Averaged control and observation of parameter-depending wave equations

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