Resumo: The dynamic programming (DP) approach provides a synthesis of optimal feedback controls for many nonlinear optimal control problems. However, once we adopt this approach and compute the value function via the numerical approximation of Hamilton-Jacobi-Bellman (HJB) equation there are two major difficulties: the solutions of an HJB equation are in general non-smooth and the approximation in high dimension requires huge memory allocations.
In this talk, I will start describing the Linear Quadratic Regulator (LQR) case for infinite horizon problems and then, switching to nonlinear evolutive Partial Differential Equations (PDEs). The discretization of a PDE leads to a very large system of ODEs and therefore model order reduction, e.g Proper Orthogonal Decomposition (POD), is crucial in order to reduce the complexity of the problem. Finally, I will discuss some numerical tests will illustrate the effectiveness of the method.