Enrique Zuazua – Minicurso X ENAMA

Greedy Control

Enrique Zuazua (Universidade de Autonoma de Madrid – Espanha)


In this series of lectures, based on recent joint work with Martin Lazar (Dubronik, Croatia) we discuss the problem of controllability for parameter-dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter depending Partial Differential Equations (PDE) or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation.

The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies. We shall also present some problems arising in the context when facing PDE. Finally, following a recent joint work with M. Choulli (Metz, France), we also consider the problem of the stability of the recovery of the coefficients of elliptic PDE in terms of their resolvents arising naturally in this context, when trying to develop greedy algorithms for robust control, independent of the data to be.