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Center for Nonlinear Studies |
In this talk we discuss a recent approach to the problem of controllability for parameter dependent systems. It is based on adaptation of reduced basis techniques, in particular of (weak) greedy algorithms, originally developed for constructing approximative solutions to parametric PDEs. The idea is to identify the most distinguished parameter values that provide an approximating space of a small dimension for all parameter dependent 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. Relation to other methods for parameter dependent control problems (averaged control, simultaneous and ensemble control) will be discussed as well.
Host: Anatoly Zlotnik