Approximate controllability of impulsive neutral stochastic functional ...

J Control Theory Appl 2013 11 (3) 351–358 DOI 10.1007/s11768-013-2061-7

Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces P. MUTHUKUMAR† , C. RAJIVGANTHI Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram-624 302, Tamilnadu, India

Abstract: Many practical systems in physical and technical sciences have impulsive dynamical behaviors during the evolution process which can be modeled by impulsive differential equations. In this paper, we prove the approximate controllability of control systems governed by a class of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces. Sufficient conditions for approximate controllability of the control systems are established under the natural assumption that the corresponding linear system is approximately controllable. The results are obtained by using semigroup theory, stochastic analysis techniques, fixed point approach and abstract phase space axioms. An example is provided to illustrate the application of the obtained results. Keywords: Approximate controllability; Hilbert space; Impulsive neutral stochastic functional differential system; Semigroup theory; Sadovskii’s fixed point theorem

1 Introduction The theory of semigroups of bounded linear operators is closely related to solving differential and integro differential equations in Banach spaces. In recent years, this theory has been applied to a large class of nonlinear differential equations in Banach spaces. Using the method of semigroups, various types of solutions to semilinear evolution equations have been discussed by Pazy [1]. Various evolutionary processes from fields as diverse as physics, population dynamics, aeronautics, economics and engineering are characterized by the fact that they undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. Because the duration of these changes are often negligible compared to the total duration of the process, such changes can be reasonably well approximated as being instantaneous changes of state, or in the form of impulses. These processes can be more suitably modeled by impulsive differential equations, which allow for discontinuities in the evolution of the state. One can refer to (see [2–7] and references therein). The notion of controllability has played a central role throughout the history of modern control theory. It is well known that controllability of deterministic systems are widely used in many fields of science and technology. In recent years, the extensions of the deterministic controllability concepts to stochastic control systems have rarely been reported. However, in many cases, the accurate analysis, design and assessment of systems subjected to realistic environments must take into account the potential of random loads and randomness in the system properties. Randomness is intrinsic to the mathematical formulation of

many phenomena such as fluctuations in the stock market, or noise in communication networks. Mathematical modeling of such systems often leads to differential equations with random parameters. The use of deterministic equations that ignore the randomness of the parameter or replace them by their mean values can result in gross errors. All such problems are mathematically modeled and described by various stochastic systems described by stochastic differential equations, stochastic delay equations, and in some cases, stochastic integrodifferential equations which are mathematical models for phenomena with irregular fluctuations. Stochastic control theory is a stochastic generalization of classical control theory (see [8–10] and references therein). From the mathematical point of view, two basic controllability can be distinguished. There are exact and approximate controllability. Roughly speaking, exact controllability enables to steer the system to arbitrary final state while approximate controllability means that system can be steered to arbitrary small neighborhood of final state by using the set of admissible controls. Bashirov et al. [11] proved the concepts of controllability for linear deterministic and stochastic systems. Controllability of nonlinear stochastic system is a well-known problem in [12]. Many authors have proved the controllability of differential systems by assuming that the controllability operator has an induced inverse on a quotient space. However, if the semigroup associated with the system is compact, then the controllability operator is also compact, and hence, the induced inverse does not exist in the infinite dimensional state space. Thus, the concept of exact controllability is too strong and the approximate controllability is more ap-

Received 21 March 2012; revised 8 October 2012. † Corresponding author. E-mail: [email protected]. Tel.: +91-451-245-2371; fax: +91-451-245-4466. This work was supported by Indo-US Science and Technology Forum (IUSSTF), New Delhi, India and UGC Special Assistance Programme (SAP) DRS-II, University Grants Commission, New Delhi, India (No. F.510/2/DRS/2009(SAP-I)). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

