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

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 the system $y_{t}-Ly=0$. When $\sqrt{\nu} \notin \mathbb{Q}_{+}^*$ and $q$ is constant, $q=1$ for instance, there exists a family of root vectors of $(L^*,\mathcal{D}(L^*))$ forming a Riesz basis of $L^{2}(0,\pi;\mathbb{R}^2 )$. Moreover in \cite{JFA14} the authors show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^*,\mathcal{D}(L^*))$, that is to say the existence of $T_0(\nu)$ such that the system is null controllable at time $T > T_0(\nu)$ and not null controllable at time $T < T_0(\nu)$. In the same paper, the authors prove that for all $\tau \in [0, +\infty]$, there exists $\nu \in ]0, +\infty[$ such that $T_0(\nu)=\tau$. 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.