Ann Univ Ferrara DOI 10.1007/s11565-015-0232-9

Approximate controllability of second order semilinear stochastic system with nonlocal conditions Anurag Shukla1 · N. Sukavanam1 · D. N. Pandey1

Received: 26 January 2015 / Accepted: 10 September 2015 © Università degli Studi di Ferrara 2015

Abstract This paper deals with the approximate controllability of second order semilinear stochastic system with nonlocal conditions. To establish sufficient conditions for approximate controllability, Banach fixed point theorem together with the theory of strongly continuous cosine family is been used. At the end, an example is given to illustrate the theory. Keywords Approximate controllability · Semilinear systems · Stochastic control system · Fixed point principle Mathematics Subject Classification

34K30 · 34K35 · 93C25

1 Introduction Controllability concepts play a vital role in deterministic control theory. It is well known that controllability of deterministic equation is widely used in many fields of science and technology. Kalman [1] introduced the concept of controllability for finite dimensional deterministic linear control systems. The basic concepts of control theory in finite and infinite dimensional spaces has been introduced in [2,3]. However, in many cases, some kind of randomness can appear in the problem, so that the system should be modeled by a stochastic form. Only few authors have studied the extensions of deterministic controllability concepts to stochastic control systems. Klamka et al. [4,5] studied the controllability of linear stochastic systems in finite dimensional spaces with delay and without delay in control as well as in state. In [6–11], Mahmudov et

B 1

Anurag Shukla [email protected] Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

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al. established results for controllability of linear and semilinear stochastic systems in Hilbert spaces. Instead of this, Sakthivel, Balachandran, Dauer and Bashirov et al. studied the approximate controllability of nonlinear stochastic systems in [12–15]. The controllability of second order stochastic systems has been discussed in [16,17]. In many cases, it is advantageous to treat the second order stochastic differential equations directly rather than to convert them to first order systems. For the basic theory of second order linear and partial differential equations, the reader can refer to [18,19]. Second order systems capture the dynamic behavior of many natural phenomena and have found applications in many fields such as in mathematical physics, biology and finance. On the other hand, Byszewski et al. [20] introduced nonlocal conditions into the initial value problems and argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the one set 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) = x0 alone. Kumar and Sukavanam [17] studied on the controllability of second order systems with nonlocal conditions in Banach spaces using Sadovskii’s fixed point theorem. Anurag et al. in [21–24] obtained some sufficient conditions for approximate and complete controllability of semilinear stochastic with delay in state and control using suitable fixed point theorems. There are no results on the approximate controllability of stochastic second order differential systems with nonlocal conditions in the literature. Therefore, it is necessary and important to consider controllability result for second order stochastic systems with nonlocal conditions. Let (, , P) be a complete space equipped with a normal filtration t , t ∈ J = [0, b]. Let H, U and E be the separable Hilbert spaces and ω be a Q-Wiener process on (, b , P) with the covariance operator Q such that tr Q < ∞ then there exists a complete orthonormal system en in E, a bounded sequence of nonnegative real numbers λn such that Qen = λn en , n = 1, 2, . . . and a sequence βn of independent Brownian motions such that ω(t), e =

∞   λn en , eβn (t), e ∈ E, t ∈ J k=1

and t = t ω , where t ω is the σ -algebra generated by ω. Let L 2 0 = L 2 (Q 1/2 E; H ) be the space of all Hilbert-Schmidt operators from Q 1/2 E to H with the inner product ψ, π  L 0 = tr [ψ Qπ ∗ ]. Let L p (b , H ) be the Banach space of all b 2 measurable pth power integrable random variables with values in the Hilbert space H . Let C([0, b]; L p (, H )) be the Banach space of continuous maps from [0, b] into L p (, H ) satisfying the condition supt∈J E||x(t)|| p < ∞. Let H2 be the closed subspace of C([0, b]; L p (, H )) consisting of measurable and t -adapted H valued processes φ ∈ C([0, b]; L p (, H )) endowed with the norm  ||φ|| H2 =

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1/ p sup

t∈[0,b]

p E||φ(t)|| H

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In this paper under suitable conditions we establish approximate controllability of the following nonlinear second order stochastic differential equations with nonlocal conditions:  d x  (t) = [Ax(t) + Bu(t) + f (t, x(t))]dt + σ (t, x(t))dω(t), t ∈ J (1.1) x(0) = x0 + g(x), x  (0) = x1 + g1 (x) where A : D(A) ⊂ H → H is a closed, linear and densely defined operator on H which generates a strongly continuous cosine family {C(t) : t ∈ J } on H . B is a bounded linear operator from the Hilbert space U into H . The control p u ∈ L  ([0, b], U ); f : J × H → H ; σ : J × H → L 02 are suitable nonlinear functions; x0 and x1 are 0 measurable H valued random variables independent of ω; g, g1 are continuous functions from C(J, H ) → H .

