scholarly journals Robustness of the controllability for the strongly damped wave equation under the influence of impulses, delays and nonlocal conditions

2019 ◽  
Vol 44 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Hugo Leiva
Keyword(s):  
Wave Equation ◽  
Approximate Controllability ◽  
Nonlocal Conditions ◽  
Time Interval ◽  
Short Time Interval ◽  
Damped Wave Equation ◽  
Qualitative Properties ◽  
Strongly Damped Wave Equation ◽  
Fixed Curve ◽  
Strongly Damped

This work proves the following conjecture: impulses, delays, and nonlocal conditions, under some assumptions, do not destroy some posed system qualitative properties since they are themselves intrinsic to it. we verified that the property of controllability is robust under this type of disturbances for the strongly damped wave equation. Specifically, we prove that the interior approximate controllability of linear strongly damped wave equation is not destroyed if we add impulses, nonlocal conditions and a nonlinear perturbation with delay in state. This is done by using new techniques avoiding fixed point theorems employed by A.E. Bashirov et al. In this case the delay help us to prove the approximate controllability of this system by pulling back the control solution to a fixed curve in a short time interval, and from this position, we are able to reach a neighborhood of the final state in time t by using that the corresponding linear strongly damped wave equation is approximately controllable on any interval {t0,T}, 0 < t0 < T.

Approximate controllability of semilinear impulsive strongly damped wave equation

2015 ◽  
Vol 21 (1) ◽  
Author(s):  
Hanzel Larez ◽  
Hugo Leiva ◽  
Jorge Rebaza ◽  
Addison Ríos
Keyword(s):  
Fixed Point ◽  
Wave Equation ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Approximate Controllability ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Damped Wave Equation ◽  
Strongly Damped Wave Equation ◽  
Strongly Damped

AbstractRothe's fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space

scholarly journals Controllability of the Strongly Damped Wave Equation with Impulses and Delay

2017 ◽  
Vol 4 (1) ◽  
pp. 31-39
Author(s):  
Hugo Leiva
Keyword(s):  
Wave Equation ◽  
Distributed Control ◽  
Fixed Point Theorems ◽  
Nonempty Subset ◽  
Damped Wave Equation ◽  
Initial State ◽  
Final State ◽  
Strongly Damped Wave Equation ◽  
Approximately Controllable ◽  
Strongly Damped

AbstractEvading 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.

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