Boundary controllability of parabolic coupled equations

Abstract

This paper is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional parabolic equations are considered. We show that, in this framework, boundary controllability is not equivalent and is more complex than distributed controllability. In our main result, we provide necessary and sufficient conditions for the null controllability.

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

“…There are few results on this framework and most of them concern the one-dimensional case (N = 1), D = Id and A a constant matrix. When D = Id and A is a constant matrix, a necessary and sufficient condition is exhibited in [21] and [6]. This condition is different from the one that characterizes the distributed null controllability of system (1.4) in the constant case (see [4]).…”

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

“…Observe that v ∈ L 2 (0, T ) is a scalar boundary control which acts on the Dirichlet boundary condition of the state at point x = 0 by means of the vector (b 1 , b 2 ) . The aim is to control the whole system (two states) with a control force v. The control problem (10) has been completely solved in [3] when d = 1. For a general system of n ≥ 2 coupled equations with M = I n , see [1].…”

“…The controllability problem for System (10) when d = 1 is more intricate and only few results are known. For b 1 = 0 and b 2 = 1: Firstly, System (10) is approximately controllable in H −1 (0, π; R 2 ) at time T if and only if √ d ∈ Q (see [3]). Secondly, there exists d ∈ (0, ∞) with √ d ∈ Q such that System (10) is not null controllable at any time T > 0 (see [4]).…”

Indirect controllability of locally coupled systems under geometric conditions

“…(4)). S'il semble résolu en dimension 1 d'espace pour a et p constants (voir [6]), il reste complètement ouvert en dimension supérieure ou pour des coefficients variables. Pour ces deux problèmes, il semblerait que la théorie deséquations paraboliques et les outils utilisés se heurtent pour le momentà de sérieuses difficultés.…”

“…The recent work [6] studies slightly more general systems in one space dimension and with constant coupling coefficients. The cases of higher space dimensions and varying coupling coefficients (and in particular when the coefficients vanish in a neighborhood of the boundary) are to our knowledge completely open.…”