Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 258, Issue 1, Pages 81-114Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2014.09.005
Keywords
Optimal control; Turnpike; Pontryagin maximum principle; Riccati equation; Direct methods; Shooting method
Categories
Funding
- EOARD-AFOSR [FA9550-14-1-0214]
- Paris City Hall Research in Paris Program
- Centre International de Mathematiques et Informatique (CIMI) of Toulouse
- European Research Council Executive Agency [NUMERIWAVES/FP7-246775, PI2010-04]
- BERC program of the Basque Government
- MINECO [MTM2011-29306-C02-00, SEV-2013-0323]
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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. (C) 2014 Elsevier Inc. All rights reserved.
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