On Turnpike and Dissipativity Properties of Continuous-Time Optimal Control Problems

TL;DR

This paper explores the deep connections between dissipativity, steady-state optimality, and the turnpike property in continuous-time optimal control, establishing new implications and converse results that enhance understanding of optimal trajectories.

Contribution

It provides novel theoretical results linking dissipativity, steady-state optimality, and turnpike properties, including converse implications in continuous-time control problems.

Findings

Strict dissipation inequality implies steady-state optimality and turnpike existence.

Existence of a turnpike implies dissipativity and steady-state optimality.

Numerical example illustrates the theoretical results.

Abstract

This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings.