Cosine operator and controllability of the wave equation with memory revisited

TL;DR

This paper revisits the controllability of wave equations with memory using cosine operator methods, improving previous results by precisely identifying the control time.

Contribution

It enhances the operator-based approach for wave equations with memory, providing more accurate control time identification compared to prior work.

Findings

Improved control time characterization for wave equations with memory

Enhanced operator method application to Gurtin-Pipkin type equations

Refined results over previous cosine operator approach

Abstract

Controllability of the heat equations with memory (of Gurtin-Pipkin type) has been studied using several methods with the following in common: the existing results on controllability of the (memoryless) wave equation are lifted to the equation with memory. Here we revisit the approach based on operator methods (cosine operators) and we improve the results obtained in a previous paper (L. Pandolfi: "The controllability of the Gurtin-Pipkin equation: a cosine operator approach:, Appl.Math. and Optim. 52, 143-165, 2005. In particular we identify the control time.