Approximate controllability of a second-order neutral stochastic differential equation with state-dependent delay ∗

Abstract. In this paper, the existence and uniqueness of mild solution is initially obtained by use of measure of noncompactness and simple growth conditions. Then the conditions for approximate controllability are investigated for the distributed second-order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. We construct controllability operators by using simple and fundamental assumptions on the system components. We use Lemma 3, which implies the approximate controllability of the associated linear system. Lemma 3 is also described as a geometrical relation between the range of the operator B and the subspaces Ni , i = 1, 2, 3, associated with sine and cosine operators in L2([0, a], X) and L2([0, a], LQ). Eventually, we show that the reachable set of the stochastic control system lies in the reachable set of its associated linear control system. An example is provided to illustrate the presented theory.


Introduction
Random noise causes fluctuations in deterministic models.So, necessarily, we move from deterministic problems to stochastic ones.Stochastic evolution equations are natural generalizations of ordinary differential equations incorporating the randomness into the equations.Thereby, making the system more realistic, [9,21] and the references therein explore the qualitative properties of solutions for stochastic differential equations.Considering the environmental disturbances, Kolmanovskii and Myshkis [22] introduced a class of neutral stochastic functional differential equations, which are applicable in several fields, such as chemical engineering, aero-elasticity and so on.In recent years, are measurable mappings in X norm, and G : J × B → L Q (K, X) is a measurable mapping in L Q (J, X) norm.L Q (J, X) is the space of all Q-Hilbert-Schmidt operators from K into X.B is a bounded linear operator from U into X.φ(t) is B-valued random variable independent of Brownian motion W (t) with finite second moment.Also, ψ(t) is a X-valued F t -measurable function.

Preliminaries
In this section, some definitions, notations and lemmas that are used throughout this paper are stated.Let (Ω, F, P) be a complete probability space endowed with complete family of right-continuous increasing sub σ-algebras {F t , t ∈ J} such that F t ⊂ F. A X-valued random variable is a F-measurable process.A stochastic process is a collection of random variables S = {x(t, w) : Ω → X, t ∈ J}.We usually suppress w and write x(t) instead of x(t, w).Now suppose β n (t), n = 1, 2, . . ., be a sequence of real-valued one dimensional standard Brownian motions mutually independent over (Ω, F, P).Let ς n be a complete orthonormal basis in K. Q ∈ L(K, K) be an operator defined by Qς n = λ n ς n with finite trace which is a K-valued stochastic process and is called a Q-Wiener process.Let F t = σ(W (s), 0 s t) be the σ-algebra generated by W and F a = F. Let φ ∈ L(K, X), and if with respect to the topology induced by norm φ 2 Q = φ, φ , is a Hilbert space.The family {C(t), t ∈ R} of operators in B(X) is a strongly continuous cosine family if the following are satisfied: x is strongly continuous for each x ∈ X. {S(t), t ∈ R} is the strongly continuous sine family associated to the strongly continuous cosine family {C(t), t ∈ R}.It is defined as The operator A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators C(t)) t∈R , and S(t) is the associated sine function.Let N , N be certain constants such that C(t) 2 N and S(t) 2 N for every t ∈ J = [0, a].For more details, see the book by Fattorini [13].In this work, we use the axiomatic definition of phase space B introduced by Hale and Kato [14].
We denote by C the closed subspace of all continuously differentiable process x ∈ C 1 (J, L 2 (Ω; X)) consisting of F t -adapted measurable processes such that φ, ψ ∈ L 0 2 (Ω; B) and seminorm Here n=1 is nonempty and compact in Y .Definition 3. Let X and Y be Banach spaces, and Φ, Ψ be the Measure of Noncompactness (MNC) in X and Y , respectively.If for any continuous function Theorem 1. (See [2].)Let Ψ be a MNC on a Banach space X.Let f be (Ψ, Ψ )-condensing operator.If f maps a nonempty, convex, closed subset M of the Banach space X into itself, then f has atleast one fixed point in M .Definition 4. The set given by R(f ) = {x(T ) ∈ X: x is a mild solution of (1)} is called reachable set of system (1) for some T > 0. R(0) is the reachable set of the corresponding linear control system (2).Definition 5. System (1) is said to be approximately controllable if R(f ) is dense in X.The corresponding linear system is approximately controllable if R(0) is dense in X. Lemma 3. (See [26].)Let X be Hilbert space, and X 1 , X 2 closed subspaces such that X = X 1 + X 2 .Then there exists a bounded linear operator P : X → X 2 such that for each x ∈ X, x = x − P x ∈ X 1 and We state the corresponding linear control system Lemma 4. (See [13].)Under the assumption that h : [0, a] → X is an integrable function such that Nonlinear Anal.Model.Control, 21(6):751-769

Main result
We define mild solution of problem (1) as follows: ), G(s, x s ) and g(s, x s ) are integrable, and for t ∈ [0, a], the following integral equation is satisfied: To prove our result, we always assume ρ : J ×B → (−∞, a] is a continuous function.The following hypotheses are used: and there exists a continuous bounded function (H G ) The function G satisfies the following conditions: (H g ) g : J × B → X satisfies the following: (i) For every x : (−∞, a] → X, x 0 ∈ B and x| J ∈ P C, the function g(•, ψ) : J → X is strongly measurable for every ψ ∈ B and g(•, t) is continuous for a.e.t ∈ J. (ii) There exists an integrable function α g : J → [0, +∞) and a monotone continuous nondecreasing function ) for all t ∈ J and v ∈ B.
http://www.mii.lt/NA(H l ) There exists a function H is a continuous, monotone, nondecreasing in second variable, also H(t, 0) ≡ 0 and Lemma 5. (See [2].)Let m be a nonnegative, continuous function, and A > 0 such that then m has no nonzero nonnegative solution.

Existence and uniqueness of mild solution
In this section, y : (−∞, a] → X is the function defined by y 0 = φ and y(t) = C(t)φ(0) + S(t)(ψ + g(0, φ)) on J. Clearly, y t B K a E y a + M a E φ B , where E y a = sup 0 t a {E y(t) , 0 s a}.
Proof.Let S(a) be the space S(a) = {x ∈ C(J, L 2 (Ω; X)): x(0) = 0} endowed with the norm of uniform convergence.x ∈ C 0 is identified with its extension to (−∞, a] by assuming x(θ) = 0 for θ < 0. Let Γ : S(a) → S(a) be the map defined by where x 0 = φ and x = x + y on J.It is easy to see that Thus, Γ is well defined and has values in S(a).Also, by axioms of phase space, the Lebesgue-dominated convergence theorem and conditions (H f ), (H G ), (H g ) it can be shown that Γ is continuous.
Step 1.We prove that there exists k > 0 such that Γ (B k ) ⊂ B k , where B k = {x ∈ S(a): E x 2 k}.In fact, if we assume that the assertion is false, then for k > 0, there exist Hence, where α = max{α g , α f , α G }. Thus, (3) is a contradiction to hypothesis (H 1 ).Hence, Step 2. We prove that Γ is a condensing map on any bounded subset of the space C(J, L 2 (Ω; X)).Let O be a bounded subset of C(J, L 2 (Ω; X)).Let M[0, a] be the partially ordered linear space of all real monotone nondecreasing functions on [0, a], and http://www.mii.lt/NAwe define a Measure of Noncompactness (MNC), Ψ : C(J, L 2 (Ω; X)) → M[0, a] by where χ t is the Hausdorff MNC in C(J, L 2 (Ω; X)) and is nondecreasing and bounded, so for all > 0, it has only a finite number of jumps of magnitude greater than .The disjoint δ 1 neighborhoods of the points corresponding to these jumps are removed from [0, a].Using points β j , j = 1, 2, . . ., m, divide the remaining part into intervals on which the oscillations of Ψ (O) is less than .These points β j are surrounded by disjoint δ 2 neighborhoods.Now consider the family o = {o k , k = 1, . . ., l} of all functions continuous with probability one such that o k coincides with an arbitrary element of [(Ψ (O))(β j ) + 1] net of the set O βj on the segment where Then By choosing δ 1 > 0 and δ 2 > 0 sufficiently small, we can make sure that Together with Lemma 5, we get that Ψ (O) ≡ 0. Similarly, we can prove that Γ is continuous.The MNC Ψ possess all required properties.The operator Γ is condensing.
Then from Theorem 1 it is implied that there exist a mild solution to problem (1).The uniqueness of mild solution follows from Lemma 5. Let m 1 , m 2 ∈ C(J, L 2 (Ω; X)) be two mild solution of Γ .Then it follows that Thus, from Lemma 5 it follows that m 1 − m 2 2 C(J,L2(Ω;X)) ≡ 0. Hence, m 1 = m 2

Approximate controllability
In this section, the approximate controllability of the distributed control system (1) is studied as an extension of co-author N. Sukavanam's method in [26].Assume that f , g, G satisfy the following conditions: (C1) The function f, g : J × B → X are continuous.For all t ∈ J and for all z 1 , z 2 ∈ L 2 (J; B), there exists constants L f , L g > 0 such that Also, y : (−∞, a] → X is the function defined by y 0 = φ and y(t) = C(t)φ(0) + S(t)(z + g(0, φ) on J. Clearly, y t B K a E y a + M a E φ B , where y a = sup 0 t a y(t) .
neutral functional differential equations can be studied.We considered the white noise as a Wiener process, but this work can be extended to incorporate other disturbances in the form of Poisson processes, etc.
m, and linear on the complementary segments.