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

TL;DR

This paper investigates the boundary null controllability of coupled parabolic equations with a focus on the simultaneous condensation of eigenvalues and eigenfunctions, revealing new phenomena and conditions for controllability based on spectral properties.

Contribution

It introduces the concept of simultaneous condensation of eigenvalues and eigenfunctions and establishes conditions for minimal control time in such systems.

Findings

Existence of a minimal control time depending on spectral condensation.

New phenomena when eigenfunctions are complete but not a Riesz basis.

Controllability depends on the condensation of eigenvalues and eigenfunctions.

Abstract

Let the matrix operator $L = D\partial_{xx} + q(x)A_0$, with $D = diag(1, \nu)$, $\nu \neq 1$, $q \in L^{\infty} (0, $\pi$)$, and $A_0$ is a Jordan block of order 1. We analyze the boundary null controllability for system $y_t - Ly = 0$. When $ \nu \notin \mathbb{Q} ^*_+ $ and $q(x) = 1$, $x $\in$ (0, $\pi$)$, there exists a family of root vectors of $(L * , D(L *)) $ forming a Riesz basis, moreover, F. Ammar Khodja, A.Benabdallah, M.Gonzalez-Burgos, L.Teresa, show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^* , D(L^*))$. But there exists $q \in L^{\infty} (0, $\pi$)$ such that the family of eigenfunctions of $(L^* , D(L^*))$ is complete but it is not a Riesz basis. In this framework new phenomena arise : simultaneous condensation of eigenvalues and eigenfunctions. We prove the existence of a minimal time $T_0 \in [0, +\infty]$ depending on the condensation of eigenvalues and associated eigenfunctions of $(L^* , D(L^*))$, such that the corresponding system is null controllable at any time $T > T_0$ and is not if $T < T_0$.

Full text

1. 1 First Section

1 First Section

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