Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations

By Antonio López, Xu Zhang, Enrique Zuazua. Journal des Mathématiques Pures et Appliquées, 2000

101 citations recorded by the source corpus.

Abstract In this paper we prove that the null controllability property of the heat equation may be obtained as limit of the exact controllability properties of singularly perturbed damped wave equations. We impose Dirichlet, homogeneous boundary conditions. The control is supported in a neighborhood of a subset of the boundary that satisfies the classical requirements to apply multiplier techniques. The proof needs an iterative argument that allows to treat separately the low and high frequencies and to make use of the dissipativity of the systems under consideration. This is combined with sharp observability estimates on the eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano, and global Carleman estimates. This proof applies in any space dimension. As a consequence of the uniform controllability we derive uniform observability estimates which can not be proved by classical methods due to the singular character of the perturbations we deal with.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5f878d08bb16f170ceb598d8ddb809b870d8b209

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, methodology): “For instance, in [138] the Carleman inequalities play a key role when deriving the continuity of controls for the following singular perturbation problem connecting dissipative wave equations with the heat equation εutt −∆u+ ut = f1ω, and in [36] where the same issue is addressed for the convective ”
Null controllability of degenerate heat equations (2005, background): “Introduction Controllability properties of nondegenerate parabolic equations have been widely studied in recent years; concerning the heat equation, see in particular [2, 6, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 34, 35, 37, 38]; concerning the Stokes and Navier-Stokes equations, see in”
Exact controllability of semilinear plate equations (2001, unknown): “For the case x 0 ∈ Ω , using the argument in [17], one can prove Theorem 4.1 similarly.”
Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn (2017, background): “See [18, 27, 32] for the case where ω is open; [1, 2, 16] for the case when ω is measurable.”
Carleman estimates for one-dimensional degenerate heat equations (2006, background): “Controllability properties of nondegenerate parabolic equations have been widely studied in recent years; concerning the heat equation, see in particular [2, 5, 10, 13, 14, 15, 16, 17, 19, 24, 25, 27, 28, 29, 32, 33]; concerning the Stokes and Navier-Stokes equations, see in particular [3, 18, 19, 2”
On the existence of time optimal controls for linear evolution equations (2007, background): “We refer [6, 7, 8, 13, 19] and the rich references cited therein for recent progress on the observability estimates (of L(2)-norm in time) for PDEs.”
A unified controllability/observability theory for some stochastic and deterministic partial differential equations (2010, background): “More precisely, it has been shown that: i) By taking the singular limit of some exactly controllable hyperbolic equations, one gives the null controllability of some parabolic equations ([32]); and ii) Controllability results of the heat equation can be derived from the exact controllability of some”