionRecentlymanyresultshavebeenobtainedfordistributedbilinearsystemsl3'7'8].In[3]thecontrollabilityofabilinearsystem%=Aw(t) p(t)Bw(t)wasstudied,whereAistheindnitesimalgeneratorofaCosemigroupofboundedlinearoperatorsT(t)onaBanachspaceX.B:X-- 5isaboundedlinearoperator,andpELI([0,T];R)isacontrol.TheconditionsforelementsofEtobeaccessiblefromagiveninitialstatecoweregiven.Itisclearthattheystudiedthebilinearsysteminaspecialcase.In[8]theystudiedthesysteminamorespecialcasebecausetheyass…