【online】Weak KAM theory for noncoercive Hamilton-Jacobi-Bellman equations. Parts I and II
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题目: 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)
时间:2022年9月26日 15:00 -17:00
方式: Umeet APP ID：159 343 7031 密码：192715
摘要：The long-time average behavior of the value function in the classical calculus of variations is known to be connected with the existence of solutions of the so-called critical equation, that is, a stationary Hamilton-Jacobi equation which includes a sort of nonlinear eigenvalue called the critical constant (or effective Hamiltonian). In these talks, we will address similar issues for the dynamic programming equation of an optimal control problem, the so-called Hamilton-Jacobi-Bellman equation, for which coercivity of the Hamiltonian is non longer true.
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 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.