Rend. Circ. Mat. Palermo DOI 10.1007/s12215-016-0235-0

Approximate controllability of second order semilinear stochastic system with variable delay in control and with nonlocal conditions Urvashi Arora1 · N. Sukavanam1

Received: 18 August 2015 / Accepted: 31 January 2016 © Springer-Verlag Italia 2016

Abstract The present paper is concerned with the study of approximate controllability of a second order semilinear stochastic system with variable delay in control and with nonlocal conditions. The control function for this system has been established with the help of infinite dimensional controllability operator. Using this control function, the sufficient conditions for the approximate controllability of the proposed system have been obtained using Banach Fixed Point theorem. At the end, two examples have been provided to show the effectiveness of the result. Keywords Approximate controllability · Stochastic control system · Variable delay · Banach fixed point theorem · Nonlocal conditions Mathematics Subject Classification

93B05 · 93C10

1 Introduction Modelling and control of dynamical systems with input/output delays arise naturally in various engineering applications. Further, satisfactory modelling of systems with variable delays is also important for the synthesis of effective control systems since they show significantly different characteristics from that of fixed time delays. In practical applications, variable delays always exist in a flexible spacecraft due to the physical structure and energy consumption of the actuators. It is essential that system models must take into account these variable delays in order to predict the true system dynamics. The presence of variable delays is often the main cause of substantial performance deterioration and even instability of the system. Moreover, a majority of processes in industrial practice have stochastic characteristics and

B

Urvashi Arora [email protected] N. Sukavanam [email protected]

1

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

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U. Arora, N. Sukavanam

systems have to be modelled in the form of stochastic differential equations. Thus, it is of theoretical and practical significance to address controllability problems for such stochastic systems with variable delays in control. Controllability is one of the most important aspects of industrial process operability because it can be used to access the attainable operation of a given process and improve the dynamic performance. It refers to the ability of a controller to arbitrarily alter the functionality of the dynamical system. The concept of controllability was introduced by Kalman [1]. Barnett [2] and Curtain [3] introduced the basic concepts of control theory in their books. Controllability of linear and nonlinear stochastic systems in finite dimensional spaces with delays in control has been extensively studied by Klamka et al. [4,5] and Shen et al. [6]. The results on controllability of linear and nonlinear stochastic systems have been a subject of intense research over the past few years(see Bashirov et al. [7], Balachandran [8], Dauer [9], Mahmudov et al. [10–15] and Sakthivel [16]). On the other hand, non local conditions introduced by Byszewski et al. [17] are the conditions given by an expression where the value of an unknown function is expressed by the value(s) of this function at regular intervals rather than continuously over the history period. He has done a great work on nonlocal condition problems and claim that these types of conditions are usually more precise for physical measurements than the classical ones as more information was taken into account at the oneset of the experiment. In recent times, many researchers have studied the existence and uniqueness of the mild solution of semilinear systems with nonlocal conditions. To discuss controllability of a second order system by converting it into a first order system may not give desired results because of the behavior of the semigroup generated by the linear part of the converted first order system. So, it is better to treat the second order system directly rather than to convert it into first order system. The basic concepts regarding the theory of second order differential equations have been described in [18] and [19]. The controllability of second order deterministic and stochastic nonlinear systems have been investigated extensively by many authors, for instance [20]. Kumar and Sukavanam [21] obtained the sufficient conditions for the controllability of second order deterministic systems with nonlocal conditions in Banach spaces using Sadovskii’s Fixed point theorem. Arora et al. [22] established the controllability of second order semilinear stochastic system with nonlocal conditions using Sadovskii’s Fixed Point theorem. However the situation is less satisfactory for the second order stochastic systems with control delays and with nonlocal conditions. In recent years, there has been an increasing interest in the controllability of stochastic systems with control delays. Inspired by the above work, this study focuses on the approximate controllability of the second order semilinear stochastic system with variable delay in control and with nonlocal conditions in separable Hilbert spaces ⎫ d y  (t) = [A(t)y(t) + B1 (t)u(t) + B2 (t)u(h(t)) + f(t, y(t))]dt + σ (t, y(t))dω(t), t ∈ J ⎬ y(0) = y0 + g(y), y  (0) = y1 + g1 (y) ⎭ and u(t) = 0 for t ∈ [h(0), 0]

(1.1) where A : D(A) ⊂ Y → Y is a closed, linear and densely defined operator on a separable Hilbert space Y which generates a strongly continuous cosine family {C(t) : t ∈ J } on Y . B1 , B2 are bounded linear operators from the Hilbert space U into Y . The control u ∈ p L ([0, b], U ); f : J × Y → Y and σ : J × Y → L02 are nonlinear suitable functions p where the space L ([0, b], U ) and L02 are specified in the next section. Let K be an another separable Hilbert space. Suppose ω(t) is a given K-valued Wiener process defined on a

123

Approximate controllability of second order. . .

complete probability space (, , P) with a finite trace nuclear covariance operator Q > 0. Here y0 and y1 are 0 measurable Y valued random variables independent of ω; g, g1 are continuous functions from C(J, Y ) → Y . h(t) = t − h 1 (t) is continuous differentiable and strictly increasing on J and h 1 (t) > 0 is a time variable point delay.

