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

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## 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 solution to the turnpike set converges 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 solution remains essentially, during the time frame, close to an optimal solution of an associated static optimal control problem, except during a subset of time frame that is of small relative Lebesgue measure as the time horizon is large. Motivated by a specific example of optimal control problem for the heat equation with control constraints, we next 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.

## Keywords

Measure-turnpike Strict dissipativity Strong duality State and control constraints## Mathematics Subject Classification

49J20 49K20 93D20## Notes

### Acknowledgements

We would like to thank Prof. Enrique Zuazua for fruitful discussions and valuable suggestions on this subject. We acknowledge the financial support by the grant FA9550-14-1-0214 of the EOARD-AFOSR. The second author was partially supported by the National Natural Science Foundation of China under grants 11501424.

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