Controllability of Second Order Impulsive Neutral ... - Semantic Scholar
J Optim Theory Appl (2012) 154:672–684 DOI 10.1007/s10957-012-0025-6
Controllability of Second Order Impulsive Neutral Functional Differential Inclusions with Infinite Delay Dimplekumar N. Chalishajar
Received: 3 November 2011 / Accepted: 7 March 2012 / Published online: 17 May 2012 © Springer Science+Business Media, LLC 2012
Abstract This paper is concerned with controllability of a partial neutral functional differential inclusion of second order with impulse effect and infinite delay. We introduce a new phase space to prove the controllability of an inclusion which consists of an impulse effect with infinite delay. We claim that the phase space considered by different authors is not correct. We establish the controllability of mild solutions using a fixed point theorem for contraction multi-valued maps and without assuming compactness of the family of cosine operators. Keywords Controllability · Second order impulsive neutral differential inclusions · Fixed point theorem for multi-valued maps · Strongly continuous cosine family 1 Introduction The problem of controllability for impulsive functional differential inclusions in Banach spaces has been studied extensively. Benchohra et al. [1] discussed the controllability of first and second order neutral functional differential and integro-differential inclusions in a Banach space with non-local conditions, without impulse effect. Chang and Li [2] obtained the controllability result for functional integro-differential inclusions on an unbounded domain without impulse term. Benchohra et al. [3] studied the existence result for damped differential inclusion with impulse effect. Hernandez et al. [4] proved the existence of solutions for impulsive partial neutral functional Communicated by Mark J. Balas. D.N. Chalishajar Department of Mathematics and Computer Science, Mallory Hall, Virginia Military Institute, Lexington, VA 24450, USA e-mail: [email protected] D.N. Chalishajar () e-mail: [email protected]
J Optim Theory Appl (2012) 154:672–684
673
differential equations for first and second order systems with infinite delay. It has been observed that the existence or the controllability results proved by different authors are through an axiomatic definition of the phase space given by Hale and Kato [5]. However, as remarked by Hino, Murakami, and Naito [6], it has come to our attention that these axioms for the phase space are not correct for the impulsive systems with infinite delay. On the other hand, researchers have been proving the controllability results using compactness assumption of semigroups and the family of cosine operators. Bing Liu [7] studied the controllability of first order impulsive neutral functional differential inclusions with infinite delay in a Banach space with the assumption of compactness of the semigroup. However, as remarked by Triggiani [8], in an infinite dimensional Banach space, the linear control system is never exactly controllable on a given interval of time, if either a bounded linear operator (from control space to state space) is compact or a semigroup is compact. According to Triggiani [8], this is a typical case for most control systems governed by parabolic partial differential equations, and hence the concept of exact controllability is very limited for many parabolic partial differential equations. Nowadays, researchers are engaged to overcome this problem, refer to [4, 9]. Very recently, Chalishajar et al. [10–12] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Ntouyas and O’Regan [13] made some remarks on controllability of evolution equations in Banach paces and proved a result without compactness assumption. In this paper, we discuss the controllability for the second order impulsive neutral functional differential inclusions, with infinite delay through the phase space defined in [6], and without compactness of the family of cosine operators. To the best of our knowledge, a controllability result has not been studied in this connection. Section 2 contains the preliminaries, which are required for further investigation of this paper. Section 3 deals with the main controllability result in a separable Banach space X using a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler [14]. Concluding remarks are given in Sect. 4.
