Averaged controllability of parameter dependent conservative semigroups

We consider the problem of averaged controllability for parameter depending (either in a discrete or continuous fashion) control systems, the aim being to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do it in the context of conservative models, both in an abstract setting and also analysing the specific examples of the wave and Schrödinger equations. Our first result is of perturbative nature. Assuming the averaging probability measure to be a small parameter-dependent perturbation (in a sense that we make precise) of an atomic measure given by a Dirac mass corresponding to a specific realisation of the system, we show that the averaged controllability property is achieved whenever the system corresponding to the support of the Dirac is controllable. Similar tools can be employed to obtain averaged versions of the socalled Ingham inequalities. Particular attention is devoted to the 1d wave equation in which the time-periodicity of solutions can be exploited to obtain more precise results, provided the parameters involved satisfy Diophantine conditions ensuring the lack of resonances. keywords: Parameter dependent systems, Averaged control, Perturbation arguments, Ingham inequalities, Non-harmonic Fourier series, Wave equation, Schrödinger equation. MSC 2010: 49J55, 93C20, 42A70 ∗LUNAM Université, IRCCyN UMR CNRS 6597 (Institut de Recherche en Communications et Cybernétique de Nantes), École des Mines de Nantes, 4 rue Alfred Kastler, 44307 Nantes, France [email protected] †DeustoTech Fundación Deusto, Avda Universidades, 24, 48007, Bilbao Basque Country Spain [email protected] ‡Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid Spain §Facultad Ingeniera, Universidad de Deusto, Avda. Universidades, 24, 48007, Basque Country Spain ¶Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France