2 Preliminaries This section is concerned with some basic definitions and lemmas which are used throughout the paper. A measurable function x : J → H is Bochner integrable if and only if |x(t)| is Lebesgue integrable. Let L 1 (J, H ) denotes the Banach space of functions x : J → H which are Bochner integrable with the norm [5].  ||x|| L 1 =

b

|x(t)|dt

0

For the more properties of Bochner integral, we refer Yosida [9] to the readers. Definition 2.1 Let L(E; H ) be the space of bounded linear operators from E into H . The one parameter family {C(t) : t ∈ R} ⊂ L(H ) is called a strongly continuous cosine family if 1. C(0) = I 2. C(t)x is continuous in t on R, for all x ∈ H 3. C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ R, is called a strongly continuous cosine family. If {C(t) : t ∈ R} is a strongly continuous cosine family on H , then the correspondingstrongly continuous sine family {S(t) : t ∈ R} ⊂ L(H ) is defined by t S(t)x = 0 C(s)xds, t ∈ R, x ∈ H . The infinitesimal generator of a strongly continuous cosine family of {C(t) : t ∈ R} d2 is the operator A : D(A) ⊂ H → H is defined by Ax = C(t)x |t=0 for all dt 2 x ∈ D(A) where D(A) = {x ∈ H : C(t)x is a twice continuously differentiable function of t}. Such cosine and corresponding sine families and their generators satisfy the following properties:

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Lemma 1 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 S(u)xdu = [C(r ) − C(s)]x for all 0 ≤ s ≤ r < ∞,  r ω|s|   3. There exists N ≥ 1 such that ||S(s) − S(r )|| ≤ N s e ds for all 0 ≤ s ≤ r < ∞. 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 Let G : J ×  → L 02 be a strongly measurable mapping such that b p 0 E||G(t)|| L 0 dt < ∞. Then 2

p  t  t p G(s)dω(s) ≤ L G E||G(s)|| L 0 ds E 2 0

0

for all t ∈ J and p ≥ 2, where L G is the constant involving p and b. Definition 2.2 A stochastic process x ∈ H2 is a mild solution of (1.1) if for each p u ∈ L  ([0, b], U ), it satisfies the following integral equation:  x(t) = C(t)(x0 + g(x)) + S(t)(x1 + g1 (x)) +  t + S(t − s)σ (s, x(s))dω(s)

t

S(t − s)[Bu(s) + f (s, x(s))]ds

0

0

Definition 2.3 System (1.1) is approximately controllable on [0, b] if (b) = L p (b , H ), where p

(b) = {x(b; u) : u ∈ L  ([0, b], U )} p

p

where L  ([0, b], U ) is the closed subspace of L  ([0, b] × , U ), consisting of all t adapted, U valued stochastic processes. In order to prove our main results, we assume the following hypotheses: (i) A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ≥ 0} on H . (ii) The function f : J × H → H and σ : J × H → L 02 satisfy linear growth and Lipschitz conditions. Moreover, there exist positive constants N1 (.) > 0, N2 (.) > 0, K 1 (.) > 0 and K 2 (.) ∈ L 1 (J, R) such that || f (t, x) − f (t, y)|| p ≤ N1 (t)||x − y|| p , || f (t, x)|| p ≤ N2 (t)(1 + ||x|| p ) p

p

||σ (t, x) − σ (t, y)|| L 0 ≤ K 1 (t)||x − y|| p , ||σ (t, x)|| L 0 ≤ K 2 (t)(1 + ||x|| p ) 2

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(iii) The functions g and g1 are continuous functions and there exists some positive constants Mg and Mg1 such that ||g(x) − g(y)|| p ≤ Mg ||x − y|| p , ||g(x)|| p ≤ Mg (1 + ||x|| p ) ||g1 (x) − g1 (y)|| p ≤ Mg1 ||x − y|| p , ||g1 (x)|| p ≤ Mg1 (1 + ||x|| p ) for all x, y ∈ C(J, H ) (iv) For each 0 ≤ t < b, the operator α(α I + tb )−1 → 0 in the strong operator topology as α → 0+ , where  tb =

b

S(b − s)B B ∗ S ∗ (b − s)ds

t

is the controllability Grammian. Observe that the linear deterministic system corresponding to (1.1) d x  (t) = [Ax(t) + Bu(t)]dt, t ∈ J x(0) = x0

 (2.1)

is approximately controllable on [t, b] iff the operator α(α I + tb )−1 → 0 strongly as α → 0+ .