2 Preliminaries and assumptions Let Y , U and K be the separable Hilbert spaces. Let (, , P) be a complete probability space equipped with a normal filtration t , t ∈ J = [0, b] such that the filtration t is a right continuous increasing family and 0 contains all P-null sets. Let ω be a Q-Wiener process on (, b , P) with the covariance operator Q such that tr Q < ∞. A Y -valued random variable is a  measurable function y(t) :  → Y and the collection of random variables S = {y(t) :  → Y |t ∈ [0, b]} is called a stochastic process. We assume that there exists a complete orthonormal system ek in K, a bounded sequence of nonnegative real numbers λk such that Qek = λk ek , k = 1, 2, . . . and a sequence Bk of independent Brownian motions such that < w(t), e >=

∞  

λk < ek , e > Bk (t), e ∈ K, t ∈ J

k=1

and t = t ω , where t ω is the σ -algebra generated by ω. In order to define stochastic integrals with respect to the Q-Wiener process ω(t), we introduce the subspace K0 = Q 1/2 (K) of K which is endowed with the inner product (y1 , y2 )K0 = (Q −1/2 y1 , Q −1/2 y2 ) and is a Hilbert space. Let L2 0 = L2 (Q 1/2 K; Y ) be the space of all Hilbert-Schmidt operators from K0 to Y with the norm   ||ϕ||2L0 = tr (ϕ Q 1/2 )(ϕ Q 1/2 )∗ , 2

for ϕ ∈

L02 .

Clearly, for any bounded linear operator ϕ ∈ L(K, H ), this norm reduces to ||ϕ||2L0 = tr (ϕ Qϕ ∗ ) = 2

∞   || λk ϕζk ||2 , k=1

L p (

Let b , Y ) be the Banach space of all b -measurable pth power integrable random p variables with values in the Hilbert space Y . Let L (J, Y ) be the space of all t adapted, Y valued stochastic processes on J × . Let C([0, b]; L p (, Y )) be the Banach space of continuous maps from [0, b] into L p (, Y ) satisfying the condition sup E||y(t)|| p < ∞. t∈J

Let H2 = C p ([0, b]; Y ) be the closed subspace of C([0, b]; L p (, Y )) consisting of measurable and t - adapted Y valued processes ϕ ∈ C([0, b]; L p (, Y )) endowed with the norm

1/ p ||ϕ||H2 =

p

sup E||ϕ(t)||Y

t∈[0,b]

where E is defined as integration with respect to probability measure P. For convenience, let us define the time lead function r (t) : [h(0), h(b)] → [0, b] such that r (h(t)) = t i.e r (t) is the inverse function of h(t).

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For simplicity, we suppose that the set of admissible controls Uad = L (J, U ). Now, we describe some basic concepts and properties about cosine family of operators. With the help of this, we will define the mild solution of (1.1). Then we will state the sufficient conditions to prove the approximate controllability. Definition 2.1 Strongly continuous cosine family and strongly continuous sine family [19]: “Let L(K; Y ) be the space of bounded linear operators from K into Y . The one parameter family {C(t) : t ∈ R} ⊂ L(Y ) is called a strongly continuous cosine family if (1) C(0) = I (2) C(t)y is continuous in t on R, for all y ∈ Y (3) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ R If {C(t) : t ∈ R} is a strongly continuous cosine family on Y , then the corresponding strongly  t

continuous sine family {S(t) : t ∈ R} ⊂ L(Y ) is defined by S(t)y =

C(s)yds, t ∈ R,

0

y ∈ Y. The infinitesimal generator of a strongly continuous cosine family of {C(t) : t ∈ R} is the d2 operator A : D(A) ⊂ Y → Y defined by Ay = 2 C(t)y |t=0 for all y ∈ D(A) dt where D(A) = {y ∈ Y : C(t)y is a twice continuously differentiable function of t}”. Such cosine and corresponding sine families and their generators satisfy the following properties: Lemma 1 [19] “Suppose that A is the infinitesimal generator of a cosine family of operators {C(t) : t ∈ R}. Then the following hold: (1) There exists M  ≥ 1 and ω ≥ 0 such that ||C(t)|| ≤ M  eω|t| and hence ||S(t)|| ≤ M eω|t| , r (2) A S(u)ydu = [C(r ) − C(s)]y for all 0 ≤ s ≤ r < ∞, s  r ω|s|   e ds for all 0 ≤ s ≤ r < ∞. (3) There exists N ≥ 1 such that ||S(s)− S(r )|| ≤ N s

