Uniform Turnpike Property and Singular Limits

TL;DR

This paper establishes a uniform turnpike property for parameter-dependent parabolic equations, including heat equations with oscillating coefficients, and demonstrates homogenization of the turnpike behavior through theoretical proofs and numerical validation.

Contribution

It proves a uniform turnpike result for a class of parameter-dependent parabolic equations and demonstrates homogenization in oscillatory media, extending previous results to more complex settings.

Findings

Uniform exponential stabilization of Riccati equations

Turnpike property holds uniformly for oscillatory media

Homogenization of the turnpike property achieved

Abstract

Motivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiments.