Averaged control and observation of parameter-depending wave equations. (Contrôle et observation en moyenne d’équations des ondes dépendant de paramètres.) (English. French summary) Zbl 1302.35043

Lazar, Martin; Zuazua, Enrique

C. R., Math., Acad. Sci. Paris 352, No. 6, 497-502 (2014).

Summary: We analyze the problem of averaged observability and control of wave equations. This topic is motivated by the control of parameter-dependent systems of wave equations. We look for controls ensuring the controllability of the averages of the states with respect to the parameter. This turns out to be equivalent to the problem of averaged observation in which one aims at recovering the energy of the initial data of the adjoint system by measurements done on its averages, under the assumption that the initial data of all the components of the adjoint system coincide. The problem under consideration is weaker than the classical notion of simultaneous observation and control. The method of proof uses propagation arguments based on H-measures or microlocal defect measures that reduce the problem to non-standard unique-continuation issues.
Using transmutation techniques, we also derive some results on the averaged observation and control of parameter-dependent heat equations.

MSC:

35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L05	Wave equation
35R30	Inverse problems for PDEs
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
93B07	Observability

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