3 Controllability results Let us recall two lemmas concerning approximate controllability, which will be used in the proof. The following lemma is required to define the control function p

Lemma 3 For any xb ∈ L p (b , H ), there exists φ ∈ L  (; L 2 (0, b; L 20 )) such that b xb = Exb + 0 φ(s)dω(s). Now for any α > 0 and xb ∈ L p (b , H ), we define the control function in the following form 

b −1 Exb −C(b)(x0 +g(x))− S(b)(x1 + g1 (x)) u (t, x) = B S (b − t) (α I + 0 )

 t b −1 + (α I + s ) φ(s)dω(s) 0  t ∗ ∗ −B S (b − t) (α I + sb )−1 S(b − s) f (s, x(s))ds 0  t −B ∗ S ∗ (b − t) (α I + sb )−1 S(b − s)σ (s, x(s))dω(s) α

∗ ∗

0

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

E||u α (t, x) − u α (t, y)|| p ≤

(3.1) (3.2)

Proof Let x, y ∈ H2 . From Hölder’s inequality, Lemma 2 and the assumptions on the data, we obtain p ∗ ∗ b −1 E B S (b−t)(α I +ψ0 ) C(b)[g(x) − g(y)] E||u (t, x)−u (t, y)|| ≤ 4 p p−1 ∗ ∗ b −1 +4 E B S (b − t)(α I + ψ0 ) S(b)[g1 (x) − g1 (y)] p  t p−1 ∗ ∗ b −1 +4 E B S (b−t) (α I + s ) S(b − s)[ f (s, x(s)) − f (s, y(s))]ds 0 p  t p−1 ∗ ∗ b −1 +4 E B S (b−t) (α I + s ) S(b − s)[σ (s, x(s)) − σ (s, y(s))]ds α

α

p

p−1

0

 t 4 p−1 p p p 2p ( p/q) M ≤ ||B|| M ||x − y|| + M ||x − y|| +b |N1 (s)| E||x(s) g g1 H2 H2 αp 0

 t −y(s)|| p ds + L G |K 1 (s)| E||x(s) − y(s)|| p ds ≤

4 p−1



||B|| M αp Mu p = p ||x − y|| H2 α p

2p

0

Mg + Mg1 + b

( p/q)

||N1 || L 1

p + L G ||K 1 || L 1 ||x − y|| H2



where Mu = 4 p−1 ||B|| p M 2 p Mg + Mg1 + b( p/q) ||N1 || L 1 + L G ||K 1 || L 1 and p, q are conjugate indices. The proof of the second inequality can be verified in a similar manner. The proof of lemma is complete.   For any α > 0, define the operator Pα : H2 → H2 by  t (Pα x)(t) = C(t)(x0 + g(x)) + S(t)(x1 + g1 (x)) + S(t − s)[Bu α (s, x) 0  t + f (s, x(s))]ds + S(t − s)σ (s, x(s))dω(s) 0

To prove the approximate controllability, we first prove in Theorem 3.1, the existence of a fixed point of the operator Pα defined above, using the contraction mapping principle. Second, in Theorem 3.2, we show that under certain assumptions the approximate

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controllability of system (1.1) is implied by the approximate controllability of the corresponding deterministic linear system (2.1). Theorem 3.1 Assume hypothesis (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 several steps. Step 1. For any x ∈ H2 , Pα (x)(t) is continuous on J in the L p sense. Proof of Step 1: Let 0 ≤ t1 ≤ t2 ≤ b. Then for any fixed x ∈ H2 , it follows from Holder’s inequality, Lemma 2 and assumptions on the theorem that  E||(C(t2 ) − C(t1 ))(x0 + g(x))|| p + E||(S(t2 ) E||(Pα x)(t2 ) − (Pα x)(t1 )|| ≤ 8  t1 p p [S(t2 − s) − S(t1 − s)] f (s, x(s))ds −S(t1 ))(x1 + g1 (x))|| + E 0  t p 2 +E S(t2 − s) f (s, x(s))ds t  1t1 p +E [S(t2 − s) − S(t1 − s)]σ (s, x(s))dω(s) 0  t2 p +E S(t2 − s)σ (s, x(s))dω(s) t  1t p 1 α +E [S(t2 − s) − S(t1 − s)]Bu (s, x)ds 0  t2 p α +E S(t2 − s)Bu (s, x)ds t

