On chaotic dynamics in optimal control problems

A classical way of solving optimal control problems is to apply Pontryagin's maximum principle, which reduces an optimal control problem to a very special dynamical system, and then find its explicit solutions. This dynamical system usually loses smoothness property and also property of uniqueness of solutions. Nonetheless there are a lot of cases where methods of classical dynamical systems theory can be applied. I plan to speak about two new results on appearance of chaotic dynamics here. The first one concerns sub-Riemannian geometry. I will show that geodesic flows on free Carnot groups of step greater then 3 are non-integrable in Liouville sense. The second one concerns problems with a drift and bounded two-dimensional control. An optimal synthesis is deterministic, i.e. for any initial point there exists a unique optimal solution starting from this point, however this solution as a function of initial data contains the whole dynamics of a classical Markov chain on any arbitrary small interval of time. This phenomenon can not be destroyed by small perturbations and consequently is generic.