2 Preliminaries The purpose of this paper is to study the controllability of impulsive partial neutral functional differential inclusions of second order with infinite delay. Specifically, we are concerned with the inclusions form ⎧ d ⎪ [y (t) − f (t, yt , y (t))] ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ∈ Ay(t) + Bu(t) + F (t, yt , y (t)); t ∈ J := [0, m]; t = tk ; ⎨ k = 1, 2, . . . , m; y|t=tk = Ik (y(tk ), y (tk )); (1) ⎪ ⎪ ⎪ ⎪ y |t=tk = Ik (y(tk ), y (tk )); k = 1, 2, . . . , m; ⎪ ⎪ ⎪ ⎩ y (0) = x0 , y(0) = φ ∈ Bh , where A : D(A) ⊂ X → X is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} defined on X, and F : J × Bh × X ⇒ X is a bounded,
674
J Optim Theory Appl (2012) 154:672–684
closed, and convex multi-valued map. Let J0 = ]−∞, 0], and non-local condition φ ∈ Bh (defined below) and x0 ∈ X be the given initial values. f : J × Bh → X is a given function, the state function y(t) takes values in X, and the control function u ∈ L2 (J, U ), a Banach space of admissible control functions with U as a Banach space. B is a bounded linear operator from U to X. X is a Banach space with norm | . |. Also, 0 < t0 < t1 < · · · < tp < tp+1 = m (→ ∞ as t → ∞); Ik , Ik ∈ C(X × X, X), k = 1, 2, . . . , p are bounded, y|t=tk = y(tk+ ) − y(tk− ), y |t=tk = y (tk+ ) − y (tk− ), and y(tk− ) and y(tk+ ), y (tk− ) and y (tk+ ) represent the left and right limits of y(t) and y (t), respectively, at t = tk . Furthermore, for any continuous function y defined on the interval J1 = ]−∞, m[ with values in X and for any t ∈ J , we denote by yt an element of C(J0 , X) defined by yt (θ ) = y(t + θ ), θ ∈ J0 . We present the abstract phase space Bh . Assume that h : ]−∞, 0] → ]0, ∞[ be a 0 continuous function with l = −∞ h(s) ds < +∞. Define Bh := φ : ]−∞, 0] → X such that, for any r > 0, φ(θ ) is bounded and measurable function on [−r, 0] and
0
−∞
h(s) sup φ(θ ) ds < +∞ . s≤θ≤0
Here, Bh is endowed with the norm 0 φ Bh = h(s) sup φ(θ ) ds, −∞
∀φ ∈ Bh .
s≤θ≤0
Then it is easy to show that (Bh , . Bh ) is a Banach space. Lemma 2.1 Suppose y ∈ Bh ; then, for each t ∈ J, yt ∈ Bh . Moreover, l y(t) ≤ yt B ≤ l sup y(s) + y0 B , h
where l :=
0
−∞ h(s) ds
h
s≤θ≤0
< +∞.
Proof For any t ∈ [0, a], it is easy to see that yt is bounded and measurable on [−a, 0] for a > 0, and 0 yt Bh = h(s) sup yt (θ ) ds −∞
=
−∞
=
−t
−∞ −t −∞
+
h(s) sup y(t + θ ) ds + θ∈[s,0]
−t
≤
θ∈[s,0]
h(s)
θ1 ∈[t+s,t]
h(s) 0
−t
sup
y(θ1 ) ds +
sup
0
−t
h(s) sup y(t + θ ) ds θ∈[s,0]
0
−t
h(s)
sup
h(s) sup y(θ1 ) ds
θ1 ∈[0,t]
y(θ1 ) ds
θ1 ∈[t+s,t]
y(θ1 ) + sup y(θ1 ) ds
θ1 ∈[t+s,0]
θ1 ∈[0,t]
J Optim Theory Appl (2012) 154:672–684
=
−t
−∞ −t
≤
−∞
≤
−∞
=
0
0
−∞
h(s)
sup
675
y(θ1 ) ds +
θ1 ∈[t+s,0]
0
−∞
h(s) ds sup y(s) s∈[0,t]
h(s) sup y(θ1 ) ds + l sup y(s) θ1 ∈[s,0]
s∈[0,t]
h(s) sup y(θ1 ) ds + l sup y(s) θ1 ∈[s,0]
s∈[0,t]
h(s) sup y0 (θ1 ) ds + l sup y(s) θ1 ∈[s,0]
= l sup y(s) + y0 Bh .
s∈[0,t]
s∈[0,t]
Since φ ∈ Bh , then yt ∈ Bh . Moreover, 0 yt Bh = h(s) sup yt (θ ) ds ≥ yt (θ ) −∞
0
−∞
θ∈[s,0]
h(s) ds = l y(t) .
The proof is complete.
