Title: Hybrid optimal synthesis and geometric time optimal control (1 hour)
Abstract: In the last five years a new paradigm has emerged for solving static Hamilton-Jacobi-Bellman equations. This paradigm uses fast shortest path algorithms rather than the traditional finite element, fixed-point approach and results in a vast speedup to obtain the numerical solution. The key idea in these new algorithms is an observation due to Tsitskilis that if a triangulation of the state space can be constructed so that the value function V evaluated on the grid points can be ordered along level sets of V, then a shortest path algorithm can be used.
In this talk I will discuss one such algorithm for solving HJB equations numerically. This algorithm exploits the geometry of certain relevant vector fields, in the spirit of numerical solvers that preserve geometric invariants of a flow. Two small examples will demonstrate the performance of the algorithm.
Next, I will discuss preliminary results on the application of this algorithm to one of the open problems in time optimal control: time optimal swing up of the 2D pendulum on a cart. Results from the geometric theory of time optimal control for 2D control affine systems due to Sussmann and Piccoli will be briefly reviewed and it will be shown how a combination of analytical and numerical analyses gives a convincing picture of the global structure of the time optimal trajectories.