Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms

TL;DR

This paper demonstrates local null controllability for a class of semilinear, nonlocally coupled parabolic systems using fixed-point methods, extending previous linear results and exploring boundary control possibilities.

Contribution

It introduces new controllability results for semilinear nonlocal parabolic systems, broadening the class of linear systems for which controllability is established.

Findings

Achieved local null controllability at fixed time T>0

Extended controllability results to broader linear systems

Discussed boundary controllability and future research directions

Abstract

This paper extends our previous controllability results for a class of coupled linear parabolic systems with nonlocal interactions, motivated by applications in finance such as generalized Black--Scholes models. We establish local null controllability at a fixed time T>0 for a class of semilinear, nonlocally coupled systems driven by a single internal control acting on one component. The proof combines Kakutani's fixed-point theorem with a controllability/observability estimate for the associated linearized dynamics. In addition, we obtain controllability for a broader class of linear systems than those considered in the first article. The paper concludes with remarks on boundary controllability within the same nonlocal framework and with perspectives for future research.