Application of a nonlinear stabilizerfor localizing search of optimal trajectoriesin control problems with infinite horizon

Alexander Tarasyev; Anastasia Usova

doi:10.3934/naco.2013.3.389

Article Contents

2013, Volume 3, Issue 3: 389-406. Doi: 10.3934/naco.2013.3.389

This issue Previous Article Next Article MAPLE code of the cubic algorithm for multiobjective optimization with box constraints

Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon

Alexander Tarasyev^1, and
Anastasia Usova^1,

1.
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S.Kovalevskaya str., 16, Ekaterinburg, 620990

Received: November 2011

Revised: February 2013

Published: June 2013

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Abstract

The research is focused on an algorithm for constructing solutions of optimal control problems with infinite time horizon arising, for example, in economic growth models.
There are several significant difficulties which complicate solution of the problem, such as: (1) stiffness of Hamiltonian systems generated by the Pontryagin maximum principle; (2) non-stability of equilibrium points; (3) lack of initial conditions for adjoint variables.
The analysis of the Hamiltonian system implemented in this paper for optimal control problems with infinite horizon provides results effective for construction of optimal solutions, namely, (1) if a steady state exists and satisfies regularity conditions then there exists a nonlinear stabilizer for the Hamiltonian system; (2) a nonlinear stabilizer generates the system with excluding adjoint variables whose trajectories (according to qualitative analysis of the corresponding differential equations) approximate solutions of the original Hamiltonian system in a neighborhood of the steady state;(3) trajectories of the stabilized system serve as first approximations and localize search of optimal trajectories.
The results of numerical experiments are presented by modeling of an economic growth system with investment in capital and enhancement of the labor efficiency.

Keywords:

Optimal control theory,
Hamiltonian trajectories in optimal control,
modeling for control optimization,
nonlinear stabilization.

Mathematics Subject Classification: Primary: 93C10, 34H05, 93D15.

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