The uniform boundedness principle together with (1) above implies that both {C(t) : t ∈ J } and {S(t) : t ∈ J } are uniformly bounded and M = M  eω|b| ”. Lemma 2 [23] “Let G : J ×  → L02 be a strongly measurable mapping such that  b p E||G(t)|| 0 ds < ∞. Then L 0

2



E

0

t

p  t p G(s)dω(s) ≤ L G E||G(s)|| 0 ds L2 0

for all t ∈ J and p ≥ 2, where L G is the constant involving p and b”. Definition 2.2 A stochastic process y ∈ H2 is a mild solution of (1.1) if for each u ∈ p L (J, U ), it satisfies the following integral equation:  t y(t) = C(t)(y0 + g(y)) + S(t)(y1 + g1 (y)) + S(t − s)[B1 (s)u(s) + f(s, y(s))]ds 0  t  t + S(t − s)B2 (s)u(h(s))ds + S(t − s)σ (s, y(s))dω(s) 0

123

0

Approximate controllability of second order. . .

Taking h(s) = τ in the above equation and using the time lead function r (t), we have s = r (τ ) and ds = r˙ (τ )dτ . Using this, this solution can be further changed into y(t) = C(t)(y0 + g(y)) + S(t)(y1 + g1 (y))  h(t) 

+ S(t − s)B1 (s) + S(t − r (s))B2 (r (s))r  (s) u(s)ds 0  t  t S(t − s)B1 (s)u(s)ds + S(t − s)f(s, y(s))ds + 

h(t) t

0

S(t − s)σ (s, y(s))dω(s)

+

0

Now, we define the following operators and sets. L b ∈ L(Uad , L p (b , Y )) defined by  h(b) Lbu = (S(b − s)B1 (s) + S(b − r (s))B2 (r (s))r  (s))u(s)ds 0

+



b

S(b − s)B1 (s)u(s)ds

h(b)

where L(Y1 , Y2 ) denotes the set of bounded linear operators from Y1 to Y2 . Then its adjoint operator L ∗b ∈ L p (b , Y ) → Uad is given by  ∗ (B1 (t)S ∗ (b − t) + B2∗ (r (t))S ∗ (b − r (t))r  (t))E{z|t }, t ∈ [0, h(b)] L ∗b z = B1∗ (t)S ∗ (b − t)E{z|t }, t ∈ (h(b), b] Now, we define the linear controllability operator 0T ∈ L(L p (b , Y ), L p (b , Y )) as follows: b0 {.} = L b (L b )∗ {.}  h(b)

 = r (t)S(b − r (t))B2 (r (t))B2∗ (r (t))S ∗ (b − r (t))r  (t) 0  +S(b − t)B1 (t)B1∗ (t)S ∗ (b − t) E{.|t }dt  b + S(b − t)B1 (t)B1∗ (t)S ∗ (b − t)E{.|t }dt h(b)

The corresponding controllability operator for deterministic system is:  h(b)

 r (s)S(b − r (s))B2 (r (s))B2∗ (r (s))S ∗ (b − r (s))r  (s)

tb = t  +S(b − s)B1 (s)B1∗ (s)S ∗ (b − s) ds  b + S(b − s)B1 (s)B1∗ (s)S ∗ (b − s)ds h(b)

Definition 2.3 The stochastic system (1.1) is approximate controllable on [0, b] if (b) =

L p (b , Y ), where (b) = {y(b; u) : u ∈ Uad }.

In order to prove main results, we assume the following hypotheses: (i) A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ≥ 0} on Y .

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(ii) The function f : J × Y → Y and σ : J × Y → L02 satisfy linear growth and Lipschitz conditions. i.e, there exist positive constants N1 , N2 , K 1 and K 2 such that ||f(t, y1 ) − f(t, y2 )|| p ≤ N1 ||y1 − y2 || p , ||f(t, y)|| p ≤ N2 (1 + ||y|| p ) p

p

||σ (t, y1 ) − σ (t, y2 )||L0 ≤ K 1 ||y1 − y2 || p , ||σ (t, y)||L0 ≤ K 2 (1 + ||y|| p ) 2

2

(iii) The functions g and g1 are continuous functions and there exist some positive constants Mg and Mg1 such that ||g(y1 ) − g(y2 )|| p ≤ Mg ||y1 − y2 || p , ||g(y)|| p ≤ Mg (1 + ||y|| p ) ||g1 (y1 ) − g1 (y2 )|| p ≤ Mg1 ||y1 − y2 || p , ||g1 (y)|| p ≤ Mg1 (1 + ||y|| p ) for all y1 , y2 ∈ C(J, Y ) (iv) For each 0 ≤ t < b, the operator α(α I + tb )−1 → 0 in the strong operator topology as α → 0+ . Observe that the linear deterministic system with delay corresponding to (1.1) dy  (t) = [A(t)y(t) + B1 (t)u(t) + B2 (t)u(h(t))]dt, t ∈ J y(0) = y0 , y  (0) = y1 and u(t) = 0 for t ∈ [h(0), 0]

