学术报告

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

发布人：发布时间： 2021-01-19

字体大小： 【小】 【中】 【大】

**题目:** Long time behaviour of solutions to Hamilton-Jacobi equations for sub-Riemannian control systems

**报告人:** Piermarco Cannarsa（University of Rome Tor Vergata）

**时间:** 2021年01月21日 16:00-17:00

**方式：**Zoom APP ID：690 2910 8040 密码：197423

**摘要：**Sub-Riemannian control systems are an important class of dynamical systems, with linear dependence on controls (but nonlinear on state variables). Controllability properties for such systems are derived by the so-called Lie Algebra rank condition on the associated family of vector fields, called Hörmander vector fields.

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.

**邀请人：**程伟 老师