Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||43|
|Published online||01 March 2021|
Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions*
Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373,
** Corresponding author: firstname.lastname@example.org
Accepted: 30 November 2020
Let the matrix operator L = D∂xx + q (x) A0, with D = diag (1, ν), ν ≠ 1, q ∈ L∞ (0, π), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for the system yt − Ly = 0. When and q is constant, q = 1 for instance, there exists a family of root vectors of forming a Riesz basis of L2(0,π;ℝ2). Moreover F. Ammar Khodja et al. [J. Funct. Anal. 267 (2014) 2077–2151] shows the existence of a minimal time of control depending on condensation of eigenvalues of , that is to say the existence of T0 (ν) such that the system is null controllable at time T > T0 (ν) and not null controllable at time T < T0 (ν). In the same paper, the authors prove that for all τ ∈ [0, + ∞], there exists ν ∈] 0, + ∞ [ such that T0 (ν) = τ. When q depends on x, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.
Mathematics Subject Classification: 93B05 / 93C20 / 93C25 / 30E05 / 35K90 / 35P10
Key words: Control theory / parabolic partial differential equations / minimal null control time
© The authors. Published by EDP Sciences, SMAI 2021
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