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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

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  • 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.


    \begin{equation} \\ \end{equation}
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  • 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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