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propriate for these control systems. In deterministic case, Sakthivel et al. [13] studied the approximate controllability of semilinear fractional differential systems by assuming that the C0 -semigroup is compact and the nonlinear function is continuous and uniformly bounded. Mahmudov [14] have proved the approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. Recently, Sakthivel et al. [15] studied the approximate controllability of fractional stochastic evolution equations. The approximate controllability of nonlinear stochastic systems in infinite dimensional spaces has been extensively studied by several authors (see [16–18] and references therein). On the other hand, stochastic differential equation with delay is a special type of stochastic functional differential equation. Stochastic functional differential equations with state-dependent delay have become more important in some mathematical models of real phenomena and for this reason the study of this type of equations has been receiving great attention in recent years. Recently, Hernandez et al. [19] studied the existence results for partial neutral functional differential equation with statedependent delay. Very recently, Sakthivel et al. [20] studied the approximate controllability concepts in fractional differential equations with state-dependent delay. Only few authors studied the existence and controllability of differential system with state-dependent delay in deterministic cases (see [4–5, 21]). On the other hand, Senguttuvan et al. [22] proved the existence of solutions to neutral stochastic impulsive differential equations with state-dependent delay. Recently, Yan et al. [6] also studied the existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay. However, up to now, approximate controllability problems for nonlinear stochastic differential systems with statedependent delay and impulses have not been considered in the literature. Therefore, this paper studies the approximate controllability problem for impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces by using the results [11,14,23]. In Section 3, we study the approximate controllability of the following impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces: ⎧ d[x(t) + F (t, xt )] ⎪ ⎪ ⎪ ⎪ ⎪ = [Ax(t) + Bu(t)]dt + G(t, xρ(t,xt ) )dW (t), ⎪ ⎪ ⎪ ⎨ t ∈ J = [0, b] \ {t1 , t2 , . . . , tm }, (1) ⎪ ⎪ x0 = φ ∈ B, ⎪ ⎪ ⎪ − ⎪ Δx|t=tk = x(t+ ⎪ k ) − x(tk ) = Ik (x(tk )), ⎪ ⎩ k = 1, 2, . . . , m,

ploying the same notation · for the norm of L(K, H), where L(K, H) denotes the space of all bounded operators from K into H, simply L(H) if K = H. The histories xs represents the function defined by xs : (−∞, 0] → H, xs (θ) = x(s + θ), belong to some abstract phase space B described axiomatically and ρ : J × B → (−∞, b] is a continuous function. Furthermore, F : J × B → H and G : J × B → LQ (K, H) are given functions. Here, LQ (K, H) denotes the space of all Q-Hilbert Schmidt operators from K into H. We define the following classes of functions: let P C(J, L2 (Ω, F, P ; H)) = {x(t) is continuous everywhere + except for some tk at which x(t− k ) and x(tk ) exist and − x(tk ) = x(tk ), k = 1, 2, . . . , m} be the Banach space of piece-wise continuous function from J into L2 (Ω, F, P ; H) with the norm xP C = sup |x(t)| < ∞. P C(J, L2 )

where the state variable x( · ) takes values in a Hilbert space H and the control function u( · ) is given in L2 (J, U ), a Banach space of admissible control functions with U as a Banach space. B is a bounded linear operator from U into H. A is the infinitesimal generator of a compact semigroup T (t) on a Hilbert space H. Let K be a another separable Hilbert space. Suppose {W (t)}t0 is a given Kvalued Brownian motion or Wiener process with a finite trace nuclear covariance operator Q 0. We are also em-

of the operator). Then, the above K-valued stochastic process W (t) is called a Q-Wiener process. We assume that Ft = σ(W (s) : 0 s t) is the σ-algebra generated by W and Fb = F. Let ϕ ∈ L(K, H) and define ∞ √ ϕ2Q = tr(ϕQϕ∗ ) = λn ϕζn 2 .

t∈J

is the closed subspace of P C(J, L2 (Ω, F, P ; H)) consisting of measurable and Ft -adapted H-valued processes x( · ) ∈ P C(J, L2 (Ω, F, P ; H)) endowed with the norm x2 = sup{Ex(t)2 , t ∈ J}. Here, choose the points t1 , t2 , . . . , tm such that 0 = t0 < t1 < . . . < tm < tm+1 = − b, and Δx|t=tk = x(t+ k ) − x(tk ) and Ik : B → H is appropriate nonlinear function. This represents the jump in the state x at time tk with Ik determining the size of the jump. The rest of this paper is organized as follows: In Section 2, we recall some basic definitions, notations, lemmas and theorems which will be needed in the sequel. In Section 3, we prove the approximate controllability of system (1). Section 4 is reserved for an application. Section 5 contains the conclusion.

2

Preliminaries

For more details of this section, see [8–12,14–18] and the references therein. Let (Ω, F, P ) be a complete probability space furnished with complete family of right continuous increasing sub σ algebras {Ft , t ∈ J} satisfying Ft ⊂ F. An H-valued random variable is an F measurable function x(t) : Ω → H, and a collection of random variable S = {x(t, ω) : Ω → H|t∈J } is called a stochastic process. Usually, we suppress the dependence on ω ∈ Ω and write x(t) instead of x(t, ω) and x(t) : J → H in the place of S. Let βn (t) (n = 1, 2, . . .) be a sequence of real valued one dimensional standard Brownian motions mutually independent ∞ √ λn βn (t)ζn , t 0 where over (Ω, F, P ). Set W (t) = n=1

λn 0 (n = 1, 2, . . .) are nonnegative real numbers and {ζn } (n = 1, 2, . . .) is complete orthonormal basis in K. Let Q ∈ L(K, K) be an operator defined by Qζn = λn ζn ∞ with finite tr(Q) = λn < ∞ (tr denotes the Trace n=1

n=1

If ϕQ < ∞, then ϕ is called a Q-Hilbert Schmidt operator. Let LQ (K, H) denote the space of all Q-Hilbert


Schmidt operators ϕ : K → H. The completion LQ (K, H) of L(K, H) with respect to the topology induces by the norm · Q where ϕ2Q = ϕ, ϕ is a Hilbert space with the above norm topology. In this work we will employ an axiomatic definition of the phase space B introduced by Hale and Kato [24]. The axioms of the space B are established for F0 -measurable functions from J0 into H, where J0 := (−∞, 0], endowed with a seminorm · B . We will assume that B satisfies the following axioms: a) If x : (−∞, b) → H, b > 0 is continuous on [0, b) and x0 ∈ B, then for each t ∈ [0, b) the following conditions hold: i) xt ∈ B; ii) x(t) K1 xt B ; and iii) xt B K2 (t)x0 B + K3 (t) sup{x(s); 0 s b}, where K1 > 0 is a constant, K2 : [0, ∞) → [0, ∞) is a locally bounded function, K3 : [0, ∞) → [0, ∞) is a continuous function. Moreover, K1 , K2 ( · ) and K3 ( · ) are independent of x( · ). b) For the function x( · ) in (a), xt is a B-valued continuous functions on [0, b). c) The space B is complete. Examples of phase space satisfying the above axioms can be found in [25–26]. The B-valued stochastic process xt : Ω → B, t 0, is defined by setting xt = {x(t + s)(w) : s ∈ (−∞, 0]} . The collection of all strongly measurable, square integrable H valued random variables, denoted by L2 (Ω, F, P ; H) ≡ L2 (Ω; H), is a Banach space equipped with norm 2 1 x( · )L2 = (Ex( ·; w) H ) 2 , where the expectation E h(w)dP . Let J1 = (−∞, b] is defined by E(h) = Ω and C(J1 , L2 (Ω; H)) be the Banach space of all continuous maps from J1 into L2 (Ω; H) satisfying the condition sup Ex(t)2 < ∞. Let C be the closed subspace of t∈J1