 (2.1)

is approximate controllable on [t, b] iff the operator α(α I + tb )−1 → 0 strongly as α → 0+ [14]. For simplicity, let us take l1 = max{||B1 (t)||, ||B2 (t)||, t ∈ [0, b]}, r = max{|r  (t)|, t ∈ [0, b]}

3 Main result Now we state two lemmas which will be useful in proving the main result. The following lemma is required to define the control function. p

Lemma 3 [14] “For any yb ∈ L p (b , Y ), there exists ϕ ∈ L (J, L02 ) such that yb =  b E yb + ϕ(s)dω(s)”. 0

Now for any α > 0 and yb ∈ L p (b , Y ), we define the control function for t ∈ [0, h(b)] α

u (t, y) =

B1∗ (t)S ∗ (b  + 0

h(t)

   b −1 − t) (α I + 0 ) E yb − C(b)(y0 + g(y)) − S(b)(y1 + g1 (y)) 

(α I

+ sb )−1 ϕ(s)dw(s)

−B1∗ (t)S ∗ (b



−B1∗ (t)S ∗ (b − t)

123

h(t)

− t) 

0 h(t) 0

(α I + sb )−1 S(b − s)f(s, y(s))ds (α I + sb )−1 S(b − s)σ (s, y(s))dω(s)

Approximate controllability of second order. . .

Similarly for t ∈ (h(b), b] u α (t, y) = (r  (t)B2∗ (r (t))S ∗ (b − r (t))    +B1∗ (t)S ∗ (b − t)) (α I + 0b )−1 Eyb − C(b)(y0 + g(y)) − S(b)(y1 + g1 (y))   t (α I + sb )−1 ϕ(s)dw(s) − (r  (t)B2∗ (r (t))S ∗ (b − r (t)) + h(t)

+B1∗ (t)S ∗ (b − t))

 t h(t)

(α I + sb )−1 S(b − r (s))f(s, y(s))ds

−(r  (t)B2∗ (r (t))S ∗ (b − r (t))  t (α I + sb )−1 S(b − r (s))σ (s, y(s))dω(s) +B1∗ (t)S ∗ (b − t)) h(t)

Lemma 4 There exists a positive constant Mu such that for all y1 , y2 ∈ H2 , we have Mu ||y1 − y2 || p αp  Mu  E||u α (t, y)|| p ≤ p 1 + ||y|| p α

E||u α (t, y1 ) − u α (t, y2 )|| p ≤

(3.1) (3.2)

Proof Let y1 , y2 ∈ H2 . Using Holder’s inequality, lemma 1, lemma 2 and the assumed conditions, we get for t ∈ [0, h(b)] E||u α (t, y1 ) − u α (t, y2 )|| p

p −1 ≤ 4 p−1 E B1 ∗ (t)S ∗ (b − t)(α I + 0 b ) C(b)[g(y1 ) − g(y2 )] p p−1 ∗ ∗ b −1 +4 E B1 (t)S (b − t)(α I + 0 ) S(b)[g1 (y1 ) − g1 (y2 )] +4

p−1

 E B1∗ (t)S ∗ (b − t)

h(t) 0

(α I + sb )−1 S(b − s)

p ×[f(s, y1 (s)) − f(s, y2 (s))]ds  h(t) p−1 ∗ ∗ +4 E B1 (t)S (b − t) (α I + sb )−1 S(b − s) 0 p × [σ (s, y1 (s)) − σ (s, y2 (s))]ds  4 p−1 p 2 p p p ≤ l M Mg ||y1 − y2 ||H2 + Mg1 ||y1 − y2 ||H2 αp 1   b  b + b( p/q) N1 E||y1 (s) − y2 (s)|| p ds + L G K 1 E||y1 (s) − y2 (s)|| p ds 0

≤

4 p−1 αp

0



p

p

p

l1 M 2 p Mg ||y1 − y2 ||H2 + Mg1 ||y1 − y2 ||H2

+ b( p/q) N1 b sup E||y1 (s) − y2 (s)|| p + L G K 1 b sup E||y1 (s) − y2 (s)|| p s∈[0,b]



s∈[0,b]

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U. Arora, N. Sukavanam

  4 p−1 p 2 p p ( p/q) l M + M + b bN + L K b ||y1 − y2 ||H2 M g g 1 G 1 1 αp 1 R1 p = p ||y1 − y2 ||H2 α ≤

Similarly for t ∈ (h(b), b] E||u α (t, y1 ) − u α (t, y2 )|| p

  4 p−1  p−1 2 p p p p p 2 ≤ M (||B1 || + r ||B2 || ) Mg ||y1 − y2 ||H2 + Mg1 ||y1 − y2 ||H2 αp   b  b ( p/q) p p +b N1 E||y1 (s) − y2 (s)|| ds + L G K 1 E||y1 (s) − y2 (s)|| ds 0

