2018, Fabrini, Giulia, Iapichino, Laura, Volkwein, Stefan

Often a dynamical system is characterized by one or more parameters describing physical features of the problem or geometrical configurations of the computational domain. As a consequence, by assuming that the system is controllable, a range of optimal controls exists corresponding to different parameter values. The goal of the proposed approach is to avoid the computation of a control function for any instance of the parameters. The greedy controllability consists in the selection of the most representative values of the parameter set that allows a rapid approximation of the control function for any desired new parameter value, ensuring that the system is steered to the target within a certain accuracy. By proposing the reduced basis (RB) method in this framework, we are able to consider linear parametrized partial differential equations (PDEs) in our setting. The computational costs are drastically reduced and the efficiency of the greedy controllability approach is significantly improved. As a numerical example a heat equation with convection is studied to illustrate our proposed RB greedy controllability strategy.

2017, Fabrini, Giulia, Falcone, Maurizio, Volkwein, Stefan

In this paper we use a reference trajectory computed by a model predic- tive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy.