Abstract
From an Ekeland-type variational principle for vector optimization problems we derive an e-minimum principle in the sense of Pontrjagin for suboptimal controls using classical results for differential equations.
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References
Aubin, J. P.; Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York, 1984.
Benker, H.; Kossert, S.: Remarks on quadratic optimal control problems in Hilbert spaces. Zeitschr. Analysis und ihre Anwendungen 1(3), 1982, 13–21.
Breckner, W. W.: Derived sets for weak multiobjective optimization problems with state and control variables (to appear), 1996.
Dentscheva, D.; Helbig, S.: On variational principles in vector optimization, Lecture on the 4th Workshop Multicriteria Decision, Holzhau, Germany, 1994.
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 1974, 324–353.
Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Mathem. Society 1(3), 1979, 443–474.
Gerth (Tammer), C.; Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67(2), 1990, 297–320.
Gorochowik, B. B.: About multiobjective optimization problems (in Russian). AN Soviet Union (6), 1972.
Gorochowik, B. B.; Kirillowa, F. M.: About scalarization of vector optimization problems (in Russian). Docl. AN Soviet Union (19), No. 7, 1975, 588–591.
Gorochowik, B. B.: Conditions for weakly efficient elements in multiobjective optimization (in Russian). Minsk, IN-T Matem. AN Soviet Union, Preprint, 1976.
Henkel, E.-C.; Tammer, C.: ε-variational inequalities in partially ordered spaces. Optimization 36, 1996, 105–118.
Isac, G.: A variant of Ekeland’s principle for Pareto ε-efficiency. In prep, 1994.
Khanh, P. Q.: On Caristi-Kirk’s theorem and Ekeland’s variational principle for Pareto extrema. Inst. of Mathem., Polish Acad. Sciences, Preprint 357, 1986.
Klötzler, R.: Multiobjective dynamic programming. Math. Operationsforschung, Statist., Ser. Opt. 9(3), 1978, 423–426.
Klötzler, R.: On a general conception of duality in optimal control. Lecture Notes in Math. 703, 1979, 183–196.
Loridan, P.: ε-solutions in vector minimization problems. Journ. Optimization Theory Applications 43(2), 1984, 265–276.
Nemeth, A. B.: A Nonconvex Vector Minimization Proble. Nonlin. Anal. Theory, Methods and Applications 10(7), 1986, 669–678.
Pallu de 1a Barriere: Cours d’automatique theorique. Dunod, 1965.
Salukvadze, M. E.: Vector optimization problems in control theory. Tiblisi, 1975.
Tammer, C.: A generalization of Ekeland’s variational principle. Optimization 25, 1992, 129–141.
Tammer, C.: Existence results and necessary conditions for ε-efficient elements. In: Brosowski, Ester, Helbig, Nehse (Eds): Multicriteria Decision. Lang Verlag Frankfurt Main Bern, 1993, 97–110.
Tammer, C.: Necessary conditions for approximately efficient solutions of vector approximation. Proc. 2nd Conf.Approximation and Optimization, Havana, 1993.
Tammer, C.: A variational principle and applications for vectorial control approximation problems. To appear: Math. Journ. Univ. Bacau (Romania), 1994.
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Tammer, C. (1998). Multiobjective Optimal Control Problems. In: Schmidt, W.H., Heier, K., Bittner, L., Bulirsch, R. (eds) Variational Calculus, Optimal Control and Applications. International Series of Numerical Mathematics, vol 124. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8802-8_11
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DOI: https://doi.org/10.1007/978-3-0348-8802-8_11
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