1  p−1 p−1 2 E||(C(t2 ) − C(t1 ))x0 || p + E||(C(t2 ) − C(t1 ))g(x)|| p ≤8 p p +E||(S(t2 ) − S(t1 ))x1 || + E||(S(t2 ) − S(t1 ))g1 (x)||  t1 p/q E||[S(t2 − s) − S(t1 − s]) f (s, x(s))|| p ds +t1 0  t2 p p/q +M (t2 − t1 ) E|| f (s, x(s))|| p ds p



p−1

t1 t1

E||(S(t2 − s) − S(t1 − s))σ (s, x(s))|| p ds  t2 +M p L G E||σ (s, x(s))|| p ds +L G

0



t1 t1

E||[S(t2 − s) − S(t1 − s])Bu α (s, x)|| p ds 0  t2 p +||B|| M p (t2 − t1 ) p/q E||u α (s, x)|| p ds p/q

+t1

t1

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Hence using the strong continuity of C(t) and S(t) , together with the properties of Lemma 1 and Lebesgue’s dominated convergence theorem, we conclude that the right hand side of the above equality tends to zero as t2 − t1 → 0. Thus we conclude Pα (x)(t) is continuous from the right in [0, b). A similar argument shows that it is also continuous from the left in (0, b]. Thus Pα (x)(t) is continuous on J in the L p sense. Step 2: We show that Pα (H2 ) ⊂ H2 . Proof of Step 2: Let x ∈ H2 , we obtain

p E||(Pα x)|| H2

 + sup t∈J

t

≤5

p−1

sup E||C(t)(x0 + g(x))|| p + sup E||S(t)(x1 + g1 (x))|| p t∈J

t∈J

E||S(t − s)[Bu α (s, x) + f (s, x(s))]|| p ds

0

 t + sup E||S(t − s)σ (s, x(s))|| p dω(s) t∈J 0

 p p p−1 p−1 p p p 2 M ||x0 || + ||x1 || + Mg (1 + ||x|| H2 ) + Mg1 (1 + ||x|| H2 ) ≤5 +

M p (b p/q )||B|| p bMu p (1 + ||x|| H2 ) αp p

+M p [b p/q ||N2 || L 1 + b 2 −1 ||K 2 || L 1 L G ](1 + ||x|| H2 )

p

p

Hence above condition implies that E||(Pα x)|| H2 < ∞. Since (Pα x)(t) is continuous on [0, b] therefore Pα maps H2 into H2 . Now we prove that for each fixed α > 0, the operator Pα has a unique fixed point in H2 . We claim that there exists a natural n such that Pαn is a contraction on H2 . To see this, let x, y ∈ H2 so for t ∈ [−h, b], we have E||(Pα x)(t) − (Pα y)(t)|| p = E C(t)[g(x) − g(y)] + S(t)[g1 (x) − g1 (y)]  t + S(t − s)B[u α (s, x) − u α (s, y)]ds 0  t + S(t − s)[ f (s, x(s)) − f (s, y(s))]ds 0 p  t S(t − s)[σ (s, x(s)) − σ (s, y(s))]dω(s) + 0  Mu p−1 p ≤5 M Mg + Mg1 + ||B|| p p b p/q b + ||N1 || L 1 b p/q α p p +||K 1 || L 1 L G b 2 −1 ||x − y|| H2

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Hence we obtain a positive real constant γ (α) such that p

E||(Pα x)(t) − (Pα y)(t)|| p ≤ γ (α)||x − y|| H2 for all t ∈ J and for any x, y ∈ H2 . For any natural number n, it follows from successive iterations of above inequality, upon taking the supremum over [0, b], p

||(Pαn x)(t) − (Pαn y)(t)|| H2 ≤

bn−1 (γ (α))n p ||x − y|| H2 (n − 1)!

