Infinite Horizon Linear Quadratic Overtaking Optimal Control Problems

TL;DR

This paper investigates infinite horizon linear quadratic control problems without standard assumptions, establishing conditions for the existence of overtaking optimal controls and illustrating the approach's capabilities and limitations.

Contribution

It extends the theory of overtaking optimal controls to cases lacking controllability and stabilizability, providing new existence results and concrete examples.

Findings

Overtaking optimal controls exist under certain conditions.

Classical methods are insufficient for these problems.

The approach reveals both possibilities and limitations.

Abstract

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.