Next, we introduce definitions, notation and preliminary facts from multi-valued analysis, which are useful for the development of this paper. Let C(J, X) be the Banach space of continuous functions from J to X with the norm x ∞ = supt∈J |x(t)|. B(X) denotes the Banach space of bounded linear operators from X to X. Let L1 (J, X) denote the Banach space of continuous m function y : J → X, which are integrable and endowed with the norm y L1 = 0 |y(t)| dt, y ∈ L1 (J, X). For a metric space (X, d), we introduce the following notations: P (X) := {Y ∈ P(X) : Y = φ}, Pcl (X) := {Y ∈ P (X) : Y is closed}, Pb (X) := {Y ∈ P (X) : Y is bounded}, Pcp (X) := {Y ∈ P (X) : Y is compact}, Pb,cl (X) := {Y ∈ P (X) : Y is bounded and closed}. We define a Hausdorff space Hd : P (X) × P (X) → R+ ∪ {∞} by Hd (A, B) := max sup d(a, B), sup d(A, b) , a∈A
b∈B
where d(A, b) := infa∈A d(a, b), d(a, B) := infb∈B d(a, b). Then (Pb,cl (X), Hd ) is a metric space, and (Pcl (X), Hd ) is a generalized (complete) metric space. We now recall some preliminaries about multi-valued maps. Let (X, . ) be a Banach space. A multi-valued map G : X ⇒ X is convex (resp. closed) iff G(x) is convex (resp. closed) in X, for all x ∈ X. The map G is bounded on bounded sets iff G(B) = Ux∈B G(x) is bounded in X for any bounded set B of X (i.e., supx∈B {sup{ y : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.) on X iff for each x0 ∈ X the set G(x0 ) is a nonempty, closed subset of X and if for each open set B of X containing G(x0 ) there exists an open neighbourhood A of x0 such that G(A) ⊆ B. The map G is said to be completely continuous iff G(B) is relatively compact for every bounded subset B ⊆ X. If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. iff G has a closed graph. That is, if xn → x0 and yn → y0 , where
676
J Optim Theory Appl (2012) 154:672–684
yn ∈ G(xn ), then y0 ∈ G(x0 ). We say that G has a fixed point iff there is x ∈ X such that x ∈ G(x). In the following, BCC(X) denotes the set of all nonempty bounded, closed and convex subsets of X. A multi-valued map G : J → BCC(X) is said to be measurable iff for each x ∈ X, the distance function Y : J → R, defined by Y (t) := d x, G(t) = inf |x − z| : z ∈ G(t) , is measurable. For more details on multi-valued maps see [16, 17]. An upper semi-continuous map G : X ⇒ X is said to be condensing iff for any subset B ⊆ X, with α(B) = 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of non-compactness. For the properties of the Kuratowski measure, we refer to Banas and Goebel [15]. We note that a completely continuous multi-valued map is the easiest example of a condensing map. For more details on multi-valued maps, see the book of Deimling [16] and the research article of Travis and Webb [18]. We say that the family {C(t) : t ∈ R} of operators in B(X) is a strongly continuous cosine family iff 1. C(0) = I , I is the identity operator on X. 2. C(t + s) + C(t − s) = 2C(t)C(s) for all s, t ∈ R. 3. The map t → C(t)x is strongly continuous for each x ∈ X. The strongly continuous sine family {S(t) : t ∈ R}, associated with the strongly continuous cosine family {C(t) : t ∈ R}, is defined by t C(s)x ds, x ∈ X, t ∈ R. S(t)x := 0
The infinitesimal generator A : X → X of a cosine family C(t), t ∈ R is defined by d2 Ax := 2 C(t)x , x ∈ D(A), dt t=0 where D(A) = {x ∈ X : C(t)x is twice continuous differentiable}. We refer to the book of Goldstein [19] for the detailed study of the family of cosine and sine operators. Definition 2.1 A multi-valued operator G : X ⇒ Pcl (X) is called (a) γ -Lipschitz iff there exists γ > 0 such that Hd (G(x), G(y)) ≤ γ d(x, y), for each x, y ∈ X; (b) a contraction iff it is γ -Lipschitz with γ < 1. Definition 2.2 The integral formulation y(t) of system (1) is given by y(t) := φ(t);
if t ∈ J0 , t y(t) := C(t)φ(0) + S(t) x0 − f (0, φ, x0 ) + C(t − s)f s, ys , y (s) ds 0 t t S(t − s)Bu(s) ds + S(t − s)v(s) ds + 0
0
J Optim Theory Appl (2012) 154:672–684
+
677
C(t − tk )Ik ytk , y (tk ) + S(t − tk )Ik ytk , y (tk ) ;
0
Controllability of Second Order Impulsive Neutral Functional Differential Inclusions with Infinite Delay Dimplekumar N. Chalishajar
