Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

TL;DR

This paper investigates the approximate controllability of a multidimensional semilinear reaction-diffusion equation using multiplicative controls, providing conditions for reaching desired states quickly or within specified time frames.

Contribution

It establishes new sufficient conditions for approximate controllability of reaction-diffusion equations with multiplicative controls, including cases with globally supported controls.

Findings

Approximate controllability achieved under certain conditions.

Controllability within any pre-specified time interval for globally supported controls.

Utilizes linear semigroup theory and uniform approximation techniques.

Abstract

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Moreover, in the case of a globally supported control, we prove the approximate controllability within any time-interval given in advance which does not depend on the initial and target states. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.