A new relation between the condensation index of complex sequences and the null controllability of parabolic systems

Abstract

In this note we present a new result that relates the condensation index of a sequence of complex numbers with the null controllability of parabolic systems. We show that a minimal time is required for controllability. The results are used to prove the boundary controllability of some coupled parabolic equations. To cite this article: F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, C. R. Acad. Sci. Paris, Ser. I 340 (2013). RésuméUne nouvelle rélation entre l'indice de condensation de séquences complexes et la nulle contrôlabilitée des systèmes paraboliques On annonce un résultat qui connecte l'indice de condensation des suites complexes et la nulle controlabilitée des systèmes paraboliques. On montre qu'un temps minimal est nécessaire pour controller, puis on voit le controle a zéro sur le bord de quelqueséquations paraboliques couplées. Pour citer cet article : F. Ammar-Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, C. R. Acad. Sci. Paris, Ser. I 340 (2013).

Recent results on the controllability of linear coupled parabolic problems: A survey

“…In this case the sequence {Λ k } k≥1 of distinct eigenvalues of the operator D∂ xx + A * satisfies the two first conditions in the statement of Theorem 6.2. These conditions imply the existence of a biorthogonal family to {e −Λ k t } k≥1 (and then to t j e −Λ k t k≥1,0≤j≤η−1 , with η ≥ 1) and thus, the moment method can be applied to System (52) providing a control v. Nevertheless, the "gap condition" in general fails and it is crucial in order to prove the estimates of the biorthogonal family (and therefore, for showing that the control v satisfies v ∈ L 2 (0, T ; C m )) (see [7] and [9]). Moreover in [9] we exhibit some examples where the system is null controllable at time T if and only if T is large enough (depending on D, A and B).…”

“…Let us see how inequality (14) implies the observability inequality (10) for the adjoint problem (9) and, in view of Theorem 2.1, the distributed null controllability result for problem (6).…”

Controllability of linear and semilinear non-diagonalizable parabolic systems

“…Almost nothing is known in this context and, in general, the null controllability of (5.3) is an open question; see however [1,2,7,13,20], for some particular results. As we said before, when n < n, even when the coupling matrix M has constant coefficients, a minimal time of controllability T 0 = T 0 (A) ∈ [0, ∞] for system (5.3) can appear (see [9]). …”

“…It is an interesting fact that, in the framework of the controllability of coupled parabolic systems, new (and possibly counter-intuitive) phenomena arise: minimal time of controllability and dependence of the controllability result on the position of the control domain ω; see [9][10][11]15].…”