Classical System Theory Revisited for Turnpike in Standard State Space Systems and Impulse Controllable Descriptor Systems

TL;DR

This paper extends classical system theory to analyze turnpike properties in standard state space and impulse controllable descriptor systems, focusing on Riccati equation convergence and conditions for turnpike in complex systems.

Contribution

It generalizes classical results to descriptor systems and provides new conditions for turnpike properties and Riccati equation convergence in these systems.

Findings

Established conditions for turnpike in nondetectable systems

Proved existence and convergence of Riccati solutions in impulse controllable descriptor systems

Extended classical system theory results to more complex system classes

Abstract

The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation.