# American Institute of Mathematical Sciences

doi: 10.3934/naco.2020040

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

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

Received  May 2020 Revised  September 2020 Published  September 2020

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.

Citation: Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020040
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The graph of $x(t,\varepsilon)$ and its approximations
The graph of $u(t,\varepsilon)$ and its approximations
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
 $\varepsilon$ $\;\;J_{\varepsilon}(\overline{u}_{0})\;\;$ $\;\;J_{\varepsilon}(\widetilde{u}_{0})\;\;$ $\;\;J_{\varepsilon}(\widetilde{u}_{1})\;\;$ $\;\;J_{\varepsilon}(u)\;\;$ 0.1 0.11594 0.11240 0.11020 0.11013 0.2 0.16038 0.15884 0.15330 0.15266
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