Mather problem and viscosity solutions in the stationary setting. (English) Zbl 1293.49061
The paper deals with existence of a stationary Mather measure which is invariant under the Euler-Lagrange flow and is supported on a graph. Precisely, the authors consider Lagrangians \(L: {\mathbb R}^n\times {\mathbb R}^n\times\Omega\to{\mathbb R},\) where \(\Omega\) is a compact metric space on which \({\mathbb R}^n\) acts through an action which leaves \(L\) invariant.
The stationary Mather problem consists in minimizing the integral \[ \int_{{\mathbb R}^n\times\Omega} L(0,v,\omega)\;d\mu(v,\omega) \] over all probability measures satisfying the so-called holonomy constraint, and the minimizer is called a stationary Mather measure.
The authors generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians and prove existence of stationary Mather measures invariant under the Euler-Lagrange flow. Several estimates are also obtained for viscosity solutions of the Hamilton-Jacobi equations for the discounted cost infinite horizon problem.
The stationary Mather problem consists in minimizing the integral \[ \int_{{\mathbb R}^n\times\Omega} L(0,v,\omega)\;d\mu(v,\omega) \] over all probability measures satisfying the so-called holonomy constraint, and the minimizer is called a stationary Mather measure.
The authors generalize the standard Mather problem for quasi-periodic and almost-periodic Lagrangians and prove existence of stationary Mather measures invariant under the Euler-Lagrange flow. Several estimates are also obtained for viscosity solutions of the Hamilton-Jacobi equations for the discounted cost infinite horizon problem.
Reviewer: Dian K. Palagachev (Bari)
MSC:
49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
35F21 | Hamilton-Jacobi equations |
35D40 | Viscosity solutions to PDEs |
37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
93E20 | Optimal stochastic control |