Recent results in the approximation of nonlinear optimal control problems

This survey paper presents recent advances for the numerical solution of Hamilton-Jacobi-Bellman equations related to optimal control problems. The Dynamic Programming approach suffers for the "curse of dimensionality" and the solution of the nonlinear partial differential equations characterizing the value function of optimal control problems in high dimension is out of reach. However, a combination of various techniques can circumvent this difficulty and find the solution of optimal control problems up to dimension 10, a range of dimensions which could be enough for many applications. We illustrate here some of these techniques: patchy domain decomposition, fast marching and fast sweeping and an acceleration method based on the coupling between value and policy iteration. Numerical examples will illustrate the main features of those methods. © 2014 Springer-Verlag.