WIAS Preprint No. 2941, (2022)

A turnpike property for optimal control problems with dynamic probabilistic constraints



Authors

  • Gugat, Martin
  • Heitsch, Holger
  • Henrion, René

2020 Mathematics Subject Classification

  • 90C15 49J55

Keywords

  • Probabilistic constraints, probabilistic robustness, here-and-now-decision, turnpike phenomenon, turnpike result, terminal constraint, probabilistic turnpike

DOI

10.20347/WIAS.PREPRINT.2941

Abstract

In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition, additive random perturbations in each step and a terminal condition for the expected values. We study optimal control problems where the objective function consists of a term of tracking type for the expected values and a control cost. 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.

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