Ergodicity of robust switching control and nonlinear system of quasi variational inequalities

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear quasi variational inequalities related to robust switching control, establishing ergodic properties under broad conditions using probabilistic methods.

Contribution

It provides a novel probabilistic approach to characterize the ergodic limits of nonlinear variational inequalities in control problems without non-degeneracy assumptions.

Findings

Long-term solutions characterized by ergodic variational inequalities

Results hold under dissipativity without diffusion non-degeneracy

Probabilistic dual game representation used for analysis

Abstract

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.