Approximate controllability of abstract semilinear stochastic control systems with nonlocal conditions

Abstract: This paper studies the approximate controllability issue of an abstract semilinear stochastic control system with nonlocal conditions. Sufficient conditions are formulated and proved for the approximate controllability of such systems by splitting the given semilinear system into two systems, namely a semilinear deterministic system and a linear stochastic system. To prove the approximate controllability of semilinear deterministic system, Schauder fixed point theorem has been used. At the end, an example has been presented to illustrate the feasibility of the proposed result.


Introduction
Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. The study of controllability plays an important role in the control theory and engineering. Originated by Kalman (1960) for finite-dimensional linear control systems, controllability study was started systematically at the beginning of sixties. Since

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Controllability is an important concept pertaining to any control system. It determines whether the state of the system can be steered to a given target state in a prescribed time interval or not. Therefore, it plays a very important role in the analysis and design of control systems. Also, the noise or stochastic perturbation is omnipresent and unavoidable in nature as well as in man made systems. So, we have to move from deterministic systems to stochastic systems. On the other hand, nonlocal conditions have a better effect on the solution and is more precise for the physical measurements than the claasical initial conditions. Therefore, in this paper, we discuss the approximate controllability of semilinear stochastic systems with nonlocal conditions using splitting technique.
then various researches have been carried out in the context of finite-dimensional deterministic systems and infinite-dimensional systems (Curtain & Zwart, 1995;Zabczyk, 1992). Approximate controllability of abstract semilinear systems has been studied by Zhou (1983) under certain inequality conditions involving the system operators. Lions (1988) has proved some exact and approximate controllability results for semilinear control systems. In Joshi and Sukavanam (1990), Naito (1987), Sukavanam (1993Sukavanam ( , 2000, sufficient conditions for the controllability of semilinear control systems have been studied.
On the other hand, stochastic differential equation is an emerging field drawing attention from both theoretical and applied disciplines. The deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, the study of stochastic problems is more applicable in dynamical system theory. Therefore, these differential equations are important from point of view of application also. The theory of stochastic differential equation can also be applied to various problems outside mathematics, for example in economics, mechanics epidemiology and several fields in engineering. Differential equations play an important role in formulation and analysis of mechanical, electrical, control engineering and physical sciences. Motivated by these facts many researchers are showing great interest to establish an appropriate system to investigate qualitative properties such as existence, uniqueness and controllability of these systems. Therefore, it becomes important to study the controllability of stochastic systems. Only few authors have studied the extensions of deterministic controllability concepts to stochastic control systems (Bashirov, 1996(Bashirov, , 1997Bashirov & Mahmudov, 1999;Mahmudov, 2001;Klamka, 2007;Klamka & Socha, 1977, Ren, Dai, & Sakthivel, 2013. In Bashirov (1996Bashirov ( , 1997, the Controllability concepts were weakened and the controllability notions for partially observed linear Gaussian stochastic systems were introduced and their relation with complete and approximate controllabilities was studied. On the other hand, the first results concerning the existence and uniqueness of mild solutions to abstract Cauchy problems with nonlocal conditions were formulated and proved by Byszewski and Lakshmikantham (1991). He argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the onset of the experiment, thereby reducing the ill effects incurred by a single initial measurement. Also, it has a better effect on the solution and is more precise for physical measurements than classical condition x(0) = x 0 alone. Chang, Nieto, and Li (2009) studied the controllability of semilinear differential systems with nonlocal initial conditions using Sadovskii's Fixed Point theorem. Kumar and Sukavanam (2014) proved the controllability of second-order systems with nonlocal conditions using Sadovskii's Fixed point theorem. Shukla, Arora, and Sukavanam (2015) established sufficient conditions for the approximate controllability of first-order semilinear retarded stochastic system with nonlocal conditions using Banach Fixed Point Theorem. Arora and Sukavanam (2015) established sufficient conditions for the approximate controllability of second-order semilinear stochastic system with nonlocal conditions using Sadovskii's Fixed point theorem. (2010) provides a better tool for discussing the controllability of the stochastic systems. This method may also be appropriate in the variational formulation to find the solutions of the equation under weaker assumptions on the data. Using this technique, Sukavanam and Kumar (2010) obtained sufficient conditions for the S-controllability of semilinear first-order system.

Splitting Technique introduced by Sukavanam and Kumar
Up to now, most of the existing articles in the literature concentrate on finding the approximate controllability of stochastic systems using Banach Fixed Point Theorem, Sadovskii's Fixed Point Theorem and many other techniques. But there is no work reported on the approximate controllability of abstract semilinear stochastic system with nonlocal conditions using Splitting technique. Motivated by the above analysis, in this paper we establish sufficient conditions for the approximate controllability of an abstract semilinear stochastic system with nonlocal conditions with the help of new strategy which depends on splitting the given system into two systems, one is semilinear deterministic system and the other one is linear stochastic system.

