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.
Perspectives
In this paper we consider systems governed by linear dynamics with a quadratic objective functional. However, the turnpike property is of a universal nature and also holds for nonlinear systems with nonconvex objective funtionals, provided that the objective functional leads to a preference of the turnpike if possible. This situation can be characterized by a strict dissipativity condition, which however is hard to verify. In the future, the nature of the turnpike property for nonlinear systems should be explored further.
Martin Gugat
Friedrich-Alexander-Universitat Erlangen-Nurnberg
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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