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

This paper investigates the relations between three different properties, which are of importance in continuous-time 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 that dissipativity with respect to a steady state implies optimal operation at this steady state and the existence of a turnpike at the same steady state. 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. Finally, we invoke an assumption on the uniqueness of the optimal steady state to establish the equivalence of the three properties. We draw upon a numerical example to illustrate our findings.