Relaxation and asymptotic expansion of controlled stiff differential equations

TL;DR

This paper investigates controlled relaxation systems of differential equations, providing asymptotic expansions and convergence results for the value function using Hamilton-Jacobi-Bellman equations, with applications to kinetic and hyperbolic PDEs.

Contribution

It introduces an asymptotic expansion approach for the value function and demonstrates convergence to a reduced control problem, advancing the analysis of relaxation-type systems.

Findings

Asymptotic expansion of the value function in the relaxation parameter

Convergence of the value function to a reduced control problem solution

Application to Jin-Xin relaxation and other examples

Abstract

The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton-Jacobi-Bellman equation. First, we recast the model as a singularly perturbed dynamics which we embed in a family of controlled systems. Then we study this dynamics together with the value function of the associated optimal control problem. We provide an asymptotic expansion in the relaxation parameter of the value function. We also show that its solution converges toward the solution of a Hamilton-Jacobi-Bellman equation for a reduced control problem. Such systems are motivated by semi-discretisation of kinetic and hyperbolic partial differential equations. Several examples are presented including Jin-Xin relaxation.