Internal controllability of parabolic systems with star and tree like couplings

TL;DR

This paper investigates the internal controllability of parabolic systems with star and tree-like couplings, establishing conditions under which local exact controllability to stationary solutions is achievable using Carleman estimates.

Contribution

It introduces new Carleman estimates tailored for coupled parabolic systems with specific coupling structures, enabling controllability analysis in an $L^ Infty$ framework.

Findings

Achieved local exact controllability for star and tree-like coupled parabolic systems.

Developed Carleman estimates with observation operators for the adjoint system.

Extended controllability results to nonlinear cases under certain hypotheses.

Abstract

We consider systems of parabolic equations coupled in zero order terms in a star-like or a tree-like shape, with an internal control acting in only one of the equations. We obtain local exact controllability to the stationary solutions of the system, under hypotheses concerning the supports of the coupling functions. The key point is establishing Carleman estimates with appropriate observation operators for the adjoint to the linearized system, which allows the study of the controllability problem, in either linear or nonlinear cases, in an $L^\infty$ framework.