Controllability of Impulsive Semilinear Evolution Equations with Memory and Delay in Hilbert Spaces

TL;DR

This paper establishes the approximate controllability of impulsive semilinear evolution equations with memory and delay in Hilbert spaces, extending previous work to more general cases relevant to reaction-diffusion models.

Contribution

It generalizes controllability results for impulsive semilinear equations with memory and delay, using semigroup theory and avoiding fixed point theorems.

Findings

System can be steered to a neighborhood of the final state using admissible controls.

Delays facilitate controllability in systems with memory.

The approach applies to reaction-diffusion models with impulsive effects.

Abstract

Inspired in our work on the controllability for the semilinear with memory \cite{Carrasco-Guevara-Leiva:2017aa, Guevara-Leiva:2016aa, Guevara-Leiva:2017aa}, we present the general cases for the approximate controllability of impulsive semilinear evolution equations in a Hilbert space with memory and delay terms which arise from reaction-diffusion models. We prove that, for each initial and an arbitrary neighborhood of a final state, one can steer the system from the initial condition to this neighborhood of the final condition with an appropriated collection of admissible controls thanks to the delays. Our proof is based on semigroup theory and A.E. Bashirov et al. technique \cite{Bashirov-Ghahramanlou:2015aa, Bashirov-Jneid:2013aa, Bashirov-Mahmudov:2007aa} which avoids fixed point theorems.