Approximate Controllability of Semilinear Impulsive Evolution Equations

and Applied Analysis 3 equation with controls acting on the whole domainΩ, so that hypotheses (A) and (B) hold: y tt = Δy + u (t, x) + f (t, y, y t , u (t)) , on (0, τ) × Ω, y = 0, on (0, τ) × ∂Ω, y (0, x) = y 0 (x) , y t (0, x) = y 1 (x) , in Ω, y t (t + k , x) = y t (t − k , x) + I k (t, y (t k , x) , y t (t k , x) , u (t k , x)) , x ∈ Ω, (7) where 0 < t 1 < t 2 < t 3 < ⋅ ⋅ ⋅ < t p < τ, Ω is a bounded domain in R, the distributed control u ∈ L 2 ([0, τ]; L 2 (Ω)), y 0 ∈ H 2 (Ω)∩H 1 0 , y 1 ∈ L 2 (Ω), and I k , f are smooth functions with f being bounded. 2. Controllability of the Linear Equation without Impulses In this section we will present some characterization of the approximate controllability of the corresponding linear equations without impulses. To this end, we note that for all z 0 ∈ Z and u ∈ L 2 (0, τ; U) the initial value problem z 󸀠 = Az + Bu (t) , z ∈ Z,


Introduction
There are many practical examples of impulsive control systems: a chemical reactor system with the quantities of different chemicals serving as the states variable, a financial system with two state variables of the amount of money in a market and the saving rates of a central bank, and the growth of a population diffusing throughout its habitat which is often modeled by reaction-diffusion equation, for which much has been done under the assumption that the system parameters related to the population environment either are constant or change continuously.However, one may easily visualize situations in nature where abrupt changes such as harvesting, disasters, and instantaneous stoking may occur.
That is, the Gramian controllability operator Abstract and Applied Analysis satisfies   > 0 for all 0 <  < , which is equivalent, according to (13) and Lemma 3(c), to the approximate controllability of linear system (8) on [ − , ], for all 0 <  < .This paper has been motivated by the works done in Bashirov and Ghahramanlou [1], Bashirov and Jneid [2], and Bashirov et al. [3], where a new technique to prove the controllability of evolution equations without impulses is used avoiding fixed point theorems, and the work done in [4].
The controllability of impulsive evolution equations has been studied recently by several authors, but most of them study the exact controllability only; it is worth mentioning that Chalishajar [5] studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay, Radhakrishnan and Balachandran [6] studied the exact controllability of semilinear impulsive integrodifferential evolution systems with nonlocal conditions, and Selvi and Mallika Arjunan [7] studied the exact controllability for impulsive differential systems with finite delay.To our knowledge, there are a few works on approximate controllability of impulsive semilinear evolution equations worth mentioning: Chen and Li [8] studied the approximate controllability of impulsive differential equations with nonlocal conditions, using measure of noncompactness and Monch fixed point theorem and assuming that the nonlinear term (, ) does not depend on the control variable, and Leiva and Merentes in [4] studied the approximate controllability of the semilinear impulsive heat equation using the fact that the semigroup generated by Δ is compact.
In this paper, we are not assuming the compactness of the semigroup {()} ≥0 generated by the unbounded operator ; when this semigroup is compact we can consider weaker condition on the nonlinear perturbation  and in the linear part of the system without impulses.Specifically, we can assume the following hypotheses: with 1/2 ≤   < 1, 1/2 ≤   < 1,  = 0, 1, 2, 3, . . ., ; (b) the linear system is approximately controllable only on [0, ].
This case is similar to the semilinear impulsive heat equations studied in [4], where the authors use conditions (a) and (b), the compactness of the semigroup generated by the Laplacian operator Δ, and Rothe's fixed point theorem to prove the approximate controllability of the system on [0, ].
When it comes to the wave equation, the situation is totally different; the semigroup generated by the linear part is not compact; it is in fact a group, which can never be compact.Furthermore, if the control acts on a portion  of the domain Ω for the spatial variable, then the system is approximately controllable only on [0, ] for  ≥ 2, which was proved in [9], where the following system governed by the wave equations was studied: where Ω is a bounded domain in R  ,  is an open nonempty subset of Ω, 1  denotes the characteristic function of the set , the distributed control  ∈  2 ([0, ];  2 (Ω)), and . However, if the control acts on the whole domain Ω, it was proved in [10] that the system is controllable [0, ], for all  > 0.More specifically, the authors studied the following system: where Ω is a bounded domain in R  , the distributed control  ∈  2 ([0, ];  2 (Ω)), and To justify the use of this new technique [1], in this paper, we consider as an application the semilinear impulsive wave equation with controls acting on the whole domain Ω, so that hypotheses (A) and (B) hold: where , and   ,  are smooth functions with  being bounded.

