Local controllability of reaction-diffusion systems around nonnegative stationary states

Abstract

We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of R N . This system models reversible chemical reactions. We act on the system through m controls (1 ≤ m < n), localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the local exact controllability to nonnegative (constant) stationary states in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space L ∞ (Ω) n . The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in a L ∞ -context. Finally, an appropriate inverse mapping theorem enables to go back to the nonlinear reaction-diffusion system.

Controllability of a 4 × 4 quadratic reaction–diffusion system

“…The article [36] by the author treats the local-controllability of (268) around nonnegative (constant) stationary states by using the same kind of change of variables as in (59) and (73). Nevertheless, the proof of observability inequalities for the linearized system cannot follow the same strategy as performed in Section 4.3.7.…”

Null-controllability of perturbed porous medium gas flow

“…The following Theorem is originally shown in [28,Proposition 2.3] (see also [24] and [3] for subsequent adaptations). We assume higher regularity for the initial datum a priori, and thus for the controlled trajectory, having in mind the fixed-point argument.…”

Control of reaction-diffusion models in biology and social sciences

“…Here we shall only consider scalar equations. For extensions to parabolic systems we refer to [46,1,33,51,62].…”

“…In the subsection below we shall see the apparence of such obstructions. Right: Sketch of the phase-plane analysis for the trajectory leading to a solution of (62) such that ∂ t w = 0 for all t) are the constant steady-states w ≡ 0, w ≡ θ, w ≡ 1 since they cancel both the bistable nonlinearity and the parabolic operator. Therefore, a path of steady states connecting any pair of steady-states cannot exist.…”