APPROXIMATE CONTROLLABILITY OF SECOND ORDER IMPULSIVE SYSTEMS WITH STATE-DEPENDENT DELAY IN BANACH SPACES

In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay.

1. Introduction. Second order differential equations emerge in many areas of science and engineering. One aspect of studying second order systems is through an equivalent formulation of first order equations, but this transformation may lack important information about the original evolution systems. Therefore, it is more advantageous to study a second order system directly. For the basic theory of second order differential equations, the interested readers are referred to see [12,21,49], etc. We observe that the strongly continuous cosine family is an important tool in studying the second order evolution equations. The theory of strongly continuous cosine family was introduced by Fattorini in [19,20]. Later, Travis and Webb have made essential additions to the theory of strongly continuous cosine family (cf. [49,50]).
There are several realistic evolutionary processes subject to abrupt changes in state occur at certain negligible time instant. These processes are mathematically modelled by impulsive differential equations (IDE). Impulsive differential equations have found a variety of applications in various fields of engineering (cf. [22,47,48], etc). On the other hand, delay differential equations also present in many evolutionary processes such as heat conduction in materials with fading memory, inferred grinding models and ecological models. In these phenomena, the current state of a system is influenced by the previous states. In the study of delay differential equations, many authors considered the time delay as constant under the state variable to make the study easier. However, state-dependent delays are more prevalent and adequate in applications, some nice examples of state-dependent delay models are given in [1,15] and the references therein.
Controllability plays a vital role in designing and analyzing control systems. The theory of controllability of the first and second order deterministic or stochastic systems in infinite dimensional spaces is well developed, and has been studied extensively, see for instance [6,7,8,10,13,16,28,31,42], etc. Controllability means that a dynamical system can be steered to the desired final state constrained to an admissible class of controls. For the infinite dimensional systems, two famous notions of controllability, namely exact controllability and approximate controllability have been studied extensively by many researchers account to its wide range of applications. Exact controllability means that a system can achieve a desired final state in finite time period, while the approximate controllability implies that a system can steer into an arbitrary small neighborhood of the final state. In the infinite dimensional setting, it has been observed that exact controllability rarely holds (see [52]). However, the approximate controllability is extensive and more appropriate in applications. Therefore, the approximate controllability of nonlinear evolution systems is still seeking attention by many researchers and has emerged as an important area of investigation (cf. [4,5,14,24,30,32,33,37,43,46], etc and the references therein).
Recently, many works reported on the approximate controllability of the impulsive dynamical control system with delay via fixed point methods, see for example, [4,5,32,38,46], etc. In [4], a set of sufficient conditions is established for the approximate controllability of the semilinear impulsive functional differential system with nonlocal initial conditions by invoking Schauder's fixed point theorem. Urvashi et.al. in [5], investigated the approximate controllability of second order semilinear stochastic systems in Hilbert spaces with variable delay in control. Li and Huang in [32], considered second order impulsive stochastic differential equations with statedependent delay in Hilbert spaces and examined the approximate controllability of the system. Mahumudov [38], discussed the approximate controllability of a class of second order evolution differential inclusions, using Bohnenblust-Karlin's fixed point theorem.
It is observed that the approximate controllability of the impulsive functional control systems in Banach spaces has not got much attention in the literature. In [37], Mahumudov examined the approximate controllability of the semilinear deterministic control systems in Banach spaces. Sumit et.al. [2,3], established sufficient conditions for the approximate controllability of impulsive evolution systems with delay in separable reflexive Banach spaces. There are a few articles, namely [30,33,44], etc investigated the approximate controllability of the second order impulsive functional differential equations in Banach spaces. We observe that the resolvent operator defined in these articles is not well defined in general Banach spaces. We also point out that if the state space is a Hilbert space (identified with its own dual), then the resolvent operator defined in the works [30,33,44], etc are well defined (see (2.1) below for the resolvent operator definition on general Banach spaces). This fact motivates us to consider the second order semilinear impulsive differential equations with state-dependent delay in Banach spaces and examine the approximate controllability of such systems via resolvent operator condition and Scauder's fixed point theorem.
Let X be a separable reflexive Banach space (having a strictly convex dual X * ) and H be a separable Hilbert space. We consider the following second order impulsive system with state-dependent delay: (1.1) where A is a linear operator on X, B is a bounded linear operator from H into X and the control function u ∈ L 2 (J; H). Also, f : J × P → X is a nonlinear function, where P is a phase space, which will be specified later. The impulsive functions I k , g k : X → X, for k = 1, . . . , m, and ∆x| t=τ , belong to the phase space P and the function ρ : This article is structured as follows: The next section presents the basic definitions and results required to develop the approximate controllability of the second order system (1.1). In section 3, the approximate controllability of the linear control system corresponding to the semilinear system (1.1) is discussed. Section 4 is devoted for establishing the approximate controllability of the system (1.1), by using the theory of strongly continuous cosine family and resolvent operators. In the final section, we discuss a concrete example to validate the theory developed in sections 3 and 4.

2.
Preliminaries. In this section, we introduce some basic notations, assumptions and definitions to be used in succeeding sections to prove the approximate controllability results for the evolution system (1.1). The norms in the state space X, its dual space X * and control space H are denoted by · X , · X * and · H , respectively. The inner product in H is represented by (·, ·) and the duality pairing between X and its topological dual X * is denoted by ·, · . The space of all bounded linear operators from H to X is denoted by L(H; X) endowed with norm · L(H;X) . The notation L(X), represents the space of all bounded linear operators on X endowed with the norm · L(X) . A function x(s) X . For convenience of notations, we use PC instead of PC(J; X) throughout the article.
2.1. The strongly continuous cosine family. In this subsection, we first construct a strongly continuous cosine family {C(t) : t ∈ R} in X, generated by the linear operator A, and then discuss some special properties of the strongly continuous cosine family. Let us impose the following assumptions on the linear operator A : D(A) → X.
Assumption 2.1. The operator A satisfies the following: (R1) The linear operator A is closed and the domain D(A) is dense in X. (R2) For real λ, λ > ω, λ 2 is in the resolvent set ρ(A) of A. The resolvent R(λ 2 ; A) exists, strongly infinitely differentiable, and satisfies d dλ   The strongly continuous sine family {S(t) : t ∈ R} associated with the strongly continuous cosine family {C(t) : t ∈ R} in X is defined as Let us assume that, there exists M, M , ω ≥ 0 such that C(t) L(X) ≤ M e ωt and S(t) L(X) ≤ M e ωt , for every t ∈ J.
Note that if C(t) is a strongly continuous cosine family of type (M, ω) with the infinitesimal generator A, then by the Theorem 2.1, Chapter 2, [21], for any f ∈ X and λ > ω, we have d dλ In this paper, we consider the second order problems only, for higher order abstract Cauchy problems, we refer the interested readers to [11]. Proposition 2.2 (Proposition 5.2, [51]). Let {C(t) : t ∈ R}, be a strongly continuous cosine family in X, with the infinitesimal generator A and associated sine family {S(t) : t ∈ R}. The following are equivalent: (i) The operator S(t) is compact for every t ∈ R.
The infinitesimal generator A of a strongly continuous cosine family {C(t) : t ∈ R} is defined by where D(A) = {x ∈ X : C(t)x is twice continuously differentiable function in t}, endowed with the norm We also define the set E = {x : C(t)x is once continuously differentiable function of t}, endowed with the norm forms a Banach space, see [29]. The operator valued function is a strongly continuous group of bounded linear operators on the space E × X, generated by the operator A = 0 I A 0 defined on D(A) × E (see Proposition 2.6, [50]). From this, it follows that AS(t) : E → X is a bounded linear operator and that AS(t)x → 0 as t → 0, for each x ∈ E.
For any φ ∈ P, the function φ t , t ≤ 0, defined as φ t (θ) = φ(t + θ), θ ∈ (−∞, 0]. Therefore, if the function x(·) satisfies the axiom (A1) with x 0 = φ, then we may extend the mapping t → x t by setting x t = φ t , t ≤ 0, to the whole interval (−∞, T ]. Moreover, let us introduce a set where the function ρ is same as the one defined in section 1. We give the following hypothesis on φ t : the function t → φ t , defined from Z(ρ − ) into P is continuous, and there exists a continuous and bounded function Θ φ : Lemma 2.1 (Lemma 2.3, [44]). Let x : (−∞, T ] → X, be a function such that x 0 = φ and x| J ∈ PC. Then 2.3. Resolvent operator and mild solution. In this subsection, we recall the duality mapping and resolvent operator in Banach spaces, which is introduced in [37], and then define the mild solution of the system (1.1).

