Mathematical Control and Related Fields (MCRF)

Observability of heat processes by transmutation without geometric restrictions

Pages: 177 - 187, Volume 1, Issue 2, June 2011      doi:10.3934/mcrf.2011.1.177

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Sylvain Ervedoza - CNRS, Institut de Mathématiques de Toulouse, UMR 5219, F-31062 Toulouse, France (email)
Enrique Zuazua - Basque Center for Applied Mathematics (BCAM), Bizkaia Technology Park, Building 500, E-48160 Derio - Basque Country, Spain (email)

Abstract: The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.

Keywords:  Heat equation, observability, transmutation.
Mathematics Subject Classification:  Primary: 35C15, 93B07; Secondary: 35K05.

Received: December 2010;      Revised: March 2011;      Available Online: June 2011.