WP4 - Perturbed systems

• State of the art

Although the general theory of optimal control is valid under very general conditions on the system and the costs, in particular without restriction on the dimension of the state space, in many practical problems the standard methods are not applicable without a considerable simplification of the model. This is often done by practitioners using empirical arguments, but there are mathematical tools that allow a rigorous study of some complex systems as perturbations of simpler ones. For instance, the macroscopic behavior of systems with very fast oscillations at a microscopic scale can be approximated by a homogenized model where the oscillating quantities are suitably averaged. Another example is the dimension reduction of systems with many state variables that evolve at different time scales. These two problems fall, respectively, within the theory of homogenization and of singular perturbations of control systems and of their associated Hamilton-Jacobi equations. Both theories rely on ideas and methods of ergodic control. Ergodic control problems have as cost functional the long time average of a given running cost. They are of independent interest for deterministic as well as stochastic systems.

• Objectives

Within this WP we aim at enlarging the asymptotic theory of Hamilton-Jacobi equations to cover problems with non-convex Hamiltonians, such as differential games, and with more complicated oscillations than in the classical periodic setting. In particular, we will analyze singularly perturbed systems with unbounded fast variables and the homogenization of Hamilton-Jacobi equations arising in the level set method in the stationary ergodic setting.
 

• Fellows projects

• Planned person-months: 108