Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions^*

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France.

^** Corresponding author: sambgoree@gmail.com

Received: 6 February 2019
Accepted: 30 November 2020

Abstract

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 y_t − Ly = 0. When $\sqrt{\nu }\notin Q_{+}^{*}$ and q is constant, q = 1 for instance, there exists a family of root vectors of $\left(L^{*},\mathcal{D}\left(L^{*}\right)\right)$ 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 $\left(L^{*},\mathcal{D}\left(L^{*}\right)\right)$, 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.