Uniform observability of the one-dimensional wave equation for non-cylindrical domains. Application to the control's support optimization

Arthur Bottois (1, 2) , Nicolae Cindea (3, 2) , Arnaud Munch (3, 2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

This work is concerned with the controllability of the one-dimensional wave equation with controls distributed over non-cylindrical domains. The controllability in that case has been obtained in [Castro-Cîndea-Münch, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim 2014] for domains satisfying the usual geometric optics condition. In the present work, we first show that the corresponding observability property holds true uniformly in a precise class of non-cylindrical domains. Within this class, we then consider, for a given initial datum, the problem of the optimization of the control support and prove its well-posedness. Numerical experiments are then discussed and highlight the influence of the initial condition on the optimal domain.

Domaines: Optimisation et contrôle [math.OC]

Arthur Bottois, Nicolae Cindea, Arnaud Munch. Uniform observability of the one-dimensional wave equation for non-cylindrical domains. Application to the control's support optimization. 2019. ⟨hal-02345036⟩