Controllability of Semilinear Stochastic Delay Evolution Equations in Hilbert Spaces

The controllability of semilinear stochastic delay evolution equations is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. An application to stochastic partial differential equations is given. 1. Introduction. The fixed point technique is widely used as a tool to study the controllability of nonlinear systems in finite-and infinite-dimensional Banach spaces, see the early survey paper by Balachandran and Dauer [5]. Also, Anichini [2] and Ya-mamoto [14] studied the controllability of the classical nonlinear system by means of Schaefer's theorem and Schauder's theorem, respectively. Several authors have extended the finite-dimensional controllability results to infinite-dimensional controlla-bility results represented by evolution equations with bounded and unbounded operators in Banach spaces (e.g., see Balachandran et al. [4] and Dauer and Balasubramaniam [7]). The semigroup theory gives a unified treatment of a wide class of stochastic para-bolic, hyperbolic, and functional differential equations. Much effort has been devoted to the study of the controllability of such evolution equations (Rabah and Karrakchou [11]). Controllability of nonlinear stochastic systems has been a well-known problem and frequently discussed in the literature (e.g., Aström [3], Wonham [13], and Zabczyk [15]). The stochastic control theory is a stochastic generalization of the classical control theory. The purpose of this paper is to consider the controllability of semilinear stochastic delay systems represented by evolution equations with unbounded linear operators in Hilbert spaces. The Banach fixed point theorem (see [1]) is employed to obtain the suitable controllability conditions. The system considered in this paper is an abstract formulation of the stochastic partial differential equation discussed by Liu [8]. For an example, a stochastic model for drug distribution was described in [12]. This model is a closed biological system with a simplified heart, a one organ or capillary bed, and recirculation of the blood with a constant rate of flow, where the heart is considered as a mixing chamber of constant volume. The drug concentration in the plasma in given areas of the system are assumed to be a random function of time. It is further assumed that for t ≥ 0, x 1 (s, t; ω) is the concentration in moles per unit volume at points (represented by s) in the capillary at time t with ω ∈ Ω, the supporting set of a complete probability measure space (Ω,A,P) with A being the σ-algebra and P the probability measure.