Received: 3 November 2011 / Accepted: 7 March 2012 / Published online: 17 May 2012 © Springer Science+Business Media, LLC 2012
Abstract This paper is concerned with controllability of a partial neutral functional differential inclusion of second order with impulse effect and infinite delay. We introduce a new phase space to prove the controllability of an inclusion which consists of an impulse effect with infinite delay. We claim that the phase space considered by different authors is not correct. We establish the controllability of mild solutions using a fixed point theorem for contraction multi-valued maps and without assuming compactness of the family of cosine operators. Keywords Controllability · Second order impulsive neutral differential inclusions · Fixed point theorem for multi-valued maps · Strongly continuous cosine family 1 Introduction The problem of controllability for impulsive functional differential inclusions in Banach spaces has been studied extensively. Benchohra et al. [1] discussed the controllability of first and second order neutral functional differential and integro-differential inclusions in a Banach space with non-local conditions, without impulse effect. Chang and Li [2] obtained the controllability result for functional integro-differential inclusions on an unbounded domain without impulse term. Benchohra et al. [3] studied the existence result for damped differential inclusion with impulse effect. Hernandez et al. [4] proved the existence of solutions for impulsive partial neutral functional Communicated by Mark J. Balas. D.N. Chalishajar Department of Mathematics and Computer Science, Mallory Hall, Virginia Military Institute, Lexington, VA 24450, USA e-mail: [email protected] D.N. Chalishajar () e-mail: [email protected]
J Optim Theory Appl (2012) 154:672–684
673
differential equations for first and second order systems with infinite delay. It has been observed that the existence or the controllability results proved by different authors are through an axiomatic definition of the phase space given by Hale and Kato [5]. However, as remarked by Hino, Murakami, and Naito [6], it has come to our attention that these axioms for the phase space are not correct for the impulsive systems with infinite delay. On the other hand, researchers have been proving the controllability results using compactness assumption of semigroups and the family of cosine operators. Bing Liu [7] studied the controllability of first order impulsive neutral functional differential inclusions with infinite delay in a Banach space with the assumption of compactness of the semigroup. However, as remarked by Triggiani [8], in an infinite dimensional Banach space, the linear control system is never exactly controllable on a given interval of time, if either a bounded linear operator (from control space to state space) is compact or a semigroup is compact. According to Triggiani [8], this is a typical case for most control systems governed by parabolic partial differential equations, and hence the concept of exact controllability is very limited for many parabolic partial differential equations. Nowadays, researchers are engaged to overcome this problem, refer to [4, 9]. Very recently, Chalishajar et al. [10–12] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Ntouyas and O’Regan [13] made some remarks on controllability of evolution equations in Banach paces and proved a result without compactness assumption. In this paper, we discuss the controllability for the second order impulsive neutral functional differential inclusions, with infinite delay through the phase space defined in [6], and without compactness of the family of cosine operators. To the best of our knowledge, a controllability result has not been studied in this connection. Section 2 contains the preliminaries, which are required for further investigation of this paper. Section 3 deals with the main controllability result in a separable Banach space X using a fixed point theorem for contraction multi-valued maps due to Covitz and Nadler [14]. Concluding remarks are given in Sect. 4.
