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.


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 Yamamoto [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 controllability 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 parabolic, 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.
The heart is considered as a mixing chamber of constant volume given by where V r is the residual volume of the heart and V e is the injection volume.It is assumed that an initial injection is given at the entrance of the heart resulting in a concentration x(t), 0 ≤ t ≤ T , of drug in plasma entering the heart, where T is the duration of injection.Let the time required for the blood to flow from the heart exit to the entrance of the organ be τ > 0, and also let τ be the time required for blood to flow from the exit of the organ to the heart entrance.Then, the drug concentration in the plasma leaving the heart x(•; ω) satisfies the integral equation (see [6]) where and x 1 (l, s; ω) = 0 if s < 0. Here, C is the constant volume flow rate of plasma in the capillary bed and x 1 (l, s; ω) is the concentration of drug in plasma leaving the organ at time s.The mild solutions of such integral equations are of the form in stochastic integral equations.Stochastic delay equations serve as an abstract formulation of many partial differential equations that arise in problems of heat flow in material with memory, viscoelasticity, and many other physical phenomena (for details, see [8,12] and the references therein).The main objective of this paper is to derive controllability conditions for semilinear stochastic delay evolution equations in Hilbert spaces.

Preliminaries. Consider the semilinear stochastic delay evolution equation
where T > 0 and A is a linear operator (in general unbounded), defined on a given Hilbert space X with an infinitesimal generator of an analytic semigroup S(t), t ≥ 0. The state x(•) takes its values in the Hilbert space X, and the control function u(•) is in L 2 (J, U ), the Hilbert space of admissible control functions with U a Hilbert space.B is a bounded linear operator from U into X.
Let K be a separable Hilbert space, and let (Ω, , t , P) be a complete probability space furnished with a complete family of right continuous increasing sigma algebras { t } satisfying t ⊂ for t ≥ 0. The process {w(t), t ≥ 0} is a K-valued, t -adapted Brownian motion with P{w(0) = 0} = 1, and ψ(•) is an X-valued 0 -measurable random variable independent of the Brownian motion w(•).
For any Banach space F , let L 2 (Ω,F) denote the space of strongly measurable, Fvalued, square integrable random variables equipped with the norm topology where E is defined as integration with respect to the probability measure P. Then L 2 (Ω,F) is also a Hilbert space since F is a Hilbert space.Let τ(•) be a continuous nonnegative function on R + and define r = sup{τ(t) , the family of all continuous square integrable stochastic processes and M(I, F) denote the space of t -adapted stochastic processes defined on I, taking values in F , having square integrable norms, that are continuous in t on I in the mean square sense.This is a Banach space with respect to the norm topology (2.3) Assume the following conditions: ] is a Banach space with respect to the graph topology induced by the graph norm (ii) the function f maps X α to X and there exists a constant C > 0 such that (iii) the function g maps X α to L(K, X) and there exists a constant C > 0 such that has an invertible operator W −1 defined on X\ ker W (see [9]) and there exist the positive constants N 1 , N 2 such that Here, L(K, X) is the family of all bounded linear operators from K into X, equipped with the usual operator norm topology, and w is a t -adapted Brownian motion having a nuclear covariance operator By the assumptions (i), (ii), and (iii), there exists a unique stochastic process x(•) ∈ M(I, X α ), that is, a solution of (2.1) (see [1,8]) such that x(•) is t -adapted, measurable, and almost surely that (2.9) Definition 2.1.The stochastic system (2.1) is said to be controllable on J, if for every continuous initial random process ψ(•) defined on [−r ,0], there exists a control u ∈ L 2 (J, U ) such that the solution of (2.1) satisfies x(T ) = x 1 , where x 1 and T are preassigned terminal state and time, respectively.If the system is controllable for all x 1 at t = T , it is called completely controllable on J.

Proof.
Using assumption (iv), define the control Now, it is shown that when using this control the operator defined by has a fixed point.This fixed point is a solution of (2.1).Clearly (Φx)(0) = ψ(0), which means that the control u(•) steers the semilinear stochastic delay differential system from the initial state ψ(•) to x 1 in time T provided the nonlinear operator Φ has a fixed point.First, it must be shown that Φ maps M(I, X α ) into M(I, X α ).Without loss of generality, assume that 0 ∈ ρ(A).Otherwise, if 0 ∉ ρ(A), for the identity operator I add the term νI to A giving A ν = A + νI, then 0 ∈ ρ(A ν ).This simplifies the graph norm to ζ α = A α ζ , for ζ ∈ D(A α ).Since S(t), t ≥ 0, is an analytic semigroup and A α is a closed operator, there exist numbers C 1 ≥ 1 and C α such that and for t ∈ J, where T r Q represents the trace of the operator Q and to complete the proof it remains to show that Φ ∈ C((0, T ), L 2 (Ω,X α )).To accomplish that, let t ∈ (0,T ), h > 0 and t + h ∈ J.For analytic semigroups, there exists a constant ν β > 0 such that and for all β ≥ 0 and all ζ ∈ X with S(t)ζ ∈ D(A β ) for t > 0 (see Pazy [10,Theorem 6.13]).Thus, for t > 0, the closedness of A α and the fact that S(t) commutes with A α on D(A α ) yields that by choosing β > 0 such that 0 ≤ α + β ≤ 1/2, we have for t ∈ (0,T ).Thus, letting h → 0, the desired continuity follows.Hence Φ maps M(I, X α ) into itself.Now, it is shown that for sufficiently small T , defining the interval I leads to a contraction in M(I, X α ).Indeed, for x, y ∈ M(I, X α ) satisfying x(t) = y(t) = ψ(t) for −r ≤ t ≤ 0 it can be easily seen that where Thus, for sufficiently small T , K α < 1 and Φ is a contraction in M(I, X α ) and so, by the Banach fixed point theorem (see [1]), Φ has a unique fixed point x ∈ M(I, X α ).Any fixed point of Φ is a solution of (2.1) on J satisfying (Φx)(t) = x(t) ∈ X, for all ψ(•) and T > 0. Thus, system (2.1) is completely controllable on J.
All the conditions stated in the Theorem 3.1 are satisfied, and so system (4.1) is completely controllable on J.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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