The turnpike property in finite-dimensional nonlinear optimal control

By Emmanuel Trélat, Enrique Zuazua. 2014

258 citations recorded by the source corpus.

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory.

Subfields: Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b70dc3729cd3221a4c3d7e6539de441ab7d984ee

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In Control and Systems Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

Minimal controllability time for the heat equation under unilateral state or control constraints (2017, background): “The turnpike phenomenon was analyzed in [23] for general nonlinear optimal control problems in finite dimension, showing that, when one steers a control system from one initial configuration to a final configuration in fixed time T , then, for T large, the optimal trajectory enjoys the above qualita”
On the role of dissipativity in economic model predictive control (2015, background): “The concept of dissipativity dates back to Willems (1972) (see also (Byrnes and Lin, 1994) for a discrete time version) and is as follows.”
Approximation Properties of Receding Horizon Optimal Control (2016, background): “Interestingly, the turnpike property, which is a classical property in optimal control [52,18,39,14] has recently attracted renewed attention, not least because of its importance for Receding Horizon Control [16,19,20,50, 42].”