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