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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 1, Pages 77–95 (Mi vuu312)

MATHEMATICS

Asymptotic properties of optimal solutions and value functions in optimal control problems with infinite time horizon

A. A. Usova

Department of Dynamical Systems, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia

Abstract: The research is devoted to the investigation of the behavior of optimal solutions and value functions in optimal control problems on infinite horizon, which arise in the economic growth models when an elasticity parameter of the Cobb–Douglas production function grows up to its limit value which is equal to unity. The solution is constructed within the framework of the Pontryagin maximum principle for problems on infinite time horizon. In the limit case the problem becomes linear and has a constant optimal control depending on model parameters only. Qualitative analysis of Hamiltonian systems outlines significant changes in solution behavior, namely, the absence of steady states in the limit case. Nevertheless, both the Hamiltonian function and the maximized Hamiltonian function save their properties of smoothness with respect to all variables, and strict concavity with respect to phase variables. Value functions are constructed for both linear and nonlinear optimal control problems. Numerical experiments are implemented for illustrating results of the sensitivity analysis.

Keywords: optimal control, Hamiltonian systems, value function, Pontryagin maximum principle.

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UDC: 517.977
MSC: 34H05, 49L20

Citation: A. A. Usova, “Asymptotic properties of optimal solutions and value functions in optimal control problems with infinite time horizon”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 1, 77–95

Citation in format AMSBIB
\Bibitem{Uso12} \by A.~A.~Usova \paper Asymptotic properties of optimal solutions and value functions in optimal control problems with infinite time horizon \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 1 \pages 77--95 \mathnet{http://mi.mathnet.ru/vuu312}