Preliminaries
First, we introduce some notations. For a given operator A, D(A), R(A) and N 0 (A) denote the domain, range and null space of A, respectively. Ē and E ⟂ are the closure and orthogonal complement of a set E, respectively. cov (x, y) is the covariance operator of the random variables x, y and cov (x) = cov (x, x).
Now, we consider the semilinear stochastic control system with nonlocal conditions where x u (t) is the state value at time t ∈ [0, T] corresponding to the control u taken from the set of admissible controls Y. x 0 is a Gaussian random variable with covx 0 = P 0 and x 0 , m are mutually in- is a nonlinear operator which satisfies Caratheodory conditions (Krasnoselskii, 1963) and B:V → V is a bounded linear operator. Here g is a continuous function from C(J, V) to V.
Definition 2.1 The set K T (f ) = {x u (T) ∈ V:x u (.) ∈ Z is a mild solution of (Equation 2.1) for u ∈ Y} is called the reachable set of the system (Equation 2.1). K T (0) denotes the reachable set of the corresponding linear system of (Equation 2.1).
Definition 2.2 The system (Equation 2.1) is said to be approximate controllable if K T (f ) is dense in V and the corresponding linear system is approximate controllable if K T (0) is dense in V.

Controllability results
In this section, sufficient conditions have been established for the approximate controllability of the system (Equation 2.1).
We define an operator F:Z → Z by where F is called the Nemytskii operator of f. Now, in order to obtain the desired result, we assume the following conditions: (a) For every p ∈ Z there exists a q ∈ R(B) such that Lp = Lq where L is the operator defined as in (3.3).

(b) A generates a compact semigroup S(t).
(c) The operator f (t, x) satisfies Lipschitz continuity in x, i.e. for some constant l > 0 .
(d) f is uniformly bounded on V, i.e., ||f (t, x)|| ≤ k, a constant. The corresponding linear system of (Equation 2.2) which is a deterministic system is given by The above system (Equation 3.1) is approximate controllable under the condition (a) (see lemma 2 Naito, 1987).
Let K be an operator from Z into itself defined as and L and N be operators from Z into V defined as

S(T − s)[Fz](s)ds
It is evident that hypothesis (a) is equivalent to the condition Z = N 0 (L) + R(B). Moreover, Z can be decomposed as Z = N 0 (L) + N ⟂ 0 (L). Also, under hypothesis (a) we can define a map P:N ⟂ 0 (L) → R(B) as follows : Let u ∈ N ⟂ 0 (L), P(u) = u 0 where u 0 is the unique minimum norm element in the set satisfying The operator P is well-defined as it follows from hypothesis (a) that for each u ∈ N ⟂ 0 (L), the set {u + N 0 (L)} ∩ R(B) is nonempty.

Lemma 1 The operator P from N ⟂ 0 (L) to R(B) is linear and continuous.
Proof See Lemma 1 (Naito, 1987). ✷ From continuity of P it follows that ||Pu|| Z ≤ c||u|| Z for some constant c ≥ 0.
converges to the zero operator as → 0 + in the strong operator topology. (iii) R( , Π T 0 ) converges to the zero operator as → 0 + in the weak operator topology.
Proof The proof is a straightforward adaptation of the proof of Theorem 4.1 (Mahmudov, 2001). ✷ Theorem 3.2 The following four conditions are equivalent.
(3) The corresponding deterministic system of (Equation 2.3) is small time approximate controllable.
Proof For the proof, refer to the proof of Theorem 4.2 (Mahmudov, 2001). ✷ From the above results, we conclude that the semilinear stochastic system (Equation 2.3) is approximate controllable since the corresponding deterministic linear system is approximate controllable .

Lemma 4 Under the assumptions (b) and (d) ( ||S(t)|| ≤ M ,a constant ) the operator f p r
has a fixed point m 0 for each realization z w (t) of (Equation 2.3).

Example
Let V = L 2 (0, ) and A = − d 2 d 2 with D(A) consisting of all y ∈ V with d 2 y dx 2 ∈ V and y(0) = 0 = y( ). Put n ( ) = (2∕ ) 1∕2 sin(n ), 0 ≤ ≤ , n = 1, 2, ⋯, then { n :n = 1, 2 ⋯} is an orthonormal basis for V and n is an eigenfunction corresponding to the eigenvalue n = −n 2 of the operator −A, n = 1, 2 ⋯ . Then the C 0 -semigroup S(t) generated by −A has e n t as the eigenvalues and n as their corresponding eigenfunctions (Joshi & Sukavanam, 1990). Now define an infinite-dimensional Define a continuous linear mapping B from V to V as follows Consider the control system governed by the semilinear heat equation The approximate controllability of the corresponding deterministic semilinear heat equation of (Equation 4.1) was considered by Naito (1987). Here the approximate controllability of stochastic semilinear heat control system with nonlocal conditions is considered. Now we define the bounded linear function B from L 2 (0, T:V) to L 2 (0, T:V) by The nonlinear operator f is assumed to satisfy the conditions (c) and (d).
To the system (Equation 4.1) we can associate two control systems. The one is the deterministic control system with nonlocal conditions and the second is the stochastic linear control system  y v (t, ) t = 2 y v (t, ) 2 + Bv(t, ) + f (y v (t, ) + z w (t, ));0 < t < T, 0 < < y v (t, 0) = y v (t, ) = 0;0 ≤ t ≤ T