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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 (Equation presented), in the space Z1/2 = D((−∆)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 Ω, 1ω 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 Ω 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.