Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

TL;DR

This paper provides an overview of the turnpike property in optimal control, highlighting its historical development, theoretical results in discrete and continuous time, and applications to long-horizon problems.

Contribution

It offers a comprehensive survey of discrete-time and continuous-time turnpike results, including dissipativity approaches and infinite-horizon applications.

Findings

Turnpike property occurs in many optimal control problems.

Dissipativity-based methods help exploit turnpike properties.

Numerical examples illustrate the practical relevance.

Abstract

The turnpike property refers to the phenomenon that in many optimal control problems, the solutions for different initial conditions and varying horizons approach a neighborhood of a specific steady state, then stay in this neighborhood for the major part of the time horizon, until they may finally depart. While early observations of the phenomenon can be traced back to works of Ramsey and von Neumann on problems in economics in 1928 and 1938, the turnpike property received continuous interest in economics since the 1960s and recent interest in systems and control. The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions. We comment on dissipativity-based approaches and infinite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons. After drawing upon numerical examples, the chapter concludes with an outlook on time-varying, discounted, and open problems.