Controllability of the Strongly Damped Wave Equation with Impulses and Delay

Abstract Evading fixed point theorems we prove the interior approximate controllability of the following semilinear strongly damped wave equation with impulses and delay in the space Z1/2 = D((−Δ)1/2)×L2(Ω),where r > 0 is the delay, Γ = (0, τ)×Ω, ∂Γ = (0, τ) × ∂Ω, Γr = [−r, 0] × Ω, (ϕ,ψ) ∈ C([−r, 0]; Z1/2), k = 1, 2, . . . , p, Ω is a bounded domain in ℝℕ(ℕ ≥ 1), ω is an open nonempty subset of , 1 ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, τ; U), with U = L2(Ω),η,γ, are positive numbers and f , Ik ∈ C([0, τ] × ℝ × ℝ; ℝ), k = 1, 2, 3, . . . , p. Under some conditions we prove the following statement: For all open nonempty subsets Ω of the system is approximately controllable on [0,τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state (ϕ (0), ψ(0)) to an ε-neighborhood of the final state z1 at time τ > 0.


Introduction
This work has been motivated by the work done in [23] where the Rothe's xed point theorem was applied to prove the interior approximate controllability of the following semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in Ω, |I k (t, w, v, u)| ≤ a k |w| α k + |v| α k + b k |u| β k + c k , k = , , , . . . , p.
≤ α k < , ≤ β k < , k = , , , , . . . , p, and Several evolutionary processes in nature are characterized by the fact that at certain moments in time they experience an abrupt change. This behavior is observed in real-life problems including: mechanics, chemotherapy, population dynamics, optimal control, ecology, industrial robotics, biotechnology, di usive processes, etc. The theory of impulsive di erential equations [6,24] provides a natural framework to mathematically describe these processes. The controllability of impulsive di erential equations has been widely studied, but the main focus has been exact controllability: impulsive partial neutral functional di erential equations with in nite delay are studied in [7], the exact controllability of semilinear impulsive integrodifferential evolution systems with nonlocal conditions are discussed in [25], and the exact controllability for impulsive di erential systems with nite delay are studied in [26]. To the best of our knowledge, there is only a few works on approximate controllability of impulsive semilinear evolution equations; worth mentioning is [9], where the authors study the approximate controllability of impulsive di erential equations with nonlocal conditions, using measure of noncompactness and Monch xed point theorem, and assuming that the nonlinear terms f , I k do not depend on the control variable.
However, for similinear systems without impulses and delay things are di erent, there are many results whose proofs are standard and the xed point methods for semilinear controllability problems are frequently used and known in the literature of control theory. Moreover, di erent kinds of controllability of many types of nonlinear and semilinear dynamical control systems have been recently considered in several articles, worth mentioning are [12][13][14][15][16][17][18][19] and the references therein. But, as we mention, here we would not use xed point theorems to get our result.
A Banach space is the natural choice when working with impulsive di erential equations.
The interior approximate controllability of the following strongly damped wave equation without impulses and delay has been proved in [20] for all τ > : in Ω.

Formulation of the Problem
Let X = L (Ω) = L (Ω, I R) and consider the linear unbounded operator A : The fractional powered spaces X r (see details in [20]) are given by: x ∈ X r , and if we let Zr = X r × X, the corresponding norm in this Proposition 2.1. The operator P j : Zr → Zr , j ≥ , de ned by is a continuous(bounded) orthogonal projections in the Hilbert space Zr.
Hence, the equations (4) can be written as an abstract second order ordinary di erential equations with impulses and delay as follows where for all x ∈ Ω, k = , , . . . , p, I e k : [ , τ] × Z / × U → X and f e : [ , τ] × C(−r, ; Z / ) × U → X are de ned by With the change of variables w ′ = v, we can write the second order equation (9) as a rst order system of ordinary di erential equations with impulses and delay in the space Z / = X / × X as follows: where u ∈ L ( , τ; U), U = L (Ω), Φ ∈ C(−r, ; Z / ) From condition (5) and the continuous inclusion X / ⊂ X one can prove the following proposition Proposition 2.2. The function F de ned above satis es the following estimate It is well known [8] that the operator A generates a strongly continuous semigroup T(t) t≥ in the space Z = Z / = X / × X, which is also analytic. Now, using Lemma 2.1 in [21], one can get the following representation for this semigroup.

Proposition 2.3. The semigroup T(t) t≥ generated by the operator A is compact and has the following representation
where P j j≥ is a complete family of orthogonal projections in the Hilbert space Z / given by (8) and Moreover, e A j t = e R j t P j , the eigenvalues of R j are:

Controllability of the Linear Equation
In this section we present some characterization of the interior approximate controllability of the linear strongly damped wave equations without impulses and delay. To this end, we note that, for all z ∈ Z / and u ∈ L ( , τ; U) the initial value problem admits only one mild solution given by De nition 3.1. For system (16) we de ne the following concept: The controllability map G τδ : L (τ−δ, τ; U)) → Z / de ned by The adjoint of this operator G * τδ : Z / → L (τ − δ, τ; U) is given by The Gramian controllability operator is given by: The following lemma holds in general for a linear bounded operator G : W → Z / between Hilbert spaces W and Z (see [1], [5], [10], [11] and [22]).
Lemma 3.1. The following statements are equivalent to the approximate controllability of the linear system (16) So, lim α→ G τδ uα = z and the error E τδ z of this approximation is given by the formula f) Moreover, if we consider for each v ∈ L (τ − δ, τ; U)) the sequence of controls given by we get that: and lim α→ G τδ uα = z, with the error E τδ z of this approximation is given by the formula Remark 3.1. The foregoing Lemma implies that the family of linear operators Γ ατδ : Z / → W, de ned for < α ≤ by is an approximate inverse for the right of the operator W, in the sense that lim α→ G τδ Γ ατδ = I. (21) in the strong topology.