# Sandberg 2007 Jan 10

### From NA-Wiki

**Speaker:** Mattias Sandberg, NADA

**Title:** Convergence of Numerical Methods for Optimal Control Using Viscosity Solutions and Differential Inclusion

**When:** The NA seminar meets om Wed Jan 10 at 15:15 in D4523

**Abstract**
Numerical methods for optimal control problems may be analyzed using the following facts:

- The value function associated with an optimal control problem is a viscosity solution of a Hamilton-Jacobi-Bellman equation.
- Optimal control problems may be (re)formulated using differential inclusions.

The Symplectic Pontryagin method will be presented. It is a Symplectic Euler method for the characteristics to the Hamilton-Jacobi-Bellman equation, using a regularized Hamiltonian. The behavior of this method may be obtained by using fact (1) above.

We will also see how fact (2) may be used to show that dynamic programming using a Forward Euler discretization converges with a linear rate, even in cases without any convexity. If time permits, I will show calculations where the Symplectic Pontryagin method was used.