Relaxation and asymptotic expansion of controlled stiff differential equations

Abstract

The control of relaxation-type systems of ordinary differential equations is investigated using the Hamilton–Jacobi–Bellman equation. Firstly, 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.

KEYWORDS:

• Stiff relaxation system
• singular perturbations
• asymptotic expansion
• Hamilton–Jacobi–Bellman equations
• Jin–Xin relaxation

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34H05
• 35F21

Acknowledgments

The authors are grateful to the referees for their careful reading, and for their comments which helped improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the financial support through 320021702/GRK2326, 333849990/IRTG-2379, B04, B05 and B06 of 442047500/SFB1481, HE5386/19-3,23-1,25-1,26-1,27-1,30-1, and received funding from the European Union's Horizon Europe research and innovation programme under the Marie Sklodowska-Curie Doctoral Network Datahyking [grant number 101072546].