Abstract
We consider an×nnonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. This system models reversible chemical reactions. We act on the system throughmcontrols (1 ≤m<n), localized in some arbitrary nonempty open subsetωof the domainΩ. We prove the local exact controllability to nonnegative (constant) stationary states in any timeT> 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 spaceL∞(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 aL∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.
Highlights
For 1 ≤ i ≤ n, let ui(t, .) : Ω → R be the concentration of the chemical component Ai at time t
Under appropriate assumptions, we prove the controllability of (NL-U), in an appropriate subspace of L∞(Ω)n, locally around U ∗, with controls in L∞((0, T ) × Ω)m
By an adequate affine transformation, the proof relies on the study of the null-controllability of an equivalent cascade system with second order coupling terms
Summary
According to the forward reaction of (1.1), when αi molecules of Ai disappear (1 ≤ i ≤ n), they are called the “reactants”, βi molecules of Ai appear (1 ≤ i ≤ n). The backward reaction of (1.1) is governed by the same law: when βi molecules of Ai disappear (1 ≤ i ≤ n), here they are the reactants, αi molecules of Ai appear (1 ≤ i ≤ n). – For at most quadratic nonlinearities, global existence of weak solutions holds (see [31], Thm. 5.12). Let us mention that global existence of renormalized solutions holds in all cases for (1.3) (see [15])
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