Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable

Nguyen Thi Hoai

doi:10.3934/naco.2020040

Article Contents

2021, Volume 11, Issue 4: 495-512. Doi: 10.3934/naco.2020040

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$ P_{\ast }\left( \kappa \right) $

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Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable

Nguyen Thi Hoai

Department of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Ha Noi, Vietnam

Received: April 30, 2020

Revised: August 31, 2020

Early access: September 2020

Published: December 2021

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Abstract

The direct scheme method is applied to construct an asymptotic approximation of any order to a solution of a singularly perturbed optimal problem with scalar state, controlled via a second-order linear ODE and two fixed end points. The error estimates for state and control variables and for the functional are obtained. An illustrative example is given.

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Mathematics Subject Classification: Primary: 34E10, 34D15; Secondary: 49K15, 41A60.

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Figure 1. The graph of $ x(t,\varepsilon) $ and its approximations

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Figure 2. The graph of $ u(t,\varepsilon) $ and its approximations

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Table 1. Table of values of performance index

$ \varepsilon $	$ \;\;J_{\varepsilon}(\overline{u}_{0})\;\; $	$ \;\;J_{\varepsilon}(\widetilde{u}_{0})\;\; $	$ \;\;J_{\varepsilon}(\widetilde{u}_{1})\;\; $	$ \;\;J_{\varepsilon}(u)\;\; $
0.1	0.11594	0.11240	0.11020	0.11013
0.2	0.16038	0.15884	0.15330	0.15266

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