2024/01/17 by Martino Bardi, Bardi, Martino
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35B27 #35D40 #49L25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Optimization and Control (math.OC) #Primary: 35F21 #Quantum chaos and dynamical systems #Secondary: 37J50 #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2401.09147
openalex publication_date 2024/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behaviour of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.