Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign

TL;DR

This paper demonstrates that a one-dimensional semilinear reaction-diffusion system can be approximately controlled to reach a target state with the same number and order of sign changes by manipulating the reaction coefficient, using a sequence of diffusion problems.

Contribution

It introduces a novel method for controlling reaction-diffusion equations via the reaction coefficient, enabling approximate controllability with finitely many sign changes.

Findings

Any target state with the same sign change structure can be approximately reached.

The control method involves shifting sign change points through diffusion processes.

The approach applies to states in the Sobolev space $H_0^1(0,1)$.

Abstract

We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $ u^*\in H_0^1 (0,1)$, with as many changes of sign in the same order as the given initial data $ u_0\in H^1_0(0,1)$, can be approximately reached in the $ L^2 (0,1)$-norm at some time $T>0$. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.