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 Z<inf>1/2</inf> = 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]; Z<inf>1/2</inf>), k = 1, 2,..., p, Ω is a bounded domain in R^N(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, I<inf>k</inf> ∈ 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 z<inf>1</inf> at time τ > 0.

Año de publicación:

2017

Keywords:

• strongly continuous semigroups
• Approximate controllability
• Impulses and delay
• Semilinear strongly damped wave equation

Fuente:

scopus

google