Slow decay and turnpike for infinite-horizon hyperbolic LQ problems

TL;DR

This paper analyzes the slow decay and turnpike phenomena in infinite-horizon hyperbolic LQ control problems, providing explicit decay rates and applying results to wave equation networks without geometric control conditions.

Contribution

It offers explicit decay rate estimates and turnpike properties for hyperbolic systems under weak observability or controllability assumptions, extending to wave network control problems.

Findings

Explicit slow decay rate of the closed-loop system derived.

Turnpike property established for the hyperbolic LQ problems.

Application to wave networks without geometric control condition.

Abstract

This paper is devoted to analysing the explicit slow decay rate and turnpike in the infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Assume that some weak observability or controllability are satisfied, by which, the lower and upper bounds of the corresponding algebraic Riccati operator are estimated, respectively. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to the LQ optimal control problems constraint to networks of one-dimensional wave equations and also some multi-dimensional ones with local controls which lack of GCC(Geometric Control Condition).