0

  4 p−1  p−1 2 p p p p ≤ M l (1 + r ) Mg ||y1 − y2 ||H2 + Mg1 ||y1 − y2 ||H2 2 1 αp + b( p/q) bN1

sup E||y1 (s) − y2 (s)|| p + L G bK 1 s∈[0,b]

≤

4 p−1



sup E||y1 (s) − y2 (s)|| p s∈[0,b]

  p p−1 2 p p ( p/q) (2 M l1 (1 + r )) Mg + Mg1 + b bN1 + L G K 1 b ||y1 − y2 ||H2

αp R2 p = p ||y1 − y2 ||H2 α

where p,q are conjugate indices. Combining both for t ∈ [0, b], we get E||u α (t, y1 ) − u α (t, y2 )|| p ≤

Mu p ||y1 − y2 ||H2 , where Mu = max{R1 , R2 }. αp

The second inequality can be proved in the similar manner by putting u α (t, y2 ) = 0. So, the proof of the lemma is completed.

Now, for any α > 0, define the operator Tα : H2 → H2 by  (Tα y)(t) = C(t)(y0 + g(y)) + S(t)(y1 + g1 (y) + 

h(t)

+

t

S(t − s)[B1 u α (s, y) + f(s, y(s))]ds

0

S(t − r (s))B2 (r (s))r  (s)u α (s, y)ds +

0



t

S(t − s)σ (s, y(s))dω(s)

0

To prove the required result, first of all the existence of a fixed point of the operator Tα defined above has been proved in Theorem 3.1 with the help of the contraction mapping principle. Then in Theorem 3.2, the approximate controllability of system (1.1) has been proved by assuming the approximate controllability of the corresponding deterministic linear system under some conditions. Theorem 3.1 If the conditions (i)–(iv) are satisfied, then the system (1.1) has a mild solution on [0, b]. Proof The proof of this theorem is divided into two steps. Step 1. For any y ∈ H2 , Tα (y)(t) is continuous on J .

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Approximate controllability of second order. . .

Let 0 ≤ t1 ≤ t2 ≤ b. Then for any fixed y ∈ H2 , it follows from Holder’s inequality, Lemmas 1, 2 and assumptions on the theorem that E||(Tα y)(t2 ) − (Tα y)(t1 )|| p

 ≤ 10 p−1 E||(C(t2 ) − C(t1 ))(y0 + g(y))|| p + E||(S(t2 ) − S(t1 ))(y1 + g1 (y))|| p p p  t  t 1 2 + E (S(t2 − s) − S(t1 − s))f(s, y(s))ds + E S(t2 − s)f(s, y(s))ds 0 t1 p  t 1 + E (S(t2 − s) − S(t1 − s))σ (s, y(s))dω(s) 0 p  t 2 +E S(t2 − s)σ (s, y(s))dω(s) t  1 t  t p p 1 2 + E (S(t2 − s) − S(t1 − s))B1 u α (s, y)ds + E S(t2 − s)B1 u α (s, y)ds 0 t1 p  h(t ) 1 + E (S(t2 − r (s)) − S(t1 − r (s)))B2 u α (s, y)ds 0 h(t2 )

p  S(t2 − r (s))B2 u (s, y)ds h(t1 )   p−1 p−1 ≤ 10 2 E||(C(t2 ) − C(t1 ))y0 || p + E||(C(t2 ) − C(t1 ))g(y)|| p  + E||(S(t2 ) − S(t1 ))y1 || p + E||(S(t2 ) − S(t1 ))g1 (y)|| p  +E

p/q



+t1

t1

α

E||(S(t2 − s) − S(t1 − s))f(s, y(s))|| p ds

0



+ M (t2 − t1 ) p

t2

p/q

E||f(s, y(s))|| p ds

t1



t1

+L G 0

+M p L G

E||(S(t2 − s) − S(t1 − s))σ (s, y(s))|| p ds



t2

t1

p/q



p

+ l1 M p (t2 − t1 ) p/q 

h(t1 )

+ (h(t1 )) p/q

t2

t1

E||(S(t2 − s) − S(t1 − s))B1 u α (s, y)|| p ds

0

E||u α (s, y)|| p ds

t1

E||(S(t2 − r (s)) − S(t1 − r (s)))B2 u α (s, y)|| p ds

0 p



E||σ (s, y(s))|| p ds + t1

+ l1 M p (h(t2 ) − h(t1 )) p/q



h(t2 ) h(t1 )

E||u α (s, y)|| p ds



Now, using the strong continuity of C(t) and S(t) , Lemmas 1, 2 and Lebesgue’s dominated convergence theorem, we observe that the right hand side of the above inequality tends to zero as t2 − t1 → 0. Thus we conclude Tα (y)(t) is continuous from the right in [0, b). Similarly, we can show that it is also continuous from the left in (0, b]. Therefore Tα (y)(t) is continuous on J in the L p sense. Step 2: We show that Tα (H2 ) ⊂ H2 .

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U. Arora, N. Sukavanam

Let y ∈ H2 , then we have p

E||(Tα y)||H2

 = E C(t)(y0 + g(y)) + S(t)(y1 + g1 (y)) +

t

0



h(t)

+ 0

S(t − r (s))B2 (r (s))r  (s)u α (s, y)ds +

S(t − s)[B1 u α (s, y) + f(s, y(s))]ds  0



t

p S(t − s)σ (s, y(s))dω(s)

≤ 6 p−1 sup E||C(t)(y0 + g(y))|| p + sup E||S(t)(y1 + g1 (y))|| p  + sup t∈J

t∈J t

E||S(t − s)[B1 u α (s, y) + f(s, y(s))]|| p ds

0



t

+ sup t∈J

t∈J

E||S(t − r (s))B2 (r (s))r  (s)u α (s, y)|| p ds

0

  t + sup E||S(t − s)σ (s, y(s))|| p dω(s) t∈J 0    p p p−1 p−1 p p p ≤6 M ||y0 || + ||y1 || + Mg (1 + ||y||H2 ) + Mg1 (1 + ||y||H2 ) 2 p

p M p b p/q l1 bMu p p (1 + ||y||H2 ) + M p [b p/q N2 + b 2 −1 K 2 bL G ](1 + ||y||H2 ) αp  p M p r p b p/q l1 bMu p (1 + ||y|| ) + H2 αp

+

p

Therefore, above condition implies that E||(Tα y)||H2 < ∞. Since (Tα y)(t) is continuous on [0, b] and so Tα maps H2 into H2 . Now we will show that, the operator Tα has a unique fixed point in H2 for each fixed α > 0. We claim that there exists a natural n such that Tαn is a contraction on H2 . For this, let y1 , y2 ∈ H2 so for t ∈ [−h, b], we have E||(Tα y1 )(t) − (Tα y2 )(t)|| p

= E C(t)(g(y1 ) − g(y2 )) + S(t)(g1 (y1 ) − g1 (y2 )) 

+

t

S(t − s)B1 [u α (s, y1 ) − u α (s, y2 )]ds

0

 +

t

S(t − s)[f(s, y1 (s)) − f(s, y2 (s))]ds

0



h(t)

S(t − r (s))B2 (r (s))r  (s)[u α (s, y1 ) − u α (s, y2 )]ds p  t + S(t − s)[σ (s, y1 (s)) − σ (s, y2 (s))]dω(s) 0  p p Mu p−1 p ≤6 M Mg + Mg1 + l1 p b p/q b + N1 b p/q b + K 1 L G b 2 −1 α  Mu p p + l1 r p p b p/q b × ||y1 − y2 ||H2 α +

0

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Approximate controllability of second order. . .

So, we get a positive real constant β(α) such that p

E||(Tα y1 )(t) − (Tα y2 )(t)|| p ≤ β(α)||y1 − y2 ||H2

for all t ∈ J and for any y1 , y2 ∈ H2 . Moreover,  E||Tα2 (y1 )(t) − Tα2 (y2 )(t)|| p ≤ β(α)



t 0 t

≤ β(α) 0

E||Tα (y1 )(s) − Tα (y2 )(s)|| p ds p

β(α)||y1 − y2 ||H2 ds p

= β 2 (α)t||y1 − y2 || H2 Using Mathematical Induction, we obtain  E||Tαn (y1 )(t) − Tαn (y2 )(t)|| p ≤ β(α)

≤

t 0

E||Tαn−1 (y1 )(s) − Tαn−1 (y2 )(s)|| p ds

(t n−1 )(β(α))n p ||y1 − y2 ||H2 (n − 1)!

In general, p

||Tαn (y1 ) − Tαn (y2 )||H2 ≤

(bn−1 )(β(α))n p ||y1 − y2 ||H2 (n − 1!)

(β(α)) For any fixed α > 0, there exists n such that b (n−1)! < 1. It implies that Tαn is a contraction mapping for sufficiently large n. Then, by using the contraction mapping principle, the operator Tα has a unique fixed point yα in H2 , which is the mild solution of (1.1).

n−1

n

Theorem 3.2 If the assumptions (i)–(iv) are satisfied and {S(t) : t ≥ 0} is compact and f, σ are uniformly bounded, then the system (1.1) is approximate controllable on [0, b]. Proof Let yα be a fixed point of Tα in H2 . By using the stochastic Fubini theorem, we get that   yα (b) = yb − α(α I + ob )−1 E yb − C(b)(y0 + g(y)) − S(b)(y1 + g1 (y))  +α

b

0

 +α

0

b

(α I + sb )−1 S(b − s)f(s, yα (s))ds (α I + sb )−1 [S(b − s)σ (s, yα (s)) − ϕ(s)] dω(s)

