# Approximate controllability of semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions

Volume 20, Issue 2, pp 108--121
Publication Date: October 28, 2019 Submission Date: June 17, 2019 Revision Date: July 22, 2019 Accteptance Date: August 07, 2019
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### Authors

Cosme Duque - Departamento de Matematicas, Universidad de Los Andes, Merida 5101, Venezuela. Jahnett Uzcategui - Departamento de Matematicas, Universidad de Los Andes, Merida 5101, Venezuela. Hugo Leiva - School of Mathematical and Computational Sciences, University YachayTech, San Miguel de Urcuqui, Imbabura, Ecuador. Oscar Camacho - Departamento de Automatizacion y Control Industrial, Escuela Politecnica Nacional, Quito, Ecuador.

### Abstract

In this paper, we prove that the interior approximate controllability of the linear strongly damped wave equation is not destroyed if we add impulses, nonlocal conditions, and a nonlinear perturbation with delay in the state. Specifically, we prove the interior approximate controllability of the semilinear strongly damped wave equation with impulses, delays, and nonlocal conditions. This is done by applying Roth's Fixed Point Theorem and the compactness of the semigroup generated by the linear uncontrolled system. Finally, we present some open problems and a possible general framework to study the controllability of impulsive semilinear second-order diffusion process in Hilbert spaces with delays and nonlocal conditions.

### Keywords

• Interior approximate controllability
• impulsive semilinear strongly damped wave equation with delays and nonlocal conditions
• strongly continuous semigroups
• Rothe's fixed point theorem

•  93B05
•  93C10

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