Abstract
Optimal control problems with nonlinear equations usually do not possess optimal solutions, so that their natural (i.e., continuous) extension (relaxation) must be done. The relaxed problem may also serve to derive first-order necessary optimality condition in the form of the Pontryagin maximum principle. This is done here for nonlinear Fredholm integral equations and problems coercive in an L p-space of controls with p<+∞. Results about a continuous extension of the Uryson operator play a key role.
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References
Aizicovici, S., and Papageorgiou, N. S., Optimal Control and Relaxation of Systems Governed by Volterra Integral Equations, World Congress of Nonlinear Analysts, Edited by V. Lakshmikantham, Walter de Gruyter, Berlin, Germany, pp. 2617–2625, 1996.
Chryssoverghi, I., Contrôle Optimal Relaxé d'Equations Intégrales et aux Derivées Partielles, Doctoral Thesis, EPF, Lausanne, Switzerland, 1976.
Warga, J., Optimal Control of Differential and Functional Equations, Academic Press, New York, New York, 1972.
RoubiČek, T., Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, Berlin, Germany, 1997.
Angell, T. S., On the Optimal Control of Systems Governed by Nonlinear Volterra Equations, Journal of Optimization Theory and Applications, Vol. 19, pp. 29–45, 1976.
Bakke, V. L., A Maximum Principle for an Optimal Control Problem with Integral Constraints, Journal of Optimization Theory and Applications, Vol. 13, pp. 32–55, 1974.
Carlson, D. A., An Elementary Proof of the Maximum Principle of Optimal Control Problems Governed by a Volterra Integral Equation, Journal of Optimization Theory and Applications, Vol. 54, pp. 43–61, 1987.
Corduneanu, C., Integral Equations and Applications, Cambridge University Press, Cambridge, England, 1991.
Gabasov, R., and Kirillova, F. M., Principle of Maximum in the Theory of Optimal Control, Nauka i Technika, Minsk, Belarus, 1974 (in Russian).
Schmidt, W. H., Durch Integralgleichungen beschriebene optimale Prozesse mit Nebenbedingungen in Banachräumen—notwendige Optimalitätsbedingungen, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 62, pp. 65–75, 1982.
Schmidt, W. H., Volterra Integral Processes with State Constraints, System Analysis Modelling and Simulation, Vol. 9, pp. 213–224, 1992.
Vinokurov, V. R., Optimal Control of Processes Described by Integral Equations, Parts 1–3, SIAM Journal on Control, Vol. 7, pp. 324–355, 1969.
Schmidt, W. H., Notwendige Optimalitätsbedingungen für Prozesse mit Zeitvariablen Integralgleichungen in Banach-Räumen, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 60, pp. 595–608, 1980.
Schmidt, W. H., Maximum Principles for Processes Governed by Integral Equations in Banach Spaces as Sufficient Optimality Conditions, Beiträge zur Analysis, Vol. 17, pp. 85–93, 1981.
Von Wolfersdorf, L., Optimal Control of a Class of Processes Described by General Integral Equations of Hammerstein Type, Mathematische Nachrichten, Vol. 71, pp. 115–141, 1976.
Bittner, L., Optimal Control of Processes Governed by Abstract Functional, Integral, and Hyperbolic Differential Equations, Mathematische Operationsforschung und Statistik, Vol. 6, pp. 107–134, 1975.
KruŽÍk, M., and RoubÍČek, T., Explicit Characterization of L p -Young Measures, Journal on Mathematical Analysis and Applications, Vol. 198, pp. 830–843, 1996.
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Young, L. C., Generalized Curves and the Existence of an Attained Absolute Minimum in the Calculus of Variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, Classe 3, Vol. 30, pp. 212–234, 1937.
Casas, E., Boundary Control of Semilinear Elliptic Equations with Pointwise State Contraits, SIAM Journal on Control and Optimization, Vol. 31, pp. 993–1006, 1993.
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Roubíček, T. Optimal Control of Nonlinear Fredholm Integral Equations. Journal of Optimization Theory and Applications 97, 707–729 (1998). https://doi.org/10.1023/A:1022650427993
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DOI: https://doi.org/10.1023/A:1022650427993