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arXiv · 1503.05238

Limit value for optimal control with general means

Xiaoxi Li·Marc Quincampoix·Jérôme Renault
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Abstract

We consider optimal control problem with an integral cost which is a mean of a given function. As a particular case, the cost concerned is the Ces\`aro average. The limit of the value with Ces\`aro mean when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit when the averaging parameter converges, for values defined with means of general types. We consider a given function and a family of costs defined as the mean of the function with respect to a family of probability measures -- the evaluations -- on R_+. We provide conditions on the evaluations in order to obtain the uniform convergence of the associated value function (when the parameter of the family converges). Our main result gives a necessary and sufficient condition in term of the total variation of the family of probability measures on R_+. As a byproduct, we obtain the existence of a limit value (for general means) for control systems having a compact invariant set and satisfying suitable nonexpansive property.

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Xiaoxi Li, Marc Quincampoix, Jérôme Renault. 2015-03-17. Limit value for optimal control with general means. https://doi.org/10.3934/dcds.2016.36.2113

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