Approximate controllability of nonlinear stochastic evolution systems with time-varying delays☆
Introduction
Random differential and evolution delay systems play an important role in characterizing many social, physical, biological and engineering problems. Evolution delay differential systems arises in many areas of applied mathematics. Semigroup theory gives a unified treatment of a wide class of stochastic parabolic, hyperbolic and functional integrodifferential equations, and much effort has been devoted to study controllability results for such evolution equations. In deterministic cases, fixed point techniques, among other methods are widely used as a tool for studying controllability of nonlinear systems. Several authors have extended finite dimensional controllability results to infinite dimensional controllability results represented by evolution equations with bounded and unbounded linear operators in Banach space (see the early survey by Balachandran and Dauer [1] and the references therein).
Stochastic control theory is a stochastic generalization of classical control theory. Controllability of linear stochastic systems is a well-known problem discussed in the literature [2], [3], [4], [5], [16], [17], [18], [19], [20]. Controllability results in infinite dimensional cases have been studied by several authors by the assumption that the semigroup , associated with the linear convolution operator has a bounded inverse operator with values in . Controllability of semilinear stochastic evolution equations with time delays has been studied by Balasubramaniam and Dauer [6] by using Carathéodory successive approximate solutions. Recently Balasubramaniam and Ntouyas [7] studied controllability of neutral stochastic functional differential inclusions with infinite delay in abstract space. In all the above cases to ensure the controllability result for the nonlinear systems the solution mapping must has only one solution for each admissible control u. For this purpose some assumptions such as the nonlinear function f be Lipschitz or strongly monotone must be imposed on the nonlinear terms in the systems and the linear operator should generate a compact evolution system or the consideration must be in the finite dimensional space. In this paper, we study the controllability problem of a general stochastic nonlinear evolution system with preassigned responses, which greatly extends the results in deterministic cases by Bian [8] to the stochastic settings with time-varying delays.
Consider the following nonlinear stochastic evolution system with time-varying delayswhere generates an evolution system in a real separable Hilbert space H, the state takes its values in H and the control function u is given in , an Hilbert space of admissible control functions with U as a separable Hilbert space. Here w is an H-valued Wiener process associated with a positive, nuclear covariance operator Q, f is an H-valued map defined on , g is an -valued map defined on (where K is a real separable Hilbert space and is the space of all bounded, linear operators from K to H, we write simply if ). The delays are continuous functions on .
In this paper, the multiple time-varying nonlinear system (1) is considered by taking the state to either equal or approximate to a preassigned response and without supposing that there exist associated linear systems. The idea is to fix a desirable state (preassigned response) and then to determine a control u so that the corresponding solution of Eq. (1) is approximately equal to desirable state. That is, the controllability problem is transformed into a functional equation problem. It is assumed that the preassigned response is an affine function and the nonlinear functions f and g satisfies some coercivity condition with respect to u. Further, no compactness assumptions are made; instead a generalized measure of noncompactness is used to get the controllability results in stochastic settings.
Section snippets
Preliminaries
For details of this section, the reader may refer [9], [10], [11] and the references therein.
Throughout this paper, and denote real separable Hilbert spaces. Let be a complete probability space equipped with a normal filtration . An H-valued random variable is an -measurable function and a collection of random variables is a stochastic process. For brevity, we shall henceforth suppress the dependence on and write in the place
Controllability of time-varying system
In this section, it is assumed that and is almost everywhere continuous from t and is the Hausdorff measure of noncompactness on H. Further, the following conditions are assumed to hold for and A:
- (C1)
all compact subsets and bounded subsets . Here are nonnegative.
- (C2)
There exist positive constants such that for every
Example
Consider the following nonlinear distributed parameter control system:Here is a bounded open set in , is standard one-dimensional Wiener process. and are nonlinear functions, measurable with respect to y, almost everywhere continuous with respect to t, and continuous in the last
References (20)
Controllability of linear stochastic systems in Hilbert spaces
J. Math. Anal. Appl.
(2001)- et al.
Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space
J. Math. Anal. Appl.
(2006) - et al.
Existences, uniqueness and asymptotic behaviour of mild solutions to stochastic functional differential equations in Hilbert spaces
J. Differential Equations
(2002) - et al.
Controllability of nonlinear systems in Banach spaces: a survey
J. Optim. Theory Appl.
(2002) Introduction to Stochastic Control Theory
(1970)- et al.
Controllability of stochastic linear systems
Syst. Control Lett.
(1982) Controllability of stochastic linear systems
Syst. Control Lett.
(1991)- et al.
Controllability of semilinear stochastic evolution equations with time delays
Publ. Math. Debrecen
(2003) Controllability of nonlinear evolution systems with preassigned responses
J. Optim. Theory Appl.
(1999)- et al.
Controllability of semilinear stochastic evolution equations in Hilbert space
J. Appl. Math. Stoch. Anal.
(2001)
Cited by (15)
Approximate controllability of mixed stochastic VolterraFredholm type integrodifferential systems in Hilbert space
2011, Journal of the Franklin InstituteCitation Excerpt :Controllability of linear stochastic systems is a well-known problem in the literature (see [1] and the references therein). Only few of the authors have studied the approximate controllability of nonlinear stochastic differential systems (see [3,8,12,14,15,18]). If the semigroup is compact, then assumptions (H2) in [5] and (A2) in [7]are valid if and only if the state space is finite dimensional.
Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay
2011, Journal of the Franklin InstituteCitation Excerpt :As one of the important topics in mathematical control theory, controllability plays an important role both in many branches of physics and technical sciences. In recent years, the problem of controllability for various kinds of functional differential and integrodifferential systems including delay systems in Banach spaces has been extensively studied by many researchers [9–15], and the references therein. Controllability of fractional differential systems in Banach space has been discussed in [16].
Properties of value function and existence of viscosity solution of HJB equation for stochastic boundary control problems
2011, Journal of the Franklin InstituteCitation Excerpt :The stochastic polynomial systems have been considered in Basin, Calderon-Alvarez [7]. For some problems of stochastic controllability, the papers of Muthukumar and Balasubramaniam [25] and Balachandran and Karthikeyan [3] considered them. The paper of Tiana, Yue and Peng [28] concerned with the random delay problem.
Control of time-delayed linear differential inclusions with stochastic disturbance
2010, Journal of the Franklin InstituteCitation Excerpt :The authors of [10] applied the convex hull Lyapunov function to investigate the LDI suffered from the disturbance with unit peak or finite energy. But in real systems, the disturbance may include stochastic disturbance, and then the research of stochastic systems has gradually become a hot topic in the area of control theory [11–16]. This paper deals with the stochastic disturbance which can be described by the Gaussian white noise.
Stability analysis of heat flow with boundary time-varying delay effect
2010, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :Recently, the stability of partial differential equations (PDEs) and controllability of stochastic PDEs with varying delay effect were analyzed in [5–7].
Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects
2010, Journal of the Franklin InstituteCitation Excerpt :Thus, stochastic differential equations, appear as a natural description of several observed phenomena of real world. In recent years, the stability and control theory investigation of stochastic impulsive differential equations has received an increasing interest (see [11–13] and references therein), and similar to stochastic delay differential equations (see [14–21] and references therein). However, as far as we know, there are few results about the stability of stochastic differential delay equations with impulsive effects (see [22–27] and references therein).
- ☆
The work was supported by UGC-SAP(DRS), Govt. of India, New Delhi, under sanctioned no. F.510/6/DRS/2004(SAP-1).