【online】Weak KAM theory for noncoercive Hamilton-Jacobi-Bellman equations. Parts I and II

• Part I: Piermarco Cannarsa (University of Rome Tor Vergata)

• Part II: Cristian Mendico (University of Rome Tor Vergata)

In Part I, we will derive the existence of the critical constant by studying the convergence of the time-averaged value function as time horizon goes to infinity. Moreover, for the important example of sub-Riemannian systems (that is, control systems associated with a family of vector fields which satisfies the Lie Algebra rank condition), we will construct a solution of the critical equation which coincides with its Lax-Oleinik evolution.

Then, in Part II, we will obtain a variational representation of the critical constant by using an adapted notion of closed measures, which is a class of measures introduced by A. Fathi and A. Siconolfi in [Existence of ${C}^{1}$ critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 2(155):363–388, 2004]. Furthermore, we introduce the Aubry set for a Sub-riemannian control system and we show that any fixed point of the Lax-Oleinik semigroup is horizontally differentiable on such a set.