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An aspect of the turnpike property. Long time horizon behavior


The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval.

In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

2010 Mathematics Subject Classification:

49N10, 90C31, 46N10


turnpike property, linear-quadratic optimal control problem, linear systems, dynamic optimal control problem

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Author Details

Martin Gugat

Friedrich-Alexander-Universität Erlangen-Nürnberg
Department of Mathematics
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics
(Alexander von Humboldt-Professur)
Cauerstr. 11, 91058 Erlangen, Germany

Jan Sokolowski

Institut Élie Cartan de Lorraine, CNRS
UMR 7502, Université de Lorraine, B.P. 70239
54506 Vandoeuvre-lés-Nancy Cedex, France
Systems Research Institute of the Polish Academy of Sciences
ul. Newelska 6, 01-447 Warszawa, Poland
Department of Scientific Computing, Informatics Center
Federal University of Paraiba, Brazil


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