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An Introduction to Singular Perturbations in Nonlinear Optimal Control

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Singular Perturbations in Systems and Control

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 280))

Abstract

Before turning to the general n-dimensional nonlinear optimal control problem, which is of primary interest in this paper, we will first consider a much simpler problem, namely the singularly perturbed, uncontrolled, autonomous, initial-value problem

EquationSource<math display=&#x2019;block&#x2019;> <mrow> <mrow><mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi>d</mi><mi>x</mi></mrow> <mrow> <mi>d</mi><mi>t</mi></mrow> </mfrac> <mo>=</mo><mi>f</mi><mo stretchy=&#x2019;false&#x2019;>(</mo><mi>x</mi><mo></mo><mi>y</mi><mo stretchy=&#x2019;false&#x2019;>)</mo><mo>;</mo><mi>x</mi><mo stretchy=&#x2019;false&#x2019;>(</mo><mi>&#x03B5;</mi><mo></mo><mn>0</mn><mo stretchy=&#x2019;false&#x2019;>)</mo><mo>=</mo><msub> <mi>x</mi> <mi>o</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&#x03B5;</mi><mfrac> <mrow> <mi>d</mi><mi>y</mi></mrow> <mrow> <mi>d</mi><mi>t</mi></mrow> </mfrac> <mo>=</mo><mi>g</mi><mo stretchy=&#x2019;false&#x2019;>(</mo><mi>x</mi><mo></mo><mi>y</mi><mo stretchy=&#x2019;false&#x2019;>)</mo><mo>;</mo><mi>y</mi><mo stretchy=&#x2019;false&#x2019;>(</mo><mi>&#x03B5;</mi><mo></mo><mn>0</mn><mo stretchy=&#x2019;false&#x2019;>)</mo><mo>=</mo><msub> <mi>y</mi> <mi>o</mi> </msub> </mrow> </mtd> </mtr> </mtable></mrow> <mo>}</mo></mrow></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left. {\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = f(x,y);x(\varepsilon ,0) = {x_o}} \\ {\varepsilon \frac{{dy}}{{dt}} = g(x,y);y(\varepsilon ,0) = {y_o}} \end{array}} \right\}$$
(1.1)

where x(ε,t) and y(ε,t) are scalars, ε < 0, and xo and yo are constants.

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Authors and Affiliations

  1. NASA Ames Research Center, M.S. 210-9, Moffett Field, Calif., 94035, USA

    M. D. Ardema

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  1. M.S. 210-9, NASA-Ames Research Center, 94035, Moffett Field, CA, USA

    M. D. Ardema

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© 1983 Springer-Verlag Wien

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Ardema, M.D. (1983). An Introduction to Singular Perturbations in Nonlinear Optimal Control. In: Ardema, M.D. (eds) Singular Perturbations in Systems and Control. International Centre for Mechanical Sciences, vol 280. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2638-7_1

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  • DOI: https://doi.org/10.1007/978-3-7091-2638-7_1

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81751-3

  • Online ISBN: 978-3-7091-2638-7

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