Controllability of a 4 × 4 quadratic reaction–diffusion system

Abstract

We consider a 4 × 4 nonlinear reaction-diffusion system posed on a smooth domain Ω of R N (N ≥ 1) with controls localized in some arbitrary nonempty open subset ω of the domain Ω. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction-diffusion system for every N ≥ 1 in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates cross diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for 1 ≤ N ≤ 2 thanks to our local controllabillity result together with a known theorem on the asymptotics of the free nonlinear reaction-diffusion system.

Local controllability of reaction-diffusion systems around nonnegative stationary states

“…. By using the return method, the author proves the local null-controllability around (0, u * 2 , 0, 0) of (40) (see [28]). More precisely, for this system, a reference trajectory is not difficult to construct.…”

“…Another strategy to get local controllability result for (NL), called the Small L ∞perturbations method is used in [1], [4], [28], [31] and [38]. This method requires the null-controllability of a family of linear parabolic systems.…”

“…By using the small L ∞ -perturbations method, we can obtain a local null-controllability result for (NL). See for instance [11], [10], [12] and [28].…”

“…It comes from the fact that in this case m = 5 < n−4 = 6. Intuitively, with the proof strategy performed in [28], we have to benefit from one coupling term of order 0 (in L ∞ ) and four coupling terms of order 2. This leads to the following open problem.…”

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

“…The proof of Theorem 1.3 is a consequence of the (global) null-controllability of the linear heat equation with a bounded potential (due to Andrei Fursikov and Oleg Imanuvilov, see [23] or [21, Theorem 1.5]) and the small L ∞ perturbations method (see [3, Lemma 6] and [1], [5], [30], [33], [40] for other results in this direction).…”