State-constrained controllability of linear reaction-diffusion systems

Abstract

We study the controllability of a coupled system of linear parabolic equations, with nonnegativity 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. 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.

Control of reaction-diffusion models in biology and social sciences

“…In the presence of positivity constraints, there is a minimal controllability time for the linear heat equation (see [71] and [69] for systems). The same occurs for the semilinear heat equation (see [93]).…”

“…(x, t) ∈ (0, l(t)) × (0, T ) u(0, t) = a(t) t ∈ (0, T ) l (t) = g(l(t), u(1, t), ∂ x u(1, t)) t ∈ (0, T ) 0 ≤ u(x, 0) ≤ 1. (69) In such setting, even the controllability without state constraints is a challenging question [44].…”

“…The constrained controllability of linear parabolic systems has been implemented in [69]. Note, however, that a phase plane analysis will be more intricate as the complexity of the nonlinear ODE system increases significantly with the dimension.…”

“…However, most of the existing results do not guarantee that state constraints requirements, such as the positivity of solutions, are met. The study of controllability with state constraints for parabolic partial differential equations is a much more recent research topic [93,71,72,98,106,84,69]. The presence of constraints requires the development of new methods for controllability.…”

Turnpike in optimal control of PDEs, ResNets, and beyond

“…It is not obvious whether the Riccati-inspired strategies can be readily extended to this context; now the control is not given linearly in terms of the adjoint state p T . Although the controllability theory under constraints on the state and/or control for linear and semilinear PDEs is now rather well established (Lohéac, Trélat and Zuazua 2017, Pighin and Zuazua 2018, Hegoburu, Magal and Tucsnak 2018, Pouchol, Trélat and Zuazua 2019, Maity, Tucsnak and Zuazua 2019, Le Balc'h 2020, Lissy and Moreau 2021…”