Definition 2.2.
A duality mapping J : X → 2 X * is defined as (see [9]) Since the space X is reflexive, X can be renormed such that X and X * becomes strictly convex (Theorem 1.1, Chapter 1, [9]). From the strict convexity of X * , we obtain that the duality mapping J : X → X * is single valued and demicontinuous (Theorem 1.2, Chapter 1, [9]), that is,

Let us define the operators
Whenever, X is a separable Hilbert space, which can be identified by its own dual (the duality mapping J becomes I, the identity operator), then one can define the resolvent operator as R(λ, Ψ T 0 ) := (λI + Ψ T 0 ) −1 , λ > 0. It is clear from the second expression in (2.1) that the operator Ψ T 0 : X * → X is a nonnegative symmetric operator.

Definition 2.3.
A function x(·; φ, ζ 0 , u) : (−∞, T ] → X is called a mild solution of (1.1), if it satisfies the following: . . , m, (iv) x(·)| J ∈ PC and the following integral equation is verified: Definition 2.4. The system (1.1) is said to be approximately controllable on J, for any initial function φ ∈ P, if the closure of reachable set is the whole space X, where the reachable set is defined as In order to establish the approximate controllability results for the system (1.1), we impose the following assumptions.
Moreover, there exist constants c k 's and d k 's such that for all x ∈ X, k = 1, . . . , m.
Remark 2.2. Since Ψ T 0 is the nonnegative symmetric operator and X is a separable reflexive Banach space, then by using the Lemma 2.2 [37], we obtain that for every h ∈ X and λ > 0, the equation 3. Linear control problem. In this section, we consider the linear problem corresponding to the system (1.1), we formulate an optimal control problem and then discuss about its connection to the approximate controllability of the linear control system. First, we consider the following system: If the function h : J → X is integrable, then a continuous function z : [0, T ] → X is said to be a mild solution of (3.1), if the equation is satisfied. Moreover, when ζ 0 ∈ E, the function z(·) is continuously differentiable andż Such a solution (3.2) is called a strong solution. Furthermore, if ζ 0 ∈ D(A), ζ 1 ∈ E and h is a continuously differentiable function, then the function z(·) become a classical solution of the system (3.1). The existence of mild as well as strong solutions for the above system has been discussed in [49].
3.1. Optimal control problem for the linear system. In this subsection, first we prove the existence of an optimal pair, which minimizes a cost functional consisting in the linear-quadratic regulator problem. The cost functional is given by where λ > 0, x T ∈ X and x(·) is the mild solution of the linear system: with the control u. We take the admissible control class as U ad = L 2 (J; H), consisting of the controls u. Since Bu ∈ L 1 (J; X), there exists a unique mild solution x(·) of the system (3.4) satisfying (see [49]) Definition 3.1 (Admissible class). The admissible class A ad of pairs (x, u) is defined as the set of states x(·) solving the system (3.4) with the control u ∈ U ad . That is, x is a unique mild solution of (3.4) with the control u ∈ U ad .
Since there exists a unique mild solution of the system (3.4) for any u ∈ U ad , it follows that the admissible class A ad is nonempty. By using the above definition of F(·, ·), we formulate the optimal control problem as: A solution to the problem (3.6) is called an optimal solution. The optimal pair will be denoted by (x 0 , u 0 ). The control u 0 is called an optimal control. Theorem 3.1 (Existence of an optimal pair). Let φ(0), ζ 0 ∈ X be given. Then there exists at least one pair (x 0 , u 0 ) ∈ A ad such that the functional F(x, u) attains its minimum at (x 0 , u 0 ), where x 0 is the unique mild solution of the system (3.4) with the control u 0 .