2 Preliminaries The purpose of this paper is to study the controllability of impulsive partial neutral functional differential inclusions of second order with infinite delay. Specifically, we are concerned with the inclusions form ⎧ d ⎪ [y (t) − f (t, yt , y (t))] ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ∈ Ay(t) + Bu(t) + F (t, yt , y (t)); t ∈ J := [0, m]; t = tk ; ⎨ k = 1, 2, . . . , m; y|t=tk = Ik (y(tk ), y (tk )); (1) ⎪ ⎪ ⎪ ⎪ y |t=tk = Ik (y(tk ), y (tk )); k = 1, 2, . . . , m; ⎪ ⎪ ⎪ ⎩ y (0) = x0 , y(0) = φ ∈ Bh , where A : D(A) ⊂ X → X is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} defined on X, and F : J × Bh × X ⇒ X is a bounded,
674
J Optim Theory Appl (2012) 154:672–684
closed, and convex multi-valued map. Let J0 = ]−∞, 0], and non-local condition φ ∈ Bh (defined below) and x0 ∈ X be the given initial values. f : J × Bh → X is a given function, the state function y(t) takes values in X, and the control function u ∈ L2 (J, U ), a Banach space of admissible control functions with U as a Banach space. B is a bounded linear operator from U to X. X is a Banach space with norm | . |. Also, 0 < t0 < t1 < · · · < tp < tp+1 = m (→ ∞ as t → ∞); Ik , Ik ∈ C(X × X, X), k = 1, 2, . . . , p are bounded, y|t=tk = y(tk+ ) − y(tk− ), y |t=tk = y (tk+ ) − y (tk− ), and y(tk− ) and y(tk+ ), y (tk− ) and y (tk+ ) represent the left and right limits of y(t) and y (t), respectively, at t = tk . Furthermore, for any continuous function y defined on the interval J1 = ]−∞, m[ with values in X and for any t ∈ J , we denote by yt an element of C(J0 , X) defined by yt (θ ) = y(t + θ ), θ ∈ J0 . We present the abstract phase space Bh . Assume that h : ]−∞, 0] → ]0, ∞[ be a 0 continuous function with l = −∞ h(s) ds < +∞. Define Bh := φ : ]−∞, 0] → X such that, for any r > 0, φ(θ ) is bounded and measurable function on [−r, 0] and
0
−∞
h(s) sup φ(θ ) ds < +∞ . s≤θ≤0
Here, Bh is endowed with the norm 0 φ Bh = h(s) sup φ(θ ) ds, −∞
∀φ ∈ Bh .
s≤θ≤0
Then it is easy to show that (Bh , . Bh ) is a Banach space. Lemma 2.1 Suppose y ∈ Bh ; then, for each t ∈ J, yt ∈ Bh . Moreover, l y(t) ≤ yt B ≤ l sup y(s) + y0 B , h
where l :=
0
−∞ h(s) ds
h
s≤θ≤0
< +∞.
Proof For any t ∈ [0, a], it is easy to see that yt is bounded and measurable on [−a, 0] for a > 0, and 0 yt Bh = h(s) sup yt (θ ) ds −∞
=
−∞
=
−t
−∞ −t −∞
+
h(s) sup y(t + θ ) ds + θ∈[s,0]
−t
≤
θ∈[s,0]
h(s)
θ1 ∈[t+s,t]
h(s) 0
−t
sup
y(θ1 ) ds +
sup
0
−t
h(s) sup y(t + θ ) ds θ∈[s,0]
0
−t
h(s)
sup
h(s) sup y(θ1 ) ds
θ1 ∈[0,t]
y(θ1 ) ds
θ1 ∈[t+s,t]
y(θ1 ) + sup y(θ1 ) ds
θ1 ∈[t+s,0]
θ1 ∈[0,t]
J Optim Theory Appl (2012) 154:672–684
=
−t
−∞ −t
≤
−∞
≤
−∞
=
0
0
−∞
h(s)
sup
675
y(θ1 ) ds +
θ1 ∈[t+s,0]
0
−∞
h(s) ds sup y(s) s∈[0,t]
h(s) sup y(θ1 ) ds + l sup y(s) θ1 ∈[s,0]
s∈[0,t]
h(s) sup y(θ1 ) ds + l sup y(s) θ1 ∈[s,0]
s∈[0,t]
h(s) sup y0 (θ1 ) ds + l sup y(s) θ1 ∈[s,0]
= l sup y(s) + y0 Bh .
s∈[0,t]
s∈[0,t]
Since φ ∈ Bh , then yt ∈ Bh . Moreover, 0 yt Bh = h(s) sup yt (θ ) ds ≥ yt (θ ) −∞
0
−∞
θ∈[s,0]
h(s) ds = l y(t) .
The proof is complete.
