What is it about?
The turnpike property clarifies how solutions of a dynamic optimal control problem that depends on initial data and possibly also on terminal conditions are related to optimal control problems that are independent of initial and terminal data, for example static problems or optimal control problems for periodic states. The turnpike result shows that for large time horizons most of the time the solutions of the dynamic optimal control problem and the static optimal control problem will be very close to each other.
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Why is it important?
The information about the structure of the optimal controls provided by the turnpike results is relevant in numerical analysis since it justifies to start an iterative procedure for the solution of a dynamic optimal control problems with a solution of the simpler e.g. static problem (the 'turnpike'). It also justifies certain suboptimal strategies that rapidly steer the system to the turnpike. The turnpike property implies that for large time horizons such a strategy is almost optimal. The turnpike property is also important to show that a moving-horizon optimal control strategy yields stabilizing controls.
Read the Original
This page is a summary of: Turnpike properties for partially uncontrollable systems, Automatica, March 2023, Elsevier, DOI: 10.1016/j.automatica.2022.110844.
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A Turnpike Property for Optimal Control Problems with Dynamic Probabilistic Constraints
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.
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