Proof. Let us first define Since, 0 ≤ F < +∞, there exists a minimizing sequence {u n } ∞ n=1 ∈ U ad such that lim n→∞ F(x n , u n ) = F, where x n (·) is the unique mild solution of the system (3.4), with the control u n and the initial data x n (0) = φ(0) andẋ n (0) = ζ 0 . Note that x n (·) satisfies (3.7) Since 0 ∈ U ad , without loss of generality, we may assume that F(x n , u n ) ≤ F(x, 0), where (x, 0) ∈ A ad . Using the definition of F(·, ·), we easily get From the above relation, it is clear that, there exist an R > 0, large enough such that 0 ≤ F(x n , u n ) ≤ R < +∞.
In particular, there exists a large C > 0, such that Moreover, from (3.7), we have for all t ∈ J. Since L 2 (J; X) is reflexive, an application of the Banach-Alaoglu theorem yields the existence of a subsequence From (3.9), we also infer that the sequence {u n } ∞ n=1 is uniformly bounded in the space L 2 (J; H). Since L 2 (J; H) is a separable Hilbert space (in fact reflexive), using the Banach-Alaoglu theorem, we can find a subsequence Since B is a bounded linear operator from H to X, the above convergence also implies for all t ∈ J. Here, we used the weak convergence given in (3.12) and the compactness of the operator (Qf )(·) = · 0 S(· − s)f (s)ds : L 2 (J; X) → C(J; X). Using the above convergence, we estimate Thus, x * ∈ C(J; X) is the unique mild solution of the equation From the convergence (3.11), we know that the weak limit is unique and is given by x 0 (·). By using the convergence (3.14), we obtain x * (t) = x 0 (t), for all t ∈ J, therefore, x 0 (·) is the unique mild solution of (3.15) and also x n k → x 0 in C(J; X), as k → ∞. Since x 0 is the unique mild solution of the equation (3.15), the whole sequence {x n } ∞ n=1 converges to x 0 . Since u 0 ∈ U ad and x 0 is the unique mild solution of (3.15) corresponding to the control u 0 , it is immediate that (x 0 , u 0 ) ∈ A ad .
Let us now show that (x 0 , u 0 ) is a minimizer, that is, F = F(x 0 , u 0 ). Since the cost functional F(·, ·) is continuous and convex (see Proposition III.1.6 and III.1.10, [18]) on L 2 (J; X) × L 2 (J; H), it follows that F(·, ·) is weakly lower semi-continuous (Proposition II.4.5, [18]). That is, for a sequence Therefore, we obtain Note that the cost functional (3.3) is convex, the constraint (3.4) is linear and the class U ad = L 2 (J; H) is convex, then the optimal control obtained in Theorem 3.1 is unique. The explicit expression of the optimal control u is given by the following lemma: Lemma 3.1. Assume that u is the optimal control satisfying (3.4) and minimizing the cost functional (3.6). Then u is given by A proof of the above Lemma can be obtained by proceeding similarly as in the proof of Lemma 3.1, [43].

3.2.
Approximate controllability of the linear system. Here, we discuss about the approximate controllability of the second order linear control system (3.4), by proving the following Theorem: Theorem 3.2. The following statements are equivalent: (i) The linear control system (3.4) is approximately controllable on J.
Proof. Claim 1. (ii)⇔(iii). Since the operator Ψ T 0 is symmetric, more precisely for all x * 1 , x * 2 ∈ X * , using Theorem 2.3 [37], we know that the operator Ψ T 0 is positive if and only if the Assumption (H0) holds, which proves our claim. Claim 2. (i)⇔(iii). Let us first assume that the Assumption (H0) holds true. We know that for every λ > 0 and x T ∈ X, the mild solution x λ ∈ C(J; X) for the system (3.4) can be written as )], and p(x(·)) = x T − (C(T )φ(0) + S(T )ζ 0 ). Using (3.16), it can be easily seen that and since C(T )φ(0) X ≤ M e ωT φ(0) X , S(T )ζ 0 X ≤ M e ωT ζ 0 X and x T ∈ X, we have Thus, it follows that the system (3.4) is approximately controllable, and hence (iii)⇒(i). Conversely, we assume that the linear control system (3.4) is approximately controllable on J. Then, for arbitrary x T ∈ X, there exist a sequence {u n } ∞ n=1 in U ad = L 2 (J; H) such that where x n (·) is the unique mild solution of the system (3.4) with the control u n ∈ U ad . For all n ≥ 1, we have where (x 0 , u 0 ) ∈ A ad is the optimal pair at which the functional (3.3) takes its minimum value. For given ε > 0, there exists a positive integer n 0 such that , for all n ≥ n 0 .
