Aubry-Mather theory for sub-Riemannian control systems
speaker: Piermarco Cannarsa (Università di Roma Tor Vergata)abstract: The long-time average behavior of the value function in the classical calculus of variation is known to be connected with the existence of solutions of the so-called critical equations, that is, a stationary Hamilton-Jacobi equation which includes a sort of nonlinear eigenvalue called the critical constant (or effective Hamiltonian). In this talks, we will address a similar issues for the dynamic programming equation of an optimal control problem, namely a control problem of sub-Riemannian type, for which coercivity of the Hamiltonian is non longer true. We introduce the Aubry set for sub-Riemannian control systems and we show that any fixed point of the Lax-Oleinik semigroup is horizontally differentiable on such a set. Furthermore, we obtain a variational representation of the critical constant by using an adapted notion of closed measures, introduced by A. Fathi and A. Siconolfi (2004). Then, defining a new class of probability measures (strongly closed measures) we define the Mather set for sub-Riemannian control systems and we prove that such a set is included in the Aubry set. We conclude this talk showing how the theory developed so far can be applied to study the well-posedness of the ergodic mean field games system defined on a sub-Riemannian structure.
timetable:
Wed 10 May, 9:30 - 10:00, Aula Dini
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