Averaging and linear programming in some singularly perturbed problems of optimal control
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Date
2014-06-11
Authors
Gaitsgory, Vladimir
Rossomakhine, Sergey
Journal Title
Journal ISSN
Volume Title
Publisher
Springer Verlag
Rights
http://www.springer.com/gp/open-access/authors-rights/self-archiving-policy/2124
Rights Holder
© Springer International Publishing AG, Part of Springer Science+Business Media
Abstract
The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples.
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Article made available in accordance with publisher policy.
Keywords
Optimisation, Linear programming, Control systems
Citation
Gaistgory, V. and Rossomakhine, S., 2015. Averaging and linear programming in some singularly perturbed problems of optimal control. Applied Mathematics and Optimization, 71(2), 195-276.