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

Let the matrix operator $L=D{\mathrm{\partial }}_{xx}+q(x)A_0$, 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 $y_t - Ly = 0$\mathrm{.}When$ \nu \notin $\mathbb{Q}$ ^*_+ $and$q(x) = 1$,$x $\in$ (0, $\pi$)$, there exists a family of root vectors of $(L\ast ,D(L\ast ))$ 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 $(L^{\ast },D(L^{\ast }))$. But there exists $q\in L^{\mathrm{\infty }}(0,$\pi$)$ such that the family of eigenfunctions of $(L^{\ast },D(L^{\ast }))$ 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 $T_0\in [0,+\mathrm{\infty }]$ depending on the condensation of eigenvalues and associated eigenfunctions of $(L^{\ast },D(L^{\ast }))$, such that the corresponding system is null controllable at any time $T>T_0$ and is not if $T<T_0$.