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

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.

Logarithmic convexity and impulsive controllability for the 1-D heat equation with dynamic boundary conditions

“…Unlike the distributed controllability for impulsive systems that has been extensively studied in the literature, see for instance [12,14,21,24,25,27] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area, we mention [7,18,32,35]. In the later type, the control is a function acting only at one instant of time τ .…”

Qualitative properties for the 1 − D impulsive wave equation: controllability and observability

“…Unlike the interior controllability for impulsive systems that have been extensively studied in the literature, see for instance [8,4,9,3,11] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area. We mention A. Khapalov [10] who proved the exact controllability of a class of second-order hyperbolic boundary problems with impulse controls using the Huygens' principle.…”

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…This type of control is very weak since it only acts in a subdomain at one instant of time, which makes the problem of controllablity for the heat equation with impulse control very challenging. Unlike the interior controllability for impulsive systems that have been extensively studied in the literature, see for instance [4,9,23,24,27] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area, we mention [6,18,29].…”