Randomised observation, control and stabilization of waves: [Based on the plenary lecture presented at the 86th Annual GAMM Conference, Lecce, Italy, March 24, 2015]. Issue 5 (14th March 2016) - Privat, Yannick; Trélat, Emmanuel; Zuazua, Enrique
Abstract : The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles The authors have addressed the problem of the optimal design of sensors for purely conservative wave models. They analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. Abstract : The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles we have addressed the problem of the optimal design of sensors for purely conservative wave models. We analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. In this way, using fine asymptotic spectral properties of the Laplacian, we disclosed the links between the geometric properties of the domain where waves propagate and the existence of optimal locations for the sensors or, by the contrary, the emergence of relaxation phenomena. Here we show that spectral randomised observability is equivalent to the property of spectral controllability by means of a discrete set of lumped controls acting everywhere on the domain, and distributed according to the shape of the eigenfunctions. Our results on optimal observation then find natural equivalents on the problem of optimal spectral control. We also give an interpretation of these results in terms of a feedback stabilization property, ensuring the exponential decay of the energy of solutions as time tends to infinity.