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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

Mathematical Citation Quotient (MCQ) 2017: 0.71

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Controllability of the Strongly Damped Wave Equation with Impulses and Delay

Hugo Leiva
  • University Yachay Tech, School of Mathematical Science and Information Technology San Miguel de Urcuqu-Imbabura - Urcuquí, Ecuador
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Published Online: 2017-06-30 | DOI: https://doi.org/10.1515/msds-2017-0004


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 ℝ(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.

Keywords : semilinear strongly damped wave equation; impulses and delay; approximate controllability; strongly continuous semigroups

MSC 2010: Primary: 93B05; secondary: 93C10


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About the article

Received: 2016-02-23

Accepted: 2017-05-02

Published Online: 2017-06-30

Published in Print: 2017-04-25

Citation Information: Nonautonomous Dynamical Systems, Volume 4, Issue 1, Pages 31–39, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2017-0004.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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