Existence Results for Some Nonconvex Optimization Problems Governed by Nonlinear Processes
Optimal control problems with nonlinear equations usually do not possess optimal solutions. Nevertheless, if the cost functional is uniformly concave with respect to the state, the solution may exist. Using the Balder’s technique based on a Young-measure relaxation, Bauer’s extremal principle and investigation of extreme Young measures, the existence is demonstrated here for the case of nonlinear ordinary and partial differential equations.
KeywordsOptimal Control Problem Extreme Point Multivalued Mapping Young Measure Pontryagin Maximum Principle
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- Castaing, C.; Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580, Springer, Berlin, 1977.Google Scholar
- Cesari, L.:An existence theorem without convexity conditions. SIAM J. Control 12 (1974), 319–331.Google Scholar
- Gabasov, R.; Kirillova, F.: Qualitative Theory of Optimal Processes. Nauka, Moscow, 1971.Google Scholar
- Olech, C.: Integrals of set-valued functions and linear optimal control problems. Clloque sur la Théorie Math. du Contrôle Optimal, C.B.R.M., Vander Louvain, 1970, pp. 109–125.Google Scholar
- Roubíček, T.; Schmidt, W. H.: Existence of solutions to nonconvex optimal control problems governed by nonlinear Fredholm integral equations. (submitted)Google Scholar
- Schmidt, W.H.: Maximum principles for processes governed by integral equations in Banach spaces as sufficient optimality conditions. Beiträge zur Analysis 17 (1981), 85–93.Google Scholar