# A POD-HJB Approach to Optimal Control Problems for PDEs

Alessandro Alla and Maurizio Falcone^{*}

The talk will be focussed on the coupling between an adaptive reduced basis representation for the solution of an evolutive partial differential equation and the Dynamic Programming method for the approximation of a finite horizon optimal control problem. The numerical procedure in based on the approximation of the evolutive Hamilton-Jacobi equation corresponding to the reduced system and allows for the computation of approximate optimal feedback controls via the value function. $\\$ Since we are able to solve an HJB in a rather low dimension, say 3-5 and the POD method often can not get all the informations of the original system with few basis elements, we need to update our POD basis. During the talk we will point out how to overcame some technical difficulties related to this approach: how to compute weak solutions for the HJ equation, how and when to up-date the POD basis, how to derive feedback controls.$\\$ We will also show some numerical tests on the advection-diffusion equation to illustrate the main features of this method.

Mathematics Subject Classification: 49L20 65M25 65K99

Keywords: Optimal Control, Proper Orthogonal Decomposition, Hamilton-Jacobi equations, advection-diffusion equations

Minisymposion: On Optimal Feedback Control for Partial Differential Equations: Theory and Numerical Methods