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A solution method for regular optimal control problems with state constraints

Journal of Optimization Theory and Applications volume 62, pages 489–513 (1989)Cite this article

Abstract

A method of region analysis is developed for solving a class of optimal control problems with one state and one control variable, including state and control constraints. The performance index is strictly convex with respect to the control variable, while this variable appears only linearly in the state equation. The convexity or linearity assumption of the performance index or the state equation with respect to the state variable is not required.

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References

  1. Anderson, B. D. O., andMoore, J. B.,Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

    Google Scholar 

  2. Bell, D. J., andJacobson, D. H.,Singular Optimal Control Problems, Academic Press, London, England, 1975.

    Google Scholar 

  3. McDanell, J. P., andPowers, W. F.,Necessary Conditions for Joining Optimal Singular and Nonsingular Subars, SIAM Journal on Control, Vol. 9, pp. 161–173, 1971.

    Google Scholar 

  4. Ioffe, A. D., andTichomirov, V. M.,Theorie der Extremalaufgaben, Translated from Russian by B. Luderer, VEB Deutscher Verlag der Wissenschaften, Berlin, GDR, 1979.

    Google Scholar 

  5. Phu, H. X.,Lineare Steuerungsprobleme mit Engen Zustandsbereichen, Optimization, Vol. 16, pp. 273–284, 1985.

    Google Scholar 

  6. Phu, H. X.,Some Necessary Conditions for Optimality for a Class of Optimal Control Problems Which Are Linear in the Control Variable, Systems and Control Letters, Vol. 8, pp. 261–271, 1987.

    Google Scholar 

  7. Phu, H. X.,A Method for Solving a Class of Optimal Control Problems Which Are Linear in the Control Variable, Systems and Control Letters, Vol. 8, pp. 273–280, 1987.

    Google Scholar 

  8. Phu, H. X.,On the Optimal Control of a Hydroelectric Power Plant, Systems and Control Letters, Vol. 8, pp. 281–288, 1987.

    Google Scholar 

  9. Phu, H. X.,Solution of Some High-Dimensional Linear Optimal Control Problems by the Method of Region Analysis, International Journal of Control, Vol. 47, pp. 493–518, 1988.

    Google Scholar 

  10. Phu, H. X.,On a Linear Optimal Control Problem of a System with Circuit-Free Graph Structure, International Journal of Control (to appear).

  11. Phu, H. X., Lösung einer Eindimensionalen Regulären Aufgabe der Optimalen Steuerung mit Engen Zustandsbereichen Anhand der Methode der Bereichsanalyse, Optimization, Vol. 16, pp. 431–438, 1985.

    Google Scholar 

  12. Phu, H. X., Einige Notwendige Optimalitätsbedingungen für Einfache Reguläre Aufgaben der Optimalen Steurung, Zeitschrift für Analysis und Ihre Anwendungen, Vol. 5, pp. 465–475, 1986.

    Google Scholar 

  13. Phu, H. X., Reguläre Aufgaben der Optimalen Steurung mit Linearen Zustandsrestriktionen, Zeitschrift für Analysis und Ihre Anwendungen, Vol. 7, pp. 431–440, 1988.

    Google Scholar 

  14. Phu, H. X., Zur Stetigkeit der Lösung der Adjungierten Gleichung bei Aufgaben der Optimalen Steuerung mit Zustandsbeschränkungen, Zeitschrift für Analysis und Ihre Anwendungen, Vol. 3, pp. 527–539, 1984.

    Google Scholar 

  15. Kneschke, A.,Differentialgleichung und Randwertprobleme, Vol. 3, B. G. Teubner Verlagsgesellschaft, Leipzig, GDR, 1962.

    Google Scholar 

  16. Feichtinger, G., andHartl, R. F.,Optimale Kontrolle Ökonomischer Prozesse, Walter de Gruyter, Berlin, West Germany, 1986.

    Google Scholar 

  17. Phu, H. X.,Investigation of Some Inventory Problems with Linear Replenishment Cost by the Method of Region Analysis, Optimal Control Theory and Economic Analysis 3, Edited by G. Feichtinger, North-Holland, Amsterdam, Holland, pp. 195–221, 1988.

    Google Scholar 

  18. Pastor, K.,A Simple Macroeconomic Model with Ecological Technical Progress, Ekonomicko-Matematický Obzor, Vol. 22, pp. 320–326, 1986.

    Google Scholar 

  19. Phu, H. X., Zur Lösung einer Regulären Aufgabenklasse der Optimalen Steuerung im Großen mittels Orientierungskurven, Optimization, Vol. 18, pp. 65–81, 1987.

    Google Scholar 

  20. Phu, H. X., Zur Lösung eines Zermeloschen Navigationsproblems, Optimization, Vol. 18, pp. 225–236, 1987.

    Google Scholar 

  21. Phu, H. X., Ein Konstruktives Lösungsverfahren für das Problem des Inpolygons Kleinsten Umfangs von J. Steiner, Optimization, Vol. 18, pp. 349–359, 1987.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Nghia Do, Hanoi, Vietnam

    H. X. Phu (Staff Member)

Authors
  1. H. X. Phu
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Additional information

Communicated by R. Bulirsch

The author would like to express his sincere gratitude to Prof. R. Klötzler, Prof. E. Zeidler, Prof. H. Schumann, Prof. J. Focke, and other colleagues of the Department of Mathematics, Karl Marx University, Leipzig, GDR, for their support during his stay in Leipzig.

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Phu, H.X. A solution method for regular optimal control problems with state constraints. J Optim Theory Appl 62, 489–513 (1989). https://doi.org/10.1007/BF00939818

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  • DOI: https://doi.org/10.1007/BF00939818

Key Words

  • Optimal control
  • regular problems
  • state constraints
  • method of region analysis