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 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 for trajectories of semilinear control systems

We treat an abstract semilinear control system and study the controllability problem for its trajectories. Assuming a range condition of the control action operator and an inequality condition on the system parameters, we can show that the reachable trajectory set of the semilinear system is equivalent to that of its corresponding linear system.The author wishes to express his deep appreciation to Prof. T. I. Seidman for his many helpful suggestions and to Prof. W. Takahashi for many stimulating conversations.     

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 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…     

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.     

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 for fractional diffusion equations by interior control

We consider the approximate controllability by interior control of a partial differential equation with time derivatives of non-integer order. First, we prove the well-posedness of the inhomogeneous problem for the controlled system. Next, we study the dual system and show a weak type of unique continuation property. Finally, we prove the approximate controllability.     

Relative approximate controllability of abstract functional systems with infinite delay and delayed control

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximate controllability of semilinear measure driven systems

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. By using the semigroup theory and Schauder fixed point theorem, sufficient conditions for approximate controllability of measure driven equations are established. The obtained results are a generalization and continuation of the recent results on this issue. Finally, an example is provided to illustrate the application of the obtained results.     

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.     

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 Riemann-Liouville fractional evolution equations with integral contractor assumption

We propose and investigate an evolution system with a Riemann-Liouville fractional derivative. With the aid of a resolvent method, we formulate a suitable notion of solutions to this system and demonstrate the corresponding existence and uniqueness of solutions under a regular integral contractor condition. Furthermore, by applying a space decomposition technique, we exhibit the approximate controllability result of the system. This paper closes with a simple example, which confirms our analytical findings.     

Approximate controllability of a parabolic integrodifferential equation

In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Approximate controllability for trajectories of a delay Volterra control system

For the abstract delay Volterra control system,

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.     

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.     

Exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay

In the present paper, with the help of the resolvent operator and some analytic methods, the exact controllability and continuous dependence are investigated for a fractional neutral integro-differential equations with state-dependent delay. As an application, we also give one example to demonstrate our results.