2018, Ciaramella, Gabriele, Fabrini, Giulia

The approximation of feedback control via the Dynamic Programming approach is a challenging problem. The computation of the feedback requires the knowledge of the value function, which can be characterized as the unique viscosity solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) equation. The major obstacle is that the numerical methods known in literature strongly suffer when the dimension of the discretized problem becomes large. This is a strong limitation to the application of classical numerical schemes for the solution of the HJB equation in real applications. To tackle this problem, a new multi-level numerical framework is proposed. Numerical evidences show that classical methods have good smoothing properties, which allow one to use them as smoothers in a multilevel strategy. Moreover, a new smoother iterative scheme based on the Anderson acceleration of the classical value function iteration is introduced. The effectiveness of our new framework is proved by several numerical experiments focusing on minimum-time control problems.

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.