Controllability of the Linear Equation without Impulses
In this section we will present some characterization of the approximate controllability of the corresponding linear equations without impulses.To this end, we note that for all  0 ∈  and  ∈  2 (0, ; ) the initial value problem admits only one mild solution given by Definition 2. For system (8), one defines the following concept: the controllability maps   :  2 ( − , ; ) → ,   :  2 (0, ; ) → , defined by satisfy the following relation: The adjoints of these operators  *  :  →  2 ( − , ; ),  *  :  →  2 (0, ; ) are given by The Gramian controllability operators are given by (3) and The following lemma holds in general for a linear bounded operator  :  →  between Hilbert spaces  and  (see [4,11,12]).
So lim  → 0     =  and the error    of this approximation is given by the formula (f) Moreover, if one considers for each V ∈  2 ( − , ; ) the sequence of controls given by one gets that with the error    of this approximation given by the formula Remark 4. The foregoing lemma implies that the family of linear operators Γ  :  → , defined for 0 <  ≤ 1 by is an approximate inverse for the right of the operator , in the sense that lim  → 0 in the strong topology.

Controllability of the Semilinear Impulsive System
In this section, we will prove the main result of this paper: the approximate controllability of the semilinear impulsive evolution equation given by (1).To this end, for all  0 ∈  and  ∈  2 (0, ; ), the initial value problem Now, we are ready to present and prove the main result of this paper, which is the approximate controllability of semilinear impulsive equation (1).
Geometrically, the proof goes as shown in Figure 3.This completes the proof of the theorem.

Applications
As an application, we will prove the approximate controllability of the following control system governed by the semilinear impulsive wave equation: where Ω is a bounded domain in R  , the distributed control  ∈  2 ([0, ];  2 (Ω)),  0 ∈  2 (Ω)∩ 1 0 ,  1 ∈  2 (Ω), and   ,  are smooth functions with  being bounded.4.1.Abstract Formulation of the Problem.In this part, we will choose the space where this problem will be set up as an abstract control system in a Hilbert space.Let  =  2 (Ω) =  2 (Ω, R) and consider the linear unbounded operator  : () ⊂  →  defined by  = −Δ, where Then the eigenvalues   of  have finite multiplicity   equal to the dimension of the corresponding eigenspace and 0 <  1 <  2 < ⋅ ⋅ ⋅ <   → ∞.Moreover, consider the following.
(a) There exists a complete orthonormal set { , } of eigenvectors of .
(b) For all  ∈ (), we have where ⟨⋅, ⋅⟩ is the usual inner product in  2 and which means the set {  } ∞ =1 is a complete family of orthogonal projections in  and  = ∑ ∞ =1   ,  ∈ .(c) − generates an analytic semigroup { − } given by With the change of variables   = V, we can write second order equation (46) as a first order system of ordinary differential equations in the Hilbert space  1/2 =  1/2 ×  as follows: where is an unbounded linear operator with domain (A) = () × ( It is well known that the operator A generates a strongly continuous group {()} ≥0 in the space  =  1/2 =  1/2 ×  (see [13]).The following representation for this group can be found in [9]  Proof.From [10], we know that the corresponding linear system without impulses

Lemma 3 .
The following statements are equivalent to the approximate controllability of the linear system(8) on [−, ].
as Theorem 2.2.The group {()} ≥0 generated by the operator A has the following representation:=1       ,  ∈  1/2 ,  ≥ 0,(51)where {  } ≥0 is a complete family of orthogonal projections in the Hilbert space  1/2 given by Approximate Controllability.Now, we are ready to formulate and prove the main result of this section, which is the approximate of the semilinear impulsive wave equation with bounded nonlinear perturbation.