(γ (α)) For any fixed α > 0, there exists n such that b (n−1)! < 1. It follows that Pαn is a contraction mapping for sufficiently large n. Then, by the contraction principle the   operator Pα has a unique fixed point xα in H2 , which is mild solution of (1.1). n−1

n

Theorem 3.2 Assume assumptions (i)–(iv) are satisfied and {S(t) : t ≥ 0} is compact. Moreover, if f and σ are uniformly bounded, then the system (1.1) is approximately controllable on [0, b]. Proof Let xα be a fixed point of Pα in H2 . By using the stochastic Fubini theorem, it is easy to see that xα (b) = xb − α(α I  +α

b

0

 +α

0

b

+ ob )−1

 Exb − C(b)(x0 + g(x)) − S(b)(x1 + g1 (x))

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

By the assumption that f and σ are uniformly bounded, there exists D > 0 such that || f (s, xα (s))|| p + ||σ (s, xα (s))|| p ≤ D in [0, b]×. Then there is a subsequence denoted by { f (s, xα (s)), σ (s, xα (s))} weakly converging to say { f (s, ω), σ (s, ω)} in H × L 02 . Now, the compactness of S(t) implies that S(b − s) f (s, xα (s)) → S(b − s) f (s) S(b − s)σ (s, xα (s)) → S(b − s)σ (s) in J × .

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Now, from the above equation, we get

b −1 Exb − C(b)[x0 + g(x))] α(α I + 0 )

p−1

E||xα (b) − xb || ≤ 6

p −S(b)[x1 + g1 (x)] p

 +6 p−1 E

b

b

||α(α I

+ sb )−1 ||

||α(α I

+ sb )−1 S(b

0

 p−1 +6 E

b

0

 +6 p−1 E

b

0

 p−1 +6 E

0

p/2

2

0

 p−1 +6 E

||α(α I + sb )−1 φ(s)||2L 0 ds

b

p

||S(b − s)[ f (s, xα (s)) − f (s)]||ds p − s) f (s)||ds

||α(α I + sb )−1 || ||S(b − s)[σ (s, xα (s)) − σ (s)]||2L 0 ds ||α(α I + sb )−1 S(b − s)σ (s)||2L 0 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||xα (b) − xb || p → 0+ . This gives the approximate controllability.  

4 Example Consider the stochastic wave control system given by 





⎫ ⎪ + Bu(t, y) + p(t, z(t, y)) ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +k(t, z(t, y))∂ω(t), t ∈ J, 0 ≤ y < π ⎪ ⎪ ⎪ ⎪ ⎪ z(t, 0) = z(t, π ) = 0, t ∈ J ⎬ n  ⎪ αi z(ti , y) = z 0 (y), t ∈ J z(0, y) + ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ k ⎪  ⎪ ⎪ ⎪ ⎪ z t (0, y) + βi z(si , y) = z 1 (y) ⎭

∂

∂z(t,y) ∂t

=

∂ 2 z(t,y) ∂ y2

(4.1)

i=1

where B is a bounded linear operator from a Hilbert space U → X and p : J ×X → X , k : J × X → L 02 are Lipschitz continuous with respect to y uniformly in t and satisfy linear growth conditions, u(t) is a feedback control and ω is a Q-Wiener process. Let X = L 2 [0, π ] and A : D(A) ⊂ X → X be defined by Az = z yy

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where   D(A) = z(.) ∈ X |z, z y are absolutely continuous, z yy ∈ X, z(0) = z(π ) = 0 Furthermore, A has discrete spectrum, the eigenvalues are −n 2 , n = 1, 2, · · · with the corresponding normalized eigenfunctions en (s) = (2/π )1/2 sin ns,then Az =

∞ 

(−n 2 ) < z, en > en , z ∈ X

n=1

It is well known that A is the generator of strongly continuous cosine family {C(t) : t ∈ R} on X given by C(t)z =

∞ 

cos nt < z, en > en , z ∈ X

n=1

and the associated sine family is given by S(t)z =

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

Let f : J × X → X be defined by f (t, z)(y) = p(t, z(y)), z ∈ X, y ∈ [0, π ]. Let σ : J × X → L 02 be defined by σ (t, z)(y) = k(t, z(y)) The functions g : C(J, X ) → X is defined as g(z)(y) =

n 

αi z(ti , y)

i=1

for 0 < ti < T and y ∈ [0, π ]. b Define tb = t S(b−s)B B ∗ S ∗ (b−s)ds. We claim that S ∗ (b−s)z = 0, t ≤ s ≤ b implies that z = 0. It is known that the deterministic linear system without delay associated with (4.1) is approximately controllable on [0, b] (see Ref. [7]). Assume that the functions f ,σ and g satisfy the requirement of hypotheses. 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). Since all the hypotheses of the Theorem 3.1 are satisfied, it implies the approximate controllability of the system (4.1) on J follows from Theorem 3.1.

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Ann Univ Ferrara Acknowledgments The authors are grateful to the Reviewers for valuable suggestions for improvements in the paper. First author is also thankful to Council of Scientific and Industrial Research, New Delhi Government of India (Grant No. 9924-11-44) for financial support and to carry out his research work.

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