Abstract
Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.
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Pontryagin, L. S., Boltyanskij, V. G., Gamkrelidze, R. V., and Mishchenko, E. F., Mathematische Theorie Optimaler Prozesse, Oldenbourg, München, Germany, 1964 (in German).
Hestenes, M. R., Calculus of Variations and Optimal Control Theory, John Wiley and Sons, New York, NY, 1966.
Jacobson, D. H., Lele, M. M., and Speyer, J. L., New Necessary Conditions of Optimality for Constrained Problems with State-Variable Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 35, pp. 255–284, 1971.
Girsanov, I. V., Lectures on Mathematical Theory of Extremum Problems, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 67, 1972.
Knobloch, H. W., Das Pontryaginsche Maximumprinzip für Probleme mit Zustandsbeschränkungen i und ii, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 55, pp. 545–556 and 621–634, 1975.
Maurer, H., On Optimal Control Problems with Boundary State Variables and Control Appearing Linearly, SIAM Journal on Control and Optimization, Vol. 15, pp. 345–362, 1977.
Maurer, H., On the Minimum Principle for Optimal Control Problems with State Constraints, Schriftenreihe des Rechenzentrums der Universität Münster, Münster, Germany, Vol. 41, 1979.
Ioffe, A. D., and Tihomirov, V. M., Theory of Extremal Problems, Studies in Mathematics and Its Applications, North-Holland Publishing Company, Amsterdam, Holland, Vol. 6, 1979.
Kreindler, E., Additional Necessary Conditions for Optimal Control with State-Variable Inequality Constraints, Journal of Optimization Theory and Applications, Vol. 38, pp. 241–250, 1982.
Neustadt, L. W., Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, New Jersey, 1976.
Zeidan, V., The Riccati Equation for Optimal Control Problems with Mixed State-Control Constraints: Necessity and Sufficiency, SIAM Journal on Control and Optimization, Vol. 32, pp. 1297–1321, 1994.
Hartl, R. F., Sethi, S. P., and Vickson, G., A Survey of the Maximum Principles for Optimal Control Problems with State Constraints, SIAM Review, Vol. 37, pp. 181–218, 1995.
Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Hemisphere Publishing Corporation, Washington, DC, 1975.
Maurer, H., First and Second-Order Sufficient Optimality Conditions in Mathematical Programming and Optimal Control, Mathematical Programming Study, Vol. 14, pp. 163–177, 1981.
Malanowski, K., Sufficient Optimality Conditions for Optimal Control Subject to State Constraints, SIAM Journal on Control and Optimization, Vol. 35, pp. 205–227, 1997.
Maurer, H., and Pickenhain, S., Second-Order Sufficient Conditions for Control Problems with Mixed Control-State Constraints, Journal of Optimization Theory and Applications, Vol. 86, pp. 649–667, 1995.
Malanowski, K., Maurer, H., and Pickenhain, S., Second-Order Sufficient Conditions for State-Constrained Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 123, pp. 595–617, 2004.
De Pinho, M., and Vinter, R. B., Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations, Journal of Mathematical Analysis and Applications, Vol. 212, pp. 493–516, 1997.
Devdariani, E. N., and Ledyaev, Y. S., Maximum Principle for Implicit Control Systems, Applied Mathematics and Optimization, Vol. 40, pp. 79–103, 1999.
Gear, C. W., Leimkuhler, B., and Gupta, G. K., Automatic Integration of Euler-Lagrange Equations with Constraints, Journal of Computational and Applied Mathematics, Vols. 12/13, pp. 77–90, 1985.
Gerdts, M., Direct Shooting Method for the Numerical Solution of Higher Index DAE Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 117, pp. 267–294, 2003.
Gerdts, M., Optimal Control and Real-Time Optimization of Mechanical Multibody Systems, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 83, pp. 705–719, 2003.
Kirsch, A., Warth, W., and Werner, J., Notwendige Optimalitätsbedingungen und ihre Anwendung, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 152, 1978.
Gerdts, M., Representation of Lagrange Multipliers for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two, Journal of Optimization Theory and Applications, Vol. 130, 2005.
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Communicated by H. J. Pesch
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Gerdts, M. Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two. J Optim Theory Appl 130, 443–462 (2006). https://doi.org/10.1007/s10957-006-9121-9
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DOI: https://doi.org/10.1007/s10957-006-9121-9