State-constrained controllability of linear reaction-diffusion systems

TL;DR

This paper investigates the controllability of linear reaction-diffusion systems with state constraints, establishing conditions for controllability to trajectories and minimal control time under various constraints.

Contribution

It provides new controllability results for coupled linear parabolic systems with state constraints, including conditions for approximate and exact nonnegativity control.

Findings

Controllability to trajectories is achievable in large time under certain conditions.

A positive minimal time exists for state-constrained controllability.

Results apply to systems with diagonal diffusion matrices and specific eigenvalue conditions.

Abstract

We study the controllability of a coupled system of linear parabolic equations, with non-negativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an "approximate" nonnegativity constraint, and a another stronger one, with "exact" nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a "staircase" method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.