Since f and σ are uniformly bounded, there exists D > 0 such that ||f(s, yα (s))|| p + ||σ (s, yα (s))|| p ≤ D in [0, b] × . Then we have a subsequence denoted by {f(s, yα (s)), σ (s, yα (s))} weakly converging to say {f(s, ω), σ (s, ω)} in Y × L02 . Now, using the compactness of S(t), we get S(b − s)f(s, yα (s)) → S(b − s)f(s), S(b − s)σ (s, yα (s)) → S(b − s)σ (s) in J × . Thus,

123

U. Arora, N. Sukavanam

we get E||yα (b) − yb || p

  p ≤ 6 p−1 α(α I + 0b )−1 E yb − C(b)[y0 + g(y)] − S(b)[y1 + g1 (y)]  + 6 p−1 E

b

+ 6 p−1 E

b

0

 + 6 p−1 E

b

0

 + 6 p−1 E

b

0

 + 6 p−1 E

0

 p/2

2

0



||α(α I + sb )−1 ϕ(s)||2L0 ds

b

||α(α I + sb )−1 || ||S(b − s)[f(s, yα (s)) − f(s)]||ds ||α(α I + sb )−1 S(b − s)f(s)||ds

p

p

||α(α I + sb )−1 || ||S(b − s)[σ (s, yα (s)) − σ (s)]||2L0 ds ||α(α I + sb )−1 S(b − s)σ (s)||2L0 ds

 p/2 

 p/2

2

2

Since by assumption (iv), for all 0 ≤ s < b the operator α(α I + sb )−1 → 0 strongly as α → 0+ and moreover ||α(α I + sb )−1 || ≤ 1. Thus by the Lebesgue dominated convergence theorem, we obtain E||yα (b) − yb || p → 0+ . So, we get the approximate controllability of the given system.

Remark 3.1 The system (1.1) is exactly controllable on [−h, b] if (b) = L p (b , Y ). When the semigroup associated with the dynamical system is compact or the control operator B1 , B2 are compact, then dynamical system is never exactly controllable at infinite dimensional state space. In order to prove the exact controllability result, the compactness of the semigroup S(t) should not be used.

4 Examples Example 1 Consider the stochastic wave equation with delay  ∂

∂z(t,y) ∂t



 =

t ∈ J, 0 ≤ y < π z(t, 0) = z(t, π ) = 0, t ∈ J = [0, b] n  z(0, y) + αi z(ti , y) = z 0 (y), t ∈ J i=1 k 

z t (0, y) +

i=1



⎫

⎪ ∂ 2 z(t,y) + u(t, y) + Bu(t/2, y) + p(t, z(t, y)) ∂t + k(t, z(t, y))∂ω(t), ⎪ ⎪ ⎪ ∂ y2 ⎪

βi z(si , y) = z 1 (y)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.1) where B is a bounded linear operator from the Hilbert space U → Y and p : J × Y → Y , k : J × Y → L02 are all continuous and uniformly bounded, u(t) is a feedback control and w is a Q-Wiener process. h(t) = t − t/2 = t/2 is continuous differentiable and strictly increasing on J and here h 1 (t) = t/2 > 0.

123

Approximate controllability of second order. . .

Let Y = L2 [0, π] and A : D(A) ⊂ Y → Y be defined by Az = z yy where D(A) = {z(.) ∈ Y |z, z y are absolutely continuous , z yy ∈ Y , z(0) = z(π) = 0} Furthermore, A has discrete spectrum, the eigen values are −n 2 , n = 1, 2, . . . with the corresponding normalized characteristic vectors en (s) = (2/π)1/2 sin ns,then Az =

∞  (−n 2 ) < z, en > en , z ∈ Y n=1

Here A is the generator of strongly continuous cosine family{C(t) : t ∈ R} on Y given by C(t)z =

∞ 

cos nt < z, en > en , z ∈ Y

n=1

and the associated sine family is given by

S(t)z =

∞  1 sin nt < z, en > en , z ∈ Y n n=1

Let f : J × Y → Y be defined by f(t, z)(y) = p(t, z(y)),

z ∈ Y , y ∈ [0, π].

Let σ : J × Y → L02 be defined by σ (t, z)(y) = k(t, z(y)) The functions g : C(J, Y ) → Y is defined as

g(z)(y) =

n  i=1

αi z(ti , y) and g1 (z)(y) =

k 

βi z(si , y)

i=1

for 0 < ti , s1 < b and y ∈ [0, π] Consider the time lead function r (t) = 2t. Here B1 = I and B2 = B. So, from the above choices of the functions and generator A, the system (4.1) can be formulated as an abstract second order semilinear system (1.1).

