The turnpike property in finite-dimensional nonlinear optimal control

Trelat E, Zuazua E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

Journal of Differential Equations Elsevier

Book Volume: 258

Pages Range: 81-114

Journal Issue: 1

DOI: 10.1016/j.jde.2014.09.005

Abstract

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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

University of Paris 6 - Pierre et Marie Curie / Université Paris VI Pierre et Marie Curie (UPMC) FR France (FR) Basque Center of Applied Mathematics (BCAM) ES Spain (ES)

How to cite

APA:

Trelat, E., & Zuazua, E. (2015). The turnpike property in finite-dimensional nonlinear optimal control. Journal of Differential Equations, 258(1), 81-114. https://dx.doi.org/10.1016/j.jde.2014.09.005

MLA:

Trelat, Emmanuel, and Enrique Zuazua. "The turnpike property in finite-dimensional nonlinear optimal control." Journal of Differential Equations 258.1 (2015): 81-114.

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