Controllability under positivity constraints of semilinear heat equations

TL;DR

This paper establishes controllability results for semilinear heat equations with positivity constraints on control inputs, demonstrating the existence of a positive minimal control time and providing numerical insights into control sparsity.

Contribution

It proves controllability under positivity constraints without sign or Lipschitz restrictions on the nonlinearity, extending to various initial states and trajectories.

Findings

Minimal controllability time is strictly positive.

Controllability achieved under broad nonlinear conditions.

Numerical simulations reveal control sparsity in minimal time.

Abstract

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.