Observability of heat processes by transmutation without geometric restrictions. (English) Zbl 1244.35028

Ervedoza, Sylvain; Zuazua, Enrique

Math. Control Relat. Fields 1, No. 2, 177-187 (2011).

Summary: The goal of this note is to explain how transmutation techniques 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 [K. D. Phung, in: Some problems on nonlinear hyperbolic equations and applications. Hackensack, NJ: World Scientific; Beijing: Higher Education Press. Series in Contemporary Applied Mathematics CAM 15, 386–412 (2010; Zbl 1223.35215)] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [L. Miller, Discrete Contin. Dyn. Syst., Ser. B 14, No. 4, 1465–1485 (2010; Zbl 1219.93017)] and new representation formulas allowing to make a link between heat and wave trajectories.

MSC:

35C15	Integral representations of solutions to PDEs
93B07	Observability
35K05	Heat equation
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

heat equation; observability; transmutation; heat and wave trajectories

Citations:

Zbl 1223.35215; Zbl 1219.93017