Next, we introduce definitions, notation and preliminary facts from multi-valued analysis, which are useful for the development of this paper. Let C(J, X) be the Banach space of continuous functions from J to X with the norm x ∞ = supt∈J |x(t)|. B(X) denotes the Banach space of bounded linear operators from X to X. Let L1 (J, X) denote the Banach space of continuous m function y : J → X, which are integrable and endowed with the norm y L1 = 0 |y(t)| dt, y ∈ L1 (J, X). For a metric space (X, d), we introduce the following notations: P (X) := {Y ∈ P(X) : Y = φ}, Pcl (X) := {Y ∈ P (X) : Y is closed}, Pb (X) := {Y ∈ P (X) : Y is bounded}, Pcp (X) := {Y ∈ P (X) : Y is compact}, Pb,cl (X) := {Y ∈ P (X) : Y is bounded and closed}. We define a Hausdorff space Hd : P (X) × P (X) → R+ ∪ {∞} by Hd (A, B) := max sup d(a, B), sup d(A, b) , a∈A
b∈B
where d(A, b) := infa∈A d(a, b), d(a, B) := infb∈B d(a, b). Then (Pb,cl (X), Hd ) is a metric space, and (Pcl (X), Hd ) is a generalized (complete) metric space. We now recall some preliminaries about multi-valued maps. Let (X, . ) be a Banach space. A multi-valued map G : X ⇒ X is convex (resp. closed) iff G(x) is convex (resp. closed) in X, for all x ∈ X. The map G is bounded on bounded sets iff G(B) = Ux∈B G(x) is bounded in X for any bounded set B of X (i.e., supx∈B {sup{ y : y ∈ G(x)}} < ∞). G is called upper semi-continuous (u.s.c.) on X iff for each x0 ∈ X the set G(x0 ) is a nonempty, closed subset of X and if for each open set B of X containing G(x0 ) there exists an open neighbourhood A of x0 such that G(A) ⊆ B. The map G is said to be completely continuous iff G(B) is relatively compact for every bounded subset B ⊆ X. If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. iff G has a closed graph. That is, if xn → x0 and yn → y0 , where
676
J Optim Theory Appl (2012) 154:672–684
yn ∈ G(xn ), then y0 ∈ G(x0 ). We say that G has a fixed point iff there is x ∈ X such that x ∈ G(x). In the following, BCC(X) denotes the set of all nonempty bounded, closed and convex subsets of X. A multi-valued map G : J → BCC(X) is said to be measurable iff for each x ∈ X, the distance function Y : J → R, defined by Y (t) := d x, G(t) = inf |x − z| : z ∈ G(t) , is measurable. For more details on multi-valued maps see [16, 17]. An upper semi-continuous map G : X ⇒ X is said to be condensing iff for any subset B ⊆ X, with α(B) = 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of non-compactness. For the properties of the Kuratowski measure, we refer to Banas and Goebel [15]. We note that a completely continuous multi-valued map is the easiest example of a condensing map. For more details on multi-valued maps, see the book of Deimling [16] and the research article of Travis and Webb [18]. We say that the family {C(t) : t ∈ R} of operators in B(X) is a strongly continuous cosine family iff 1. C(0) = I , I is the identity operator on X. 2. C(t + s) + C(t − s) = 2C(t)C(s) for all s, t ∈ R. 3. The map t → C(t)x is strongly continuous for each x ∈ X. The strongly continuous sine family {S(t) : t ∈ R}, associated with the strongly continuous cosine family {C(t) : t ∈ R}, is defined by t C(s)x ds, x ∈ X, t ∈ R. S(t)x := 0
The infinitesimal generator A : X → X of a cosine family C(t), t ∈ R is defined by d2 Ax := 2 C(t)x , x ∈ D(A), dt t=0 where D(A) = {x ∈ X : C(t)x is twice continuous differentiable}. We refer to the book of Goldstein [19] for the detailed study of the family of cosine and sine operators. Definition 2.1 A multi-valued operator G : X ⇒ Pcl (X) is called (a) γ -Lipschitz iff there exists γ > 0 such that Hd (G(x), G(y)) ≤ γ d(x, y), for each x, y ∈ X; (b) a contraction iff it is γ -Lipschitz with γ < 1. Definition 2.2 The integral formulation y(t) of system (1) is given by y(t) := φ(t);
if t ∈ J0 , t y(t) := C(t)φ(0) + S(t) x0 − f (0, φ, x0 ) + C(t − s)f s, ys , y (s) ds 0 t t S(t − s)Bu(s) ds + S(t − s)v(s) ds + 0
0
J Optim Theory Appl (2012) 154:672–684
+
677
C(t − tk )Ik ytk , y (tk ) + S(t − tk )Ik ytk , y (tk ) ;
0
