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Journal of Convex Analysis 30 (2023), No. 3, 1025--1052
Copyright Heldermann Verlag 2023

A Turnpike Property for Optimal Control Problems with Dynamic Probabilistic Constraints

Martin Gugat
Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Holger Heitsch
Weierstrass Institute, Berlin, Germany

René Henrion
Weierstrass Institute, Berlin, Germany

We consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.

Keywords: Probabilistic constraints, chance constraints, probabilistic robustness, here-and-now-decision, turnpike phenomenon, turnpike result, terminal constraint, probabilistic turnpike.

MSC: 90C20, 90C31.

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