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U. Arora, N. Sukavanam

 Define tb =

t

b

S(b − s)B1 B1∗ S ∗ (b − s)ds =



b

S(b − s)S ∗ (b − s)ds. We claim that

t

S ∗ (b − s)z = 0, t ≤ s ≤ b implies that z = 0 [11]. Indeed, 

∗

S (b − s)z = 0, t ≤ s ≤ b ⇒ ⇒

b

t ∞  n=1

||S ∗ (b − s)z||2 ds =< tb z, z >= 0 1 n2



b

sin2 (n(b − s))ds(z, en )2 = 0

t

  ∞  1 1 − sin 2(n(b − s)) s=b (z, en )2 = 0 ⇒ 2n 2 2n s=t n=1   ∞  1 sin 2(n(b − t)) b − t + (z, en )2 = 0 ⇒ 2n 2 2n n=1

⇒ (z, en )2 = 0 for all n ≥ 1 ⇒z=0 It follows that the operator α(α I + tb )−1 → 0 strongly as α → 0+ [14]. It implies that the deterministic linear system without delay associated with (4.1) is approximate controllable on [0, b] (see [20]). From this, we can prove that the linear deterministic system with delay is also approximate controllable on [0, b] by the method of steps [24]. Therefore (iv) condition of the Theorem (3.1) is satisfied. Now if we assume that the functions f,σ , g and g1 satisfy the requirement of hypotheses, then the approximate controllability of the system (4.1) on J follows from Theorems (3.1) and (3.2). Example 2 Consider the following finite dimensional second order stochastic system:  ⎫ x1 (t) cos x2 (t) ⎬ = x1 (t) + u 1 (t) + u 2 (t) + u 1 (0.75t) + dt + dω1 (t), ⎪ 3 x2 (t) sin x1 (t) ⎪ ⎭ dω2 (t) d x2 (t) = [x2 (t) + u 1 (t)]dt + 4

d x1 (t)



(4.2) which can be formulated in the form of (1.1) with    u 1 (t) x1 (t) , u(t) = , x2 (t) u 2 (t)     10 11 A = , B1 = , 01 10     10 ω1 (t) , ω(t) = , B2 = ω2 (t) 00 ⎤ ⎡ x1 (t) cos x2 (t) 0 ⎥ ⎢ 3 σ (t, x(t)) = ⎣ x2 (t) sin x1 (t) ⎦ 0 4 

x(t) =

Moreover h(t) = 0.75t for t ∈ [0, 2]. Consider the lead function r (t) = 43 t.

123

Approximate controllability of second order. . .



ei At + e−i At Here C(t) = cos At = 2   ei At − e−i At sin t 0 = . Consider 0 sin t 2i 

b

S(b 0

− s)B1 (s)B1∗ (s)S ∗ (b

=

 − s) =

2

0



h(2)

0

 and S(t) = sin At =

sin A(2 − s)B1 (s)B1∗ (s) sin A(2 − s)

= 1− Also

cos t 0 0 cos t

1 sin2 1 > 0 16

r  (s)S(b − r (s))B2 (r (s))B2∗ (r (s))S ∗ (b − r (s))ds = 0

Thus we have the controllability Grammian matrix  h(b)

 b

0 = r (s)S(b − r (s))B2 (r (s))B2∗ (r (s))S ∗ (b − r (s))r  (s) 0

 + S(b − s)B1 (s)B1∗ (s)S ∗ (b − s) ds +  = 0

h(b)

 0

= 1−

b h(b)

S(b − s)B1 (s)B1∗ (s)S ∗ (b − s)ds

[r  (s)S(b − r (s))B2 (r (s))B2∗ (r (s))S ∗ (b − r (s))r  (s)]ds

b

+



S(b − s)B1 (s)B1∗ (s)S ∗ (b − s)ds

1 sin2 1 > 0 16

So, the controllability Grammian matrix 0b is nonsingular. Hence the linear deterministic system with delay corresponding to (4.2)   ⎫ ⎬  d x1 (t) = x1 (t) + u 1 (t) + u 2 (t) + u 1 (0.75t) + dt (4.3) ⎭ d x2 (t) = [x2 (t) + u 1 (t)]dt is controllable. Also ||σ (t, x(t))||2 ≤

1 (1 + ||x||2 ). 9

One can see that all the other conditions of Theorems (3.1) and (3.2) are satisfied. Hence the system is controllable on [0, 2].

5 Conclusion In the present problem, we have established the approximate controllability result for a class of second order semilinear stochastic systems with variable delay in control and with nonlocal conditions in Hilbert spaces. By employing Banach Fixed Point theorem and solution operator theory, sufficient conditions for the approximate controllability of the proposed system are formulated and proved under the assumption that the associated deterministic linear system is approximate controllable.

123

U. Arora, N. Sukavanam Acknowledgments The authors are grateful to RCMP and the anonymous reviewers for their detailed comments and suggestions which have helped to improve the quality of this paper and the first author is thankful to “Ministry of Human Resources Development(MHRD)” India for financial support to carry out her research work.

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