【online】Long time behaviour of solutions to Hamilton-Jacobi equations for......

We will discuss the long-time average behaviour of the value function of optimal control problems for sub-Riemannian systems, which cannot be addressed by the classical week KAM theory as the Hamiltonian fails to be coercive in the momentum variable.

Nevertheless, by a dynamical approach, we will show how to prove the existence of a unique critical constant such that the ergodic Hamilton-Jacobi equation admits solutions. Then, we will use such a constant to obtain the limit profile of solutions to the Cauchy problem as the time horizon goes to infinity. We will also provide a representation formula for the critical constant and study the associated Aubry set, giving conditions for such a set to be compact. Finally, we will prove horizontal differentiability for critical solutions to the ergodic Hamilton-Jacobi equation on the Aubry set.