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.
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Stepwise procedure
Illustration of the proof of Theorem 1.3 in two steps: Stabilization + Control
Final data for the adjoint system
Evolution of the adjoint heat equation with final datum
graph of the control in the minimal time