Article Contents
Article Contents

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

• 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.

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

 Citation:

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

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

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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