Greedy control: Application to control theory

Evolution of the solution to the semi-discretised problem for heat eq.
Evolution of the solution to the semi-discretised problem for heat eq.

Monday, June 26th, 2017, 12:00h.,
Central Meeting Room at DeustoTech

Martin Lazar

University of Dubrovnik, ‚Dubrovnik, Croatia


In this talk we discuss a new approach to the problem of controllability for parameter dependent systems. It is based on adaptation of (weak) greedy algorithms, originally developed for constructing an approximative solutions to parametric PDEs. The idea is to identify the most distinguished parameter values describing the whole range of admissible controls. The algorithm consists of the (possible expensive) offline part devoted to the selection of parameter representatives and the online one enabling a fast computation of an approximative control for a given value of the parameter within a prescribed accuracy. Our results lead to optimal approximation rates expressed in terms of Kolmogorov widths.
These results are applied to 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.

This lecture will be orientated toward their application to control theory. Related numerical cost and comparision with other approaches will be discussed as well.