all x belongs to the space C(J1 , L2 (Ω; H)) consisting of Ft -adapted measurable process and F0 -adapted processes φ ∈ L2 (Ω, B). Let · C be a seminorm in C defined by Ex2C = sup Ext 2B , (2) t∈J

where ¯ 2 Eφ2 + K ¯ 3 sup{Ex(s)2 : 0 s b}, Ext 2B K B ¯ 2 = sup{K2 (t)}, K ¯ 3 = sup{K3 (t)}. K t∈J

t∈J

It is easy to verify that C furnished with the norm topology as defined above is a Banach space. The reader may refer to Examples (2.1) and (2.2) in [19] for the phase space P Ch (H), P Cg0 (H), P Cr × L2 (g, H). To prove the main results, we assume the following hypotheses. H1 ) There exists a Hilbert space (Y, · Y ) continuously included in H such that the following conditions are verified. For every y0 ∈ Y , the function t → T (t)y0 is continuous from [0, ∞) into Y . Moreover, T (t)Y ⊂ D(A) for every t > 0 and there exists a positive function γ ∈ L1 ([0, b]) such that AT (t)2L(Y ;H) γ(t) for every t ∈ J. H2 ) The function F is Y -valued, F : J × B → Y is con-

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tinuous and there exist a positive constants c1 , c2 such that F (t, ψ)2Y c1 ψ2B + c2 . H3 ) The function F is Y -valued, F : J × B → Y is continuous and there exists a positive constant LF > 0 such that F (t, ψ1 ) − F (t, ψ2 )2Y LF ψ1 − ψ2 2B . H4 ) The function t → φt is well defined from Z(ρ− ) = {ρ(s, τ ); (s, τ ) ∈ J × B, ρ(s, τ ) 0} into B and there exists a continuous and bounded function H φ : Z(ρ− ) → (0, ∞) such that Eφt 2B H φ (t)Eφ2B for every t ∈ Z(ρ− ). H5 ) The function G : J × B → LQ (K, H) satisfies the following conditions: i) let x : (−∞, b] → H be such that x0 = φ and x ∈ P C(J, L2 ). The function t → G(s, xρ(t,xt ) ) is measurable on J, the t → G(s, xt ) is continuous on Z(ρ− ) ∪ J, for every s ∈ J; ii) there exists an integrable function MG : J → [0, ∞), and a continuous non decreasing function ΩG : [0, ∞) → (0, ∞) such that G(t, ψ)2Q MG (t)ΩG (ψ2B ); (t, ψ) ∈ J × B; and iii) the function G is continuous and there exists a constant LG ∈ L1 (J, R+ ) such that G(t, ψ1 ) − G(t, ψ2 )2Q LG (t)ψ1 − ψ2 2B ; t ∈ J, ψ1 , ψ2 ∈ B. H6 ) A is the infinitesimal generator of a compact semigroup of bounded linear operators T (t) in H such that T (t) M for some M 1. H7 ) B is a bounded linear operator from U into H, such that B = N , for a constant N 0. H8 ) The maps Ik are continuous and there exist a positive constants c3 , c4 , Lk such that Ik (ψ)2 c3 ψ2B + c4 , Ik (ψ1 ) − Ik (ψ2 )2 Lk ψ1 − ψ2 2B , ψ1 , ψ2 ∈ B, k = 1, 2, . . . , m. H9 ) For each 0 t < b, the operator λR(λ, Πtb ) = λ(λI + Πtb )−1 → 0 in the strong operator topology as λ → 0+ , where the controllability operator Πtb ∈ L(L2 (Ω, Fb , H), L2 (Ω, Fb , H)), associated with (1), is defined as b Πtb { · } = T (b − s)BB ∗ T ∗ (b − s)E{·|Fs }ds. t