Now we can choose a δ > 0 sufficiently small such that for all 0 < λ < δ. Taking n = n 0 in (3.19), we get for all 0 < λ < δ. Invoking Lemma 3.1, we know that Proceeding in a similar way as in the proof of the equality (3.17), we obtain Since x T ∈ X is arbitrary, using the inequality (3.20) together with the expression (3.21), we deduce that the Assumption (H0) follows.
Remark 3.1. Note that for x * ∈ X * and u ∈ L 2 (J; H), we have therefore, it is immediate that (L T ) * = B * S * (T − t). Note that B * S * (T − t)x * = 0 on J implies x * = 0, and then by the above fact, we easily obtain that the operator Ψ T 0 is positive and vice versa.
4. Approximate controllability of the nonlinear system. In the present section, we investigate the approximate controllability of the system (1.1). In order to do this, we first show that, for every λ > 0 and x T ∈ X, the system (1.1) has at least one mild solution with the control where withx : (−∞, T ] → X such thatx(t) = φ(t) on t ∈ (−∞, 0) andx = x on J. Proof. Let Z = {x ∈ PC : x(0) = φ(0)} be the space endowed with the norm · PC . We consider a set Q r = {x ∈ Z : x PC ≤ r}.
For λ > 0, we define an operator Φ λ : Z → Z as where It is clear from the definition Φ λ , for λ > 0, that the fixed point of Φ λ is a mild solution of system (1.1). We prove that the operator Φ λ has a fixed point in the following steps.
Step 1. Φ λ (Q r ) ⊂ Q r , for some r. Indeed, suppose that our assertion is false. Then for any λ > 0 and for all r > 0, there exists x r (·) ∈ Q r , such that (Φ λ x r )(t) X > r, for some t ∈ J, where t may depend upon r. First, by using the expressions defined in (4.1), (2.4), and Assumption 2.2, we evaluate For any x ∈ Q r , by applying Lemma 2.1, we compute Using (4.5) and (4.6), and the Assumption 2.2, we obtain From the Assumption 2.2 (H3), it is immediate that Dividing by r in the inequality (4.7) and then passing r → ∞, we get which is a contradiction to (4.3).
Step 2. Our next aim is to show that the operator Φ λ is a continuous operator.
In order to do this, we consider a sequence → 0 as n → ∞, for all s ∈ J and k ∈ N.
In particular, we take k = n and using the above convergence together with the Assumption 2.2 (H3) and evaluate → 0 as n → ∞, uniformly for all s ∈ J.
Step 3. Φ λ is a compact operator. To prove this, first we show that Φ λ (Q r ) is equicontinuous. For 0 ≤ t 1 ≤ t 2 ≤ T and x ∈ Q r , we compute (4.14) Clearly, for arbitrary small ε, the right hand side of (4.14) converses to zero uniformly for x ∈ Q r , as |t 2 − t 1 | → 0, by invoking the facts, the operator C(t)φ(0) and S(t)ζ 0 , are uniformly continuous on J and the operator S(t) is continuous in the uniform operator topology. Therefore, Φ λ (Q r ) is equicontinuous.
Next, we show that for each λ > 0, the operator Φ λ maps Q r into a relative compact subset of Q r . In order to do this, we prove that for every t ∈ J, the set x ∈ Q r } is precompact in X. By the Assumption 2.2 (H1 ) and (H4 ), we know that the operators S(t), for t ∈ J and I k , for k = 1, ..., m, are compact. Thus, the precompactness of V (t), for each t ∈ J, follows from the compactness of the operators S(t), for t ∈ J and I k 's, for k = 1, . . . , m, and also the compactness of the operator (Qf )(·) = · 0 S(· − s)f (s)ds : L 2 (J; X) → C(J; X) (Lemma 3.2, Corollary 3.3, Chapter 3, [35]). By the Arzelá-Ascoli Theorem, it follows that the operator Φ λ is compact. Thus, invoking Schauder's fixed point theorem, Φ λ has a fixed point in Q r .
In order to establish the approximate controllability of the system (1.1), the following assumption is imposed on f (·, ·). Assumption 4.1. (H5) The function f : J × P → X satisfies the Assumption H3-(i) and uniformly bounded, that is, there exists a constantÑ > 0 such that Proof. By using the Theorem 4.1, for every λ > 0 and x T ∈ X, there exists a mild solution x λ (·) such that with the control where Using the control given in (4.15), it is easy to verify that By the Assumption 4.1 (H5), we have That is, the sequence f ·,x λ ρ(s,x s λ ) : λ > 0 is a bounded sequence in L 2 (J; X). Then, by the Banach-Alaoglu theorem, there exists a subsequence, relabeled as f ·,x λ ρ(s,x s λ ) : λ > 0 , which is weakly convergent to, say f ∈ L 2 (J; X). Furthermore, using the Assumption 2.2 (H4), we obtain for each k = 1, . . . , m. Thus, the sequences {I k (x λ (τ k )) : λ > 0} and {g k (x λ (τ k )) : λ > 0} are bounded in X. Once again using the Banach-Alaoglu theorem, we can find weakly convergent subsequences relabeled as {I k (x λ (τ k )) : λ > 0} and {g k (x λ (τ k )) : λ > 0}, with pointwise weak limits µ k and ν k respectively, for each k = 1, . . . , m. We now estimate Further, using the Assumption 2.2 and bounds of C(t), S(t), for all t ∈ J, we have Next, we evaluate In the right hand side of (4.17), the first term goes to zero using the compactness of the operator (Qf )(·) = · 0 S(· − t)f (s)ds : L 2 (J; X) → C(J; X) (see Lemma 3.2, Chapter 3, [35]), the second term tends to zero making the use of the Hypothesis (H0) and the final term tends to zero by using the compactness of the sine family S(·). Finally, by using the result (4.17) and Assumption 2.2 (H0), we obtain X → 0 as λ → 0, and hence, the control system (1.1) is approximately controllable on J.

5.
Application. The Cauchy problem for one dimensional wave equation has been studied extensively in the literature, see for instance [17,23,27,34], etc. In this section, as an application of the theory developed in the previous sections, we investigate the approximate controllability of an impulsive wave equation with statedependent delay.
Step 1. Resolvent operator and strongly continuous families. Let X = L p ([0, π]; R), for p ∈ [2, ∞), and U = L 2 ([0, π]; R). The operator A : D(A) ⊂ X → X is define as Note that C ∞ 0 ([0, π]; R) ⊂ D(A) and hence D(A) is dense in X. Also, it is easy to verify that A is closed, and hence the condition (R1) of Assumption 2.1 is fulfilled. The spectrum of the operator A defined in (5.2) is given by σ(A) = {−n 2 : n ∈ N}.
Moreover, for every f ∈ D(A), the operator A can be written as where w n (ξ) = 2 π sin(nξ) are the normalized eigenfunctions of the operator A corresponding to the eigenvalues −n 2 , n ∈ N. Next, we show that the Assumption 2.1 (R2) holds true. For any f ∈ X, the resolvent operator of A can be written as Clearly, from the above expression, for any λ > 0, λ 2 ∈ ρ(A) = C\σ(A). Now, we prove that for any λ > 0, we have with M = 1 and ω = 0. One can use the orthogonality of eigenfunctions to obtain the above result in Hilbert spaces (for example, in our case L 2 ([0, π]; R)). But in Banach spaces (in our context L p ([0, π]; R), 2 < p < ∞), establishing the estimate (5.4) is not easy. For completeness, we provide a proof for the first few values of k.
In order to verify that the space P is a phase space, first we show that it satisfies the axiom (A1). In the context of our example, we choose σ = T, µ = 0. Let x(·) be a function with x 0 ∈ P and x| J ∈ PC(J; X). For t ∈ J, we verify that x t is in P. We know that the function x t : (−∞, 0] → X is defined as x t (θ) = x(t + θ), for each t ∈ J.