Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

Abstract

Abstract. In this paper we consider the boundary null-controllability of a system of n parabolic equations on domains of the form Ω = (0, π) × Ω 2 with Ω 2 a smooth domain of R N −1 , N > 1. When the control is exerted on {0} × ω 2 with ω 2 ⊂ Ω 2 , we obtain a necessary and sufficient condition that completely characterizes the nullcontrollability. This result is obtained through the Lebeau-Robbiano strategy and require an upper bound of the cost of the one-dimensional boundary null-control on (0, π). This latter is obtained using the moment method and it is shown to be bounded by Ce C/T when T goes to 0 + . Key words. Parabolic systems, Boundary Controllability, Biorthogonal families, Kalman Rank condition.AMS subject classifications. 93B05, 93C05, 35K05.1. Introduction. The controllability of systems of n partial differential equations by m < n controls is a relatively recent subject. We can quote [LZ98], [dT00], [BN02] among the first works. More recently in [AKBDGB09b], with fine tools of partial differential equations, the socalled Kalman rank condition, which characterizes the controllability of linear systems in finite dimension, has been generalized in view of the distributed null-controllability of some classes of linear parabolic systems. On the other hand, while for scalar problems the boundary controllability is known to be equivalent to the distributed controllability, it has been proved in [FCGBdT10] that this is no more the case for systems. This reveals that the controllability of systems is much more subtle. In [AKBGBdT12], it is even showed that a minimal time of control can appear if the diffusion is different on each equation, which is quite surprising for a system possessing an infinite speed of propagation. It is important to emphasize that the previous quoted results concerning the boundary controllability were established in space dimension one. They used the moment method, generalizing the works of [FR71,FR75] concerning the boundary controllability of the one-dimensional scalar heat equation. We refer to [AKBGBdT11b] for more details and a survey on the controllability of parabolic systems.In higher space dimension the boundary controllability of parabolic systems remains widely open and it is the main purpose of this article to give some partial answers. To our knowledge, the only results on this issue are the one of [ABL12] and [AB12]. Let us also mention [Oli13] for related questions for the approximate controllability problem. In [ABL12, AB12] the results for parabolic systems are deduced from the study of the boundary control problem of two coupled wave equations using transmutation techniques. As a result there are some geometric constraints on the control domain. We will see that this restriction is not necessary.In the present work, we focus on the boundary null-controllability of the following n coupled parabolic equations by m controls in dimension N > 1

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

“…Finally, in [12] the authors extend the one-dimensional boundary null controllability results from [21] and [6] to the N -dimensional case when the domain Ω is a cylindrical domain.…”

Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control

“…This is a complex question; however, something can be said, at least when N = 1. This will be the goal of a forthcoming paper (see [4,6] for some related results).…”

Controllability of linear and semilinear non-diagonalizable parabolic systems

“…However, when n < n, this is a much more complex question. 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]).…”

“…In the one-dimensional case, when the matrix M does not depend on t, a possible alternative is the reformulation of the null controllability problem for (1.1) as a moment problem. This has been done in some recent papers (see for instance [7,11,13,20]) given boundary controllability characterizations for some coupled parabolic systems.…”

“…In [4,5], the authors provide necessary and sufficient conditions for the controllability of n × n parabolic linear systems with constant or time-dependent coefficients. The analysis of similar boundary controllability problems has been the objective of [7,13,20]. A review of all these results can be found in [8].…”