Definition 1 System (1) admits a mild solution x( · ) ∈ P C(J, L2 ) if x0 = φ, xρ(s,xs ) ∈ B, for every s ∈ J, the function t → AT (t − s)F (s, xs ) is integrable on [0, t), for every t ∈ [0, b] and x(t) = T (t)[φ(0) + F (0, φ)] − F (t, xt ) t t − AT (t−s)F (s, xs )ds+ T (t−s)Bu(s)ds 0 0t + T (t − s)G(s, xρ(s,xs ) )dW (s) 0 + T (t − tk )Ik (x(tk )), t ∈ J. (3) 0 0 such that for every r > 0, there exist xr ∈ Br (φ|J , D) and tr ∈ J such that r < Φxr (tr )2 , then ¯r )2 Euλ (tr , x 6M 2 N 2 ¯ 2 ¯2 {h + M 2 φ(0) + F (0, φ)2 + c1 {(H λ2 ¯ 3 r}+c2 +[c1 (H ¯ 2 +H)φ2 +c1 H ¯ 3r +H)φ2B + H B b ¯ 2 + H) +c2 ] γ(b − s)ds + tr(Q)M 2 ΩG ((H 0 b m ¯ 3 r) MG (s)ds + M 2 {Lk ×φ2B + H 0

k=1

¯ 2 + H)φ2 + H ¯ 3 r) + Ik (0)2 }}, ×((H B r r 2 EΦx (t ) 6M 2 [φ(0) + F (0, φ)]2 + 6{c1 ¯ xrtr 2 r t +c2 } + 6 {c1 ¯ xrs 2 + c2 }γ(tr − s)ds 0

¯ 2 + H)φ2 + H ¯ 3 r) +6tr(Q)M 2 ΩG ((H B tr m ¯ 2 + H)φ2 × MG (s)ds + 6M 2 {Lk ((H B 0

k=1

4 4 ¯ 2 + M2 ¯ 3 r) + Ik (0) } + 36bM N {h +H 2 λ ¯ 2 + H)φ2 + H ¯ 3 r} ×φ(0) + F (0, φ)2 + c1 {(H B 2 ¯ 2 + H)φ + c1 H ¯ 3 r + c2 ] +c2 + [c1 (H B 2

355


b

¯ 2 + H) γ(b − s)ds + tr(Q)M 2 ΩG ((H 0 b m ¯ 3 r) MG (s)ds + M 2 Lk ×φ2B + H

×

0

k=1

¯ 2 + H)φ2 + H ¯ 3 r) + Ik (0)2 }}, ×((H B 2 ¯ 2 + H)φ2 6M [φ(0) + F (0, φ)]2 + 6{c1 {(H B ¯ 3 r} + c2 } + 6{c1 {(H ¯ 2 + H)φ2 + H ¯ 3 r} +H B tr ¯ 2 + H) +c2 } γ(tr − s)ds + 6tr(Q)M 2 ΩG ((H 0 tr m ¯ 3 r) ¯2 ×φ2B + H MG (s)ds + 6M 2 {Lk ((H 0

k=1

4 4 ¯ 2 ¯ 3 r) + Ik (0)2 } + 36bM N {h +H)φ2B + H λ2 ¯ 2 + H)φ2 +M 2 φ(0) + F (0, φ)2 + c1 {(H B 2 ¯ ¯ ¯ 3 r + c2 ] +H3 r} + c2 + [c1 (H2 + H)φB + c1 H b ¯ 2 + H)φ2 × γ(b − s)ds + tr(Q)M 2 ΩG ((H B 0 b m ¯ 3 r) ¯ 2 + H) +H MG (s)ds + M 2 {Lk ((H 0

×φ2B

k=1

¯ 3 r) + Ik (0)2 }}, +H

and hence,

b 4 4 ¯ 3 {1 + 6bM N }{1 + γ(s)ds} 1 6c1 H 0 λ2 4 4 ¯ 3 {1 + 6bM N } +6tr(Q)M 2 H λ2 ΩG (ξ) b × lim inf MG (s)ds 0 ξ→∞ ξ m 6bM 4 N 4 2 ¯ +{1 + }6M Lk , H 3 λ2 k=1 which contradicts to our assumption. Therefore, Φ(Br (φ|J , D)) ⊂ Br (φ|J , D). Now, we define the operators (Φ1 x)(t) = T (t)[φ(0) + F (0, φ)] − F (t, x ¯t ) t ¯s )ds − AT (t − s)F (s, x 0t ¯)ds + T (t − s)Buλ (s, x 0 + T (t − tk )Ik (¯ x(tk )), 0 0, we can fix N ∈ N such that ρ(s, x ¯ns ) > 0, for every n > N . In this case, for n > N ,

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we see that E¯ xnρ(s,¯xns ) − x ¯ρ(s,¯xs ) 2B

E¯ xnρ(s,¯xns ) − x ¯ρ(s,¯xns ) 2B + E¯ xρ(s,¯xns ) − x ¯ρ(s,¯xs ) 2B , ¯ 3 xn (θ) − x(θ)2 n + E¯ H xρ(s,¯xn ) − x ¯ρ(s,¯x ) 2 , ρ(s,¯ xs )

s

s

B

¯ 3 xn − x2 + E¯ H xρ(s,¯xns ) − x ¯ρ(s,¯xs ) 2B , b n which prove that x ¯ρ(s,¯xn ) → x ¯ρ(s,¯xs ) in B as n → ∞, s for every s ∈ J such that ρ(s, x ¯s ) > 0. Similarly, if ¯ns ) < 0, for ρ(s, x ¯s ) < 0 and N ∈ N is such that ρ(s, x every n > N , we get E¯ xnρ(s,¯xns ) − x ¯ρ(s,¯xs ) 2B = φρ(s,¯xns ) − φρ(s,¯xs ) 2B ,

¯ρ(s,¯xs ) in B as n → ∞, which also shows that x ¯nρ(s,¯xn ) → x s for every s ∈ J such that ρ(s, x ¯s ) < 0. Combining the previous arguments, we can prove that x ¯nρ(s,¯xn ) → φ for every s s ∈ J such that ρ(s, x ¯s ) = 0. From the previous remarks ¯ρ(s,¯xs ) ) for every s ∈ J. Now the G(s, x ¯nρ(s,¯xn ) ) → G(s, x s assumption H5 ) and the Lebesgue dominated convergence theorem permits us to assert that Φxn → Φx in D. Thus Φ( · ) is continuous, which completes the proof that Φ2 ( · ) is completely continuous. These arguments enable us to deduce that Φ = Φ1 + Φ2 is a condensing mapping on Br (φ|J , D). Hence from the Sadovskii’s fixed point theorem Φ has a fixed point and system (1) has a solution on J. This completes the proof. Theorem 3 Under the hypotheses H1 )–H9 ) and the assumptions of Theorem 2 are holds, the functions F and G are uniformly bounded in H and L(K, H) respectively, and then, system (1) is approximately controllable on J. Proof Let x ¯λ be a fixed point of Φ in Br (φ|J , D). Using Fubini theorem, any fixed point of Φ is a mild solution ¯) satisfies to (1) under the control uλ (t, x x ¯λ (b) b = T (b)(φ(0) + F (0, φ)) − F (b, x ¯λb ) − AT (b − s) 0 b T (b − s)B{B ∗ T ∗ (b − s) ×F (s, x ¯λs )ds + 0

¯ − T (b)(φ(0) + F (0, φ)) + F (b, x ×{R(λ, Γob )[Eh ¯λb )] s + R(λ, Γτb )z(τ )dW (τ )} + B ∗ T ∗ (b − s) 0 s R(λ, Γτb )AT (b − τ )F (τ, x ¯λτ )dτ − B ∗ T ∗ (b − s) × o s R(λ, Γτb )T (b − τ )G(τ, x ¯λρ(τ,¯xλτ ) )dW (τ ) − B ∗ × o ×T ∗ (b − s)R(λ, Γtb ) T (b − tk )Ik (¯ xλ (tk ))}ds + +

0