# Integral and measure-turnpike properties for infinite-dimensional optimal control systems

## TL;DR

This paper establishes integral and measure-turnpike properties for infinite-dimensional optimal control systems, linking strict dissipativity and strong duality, and providing insights into long-term behavior of optimal trajectories.

## Contribution

It introduces the first general integral-turnpike result for infinite-dimensional systems with terminal constraints and connects strict dissipativity with measure-turnpike properties.

## Key findings

- Integral-turnpike property holds around a set for infinite-dimensional control problems.
- Strict dissipativity implies measure-turnpike property in constrained systems.
- Strong duality ensures strict dissipativity and measure-turnpike properties.

## Abstract

We first derive a general integral-turnpike property around a set for infinite-dimensional non-autonomous optimal control problems with any possible terminal state constraints, under some appropriate assumptions. Roughly speaking, the integral-turnpike property means that the time average of the distance from any optimal trajectory to the turnpike set con- verges to zero, as the time horizon tends to infinity. Then, we establish the measure-turnpike property for strictly dissipative optimal control systems, with state and control constraints. The measure-turnpike property, which is slightly stronger than the integral-turnpike property, means that any optimal (state and control) solution remains essentially, along the time frame, close to an optimal solution of an associated static optimal control problem, except along a subset of times that is of small relative Lebesgue measure as the time horizon is large. Next, we prove that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike. Finally, we conclude the paper with several comments and open problems.