Summary
In this paper we establish the existence of extremal solutions for a class of nonlinear evolution inclusions defined on an evolution triple of Hilbert spaces. Then we show that these extremal solutions are in fact dense in the solutions of the original system. Subsequently we use this density result to derive nonlinear and infinite dimensional versions of the “bang-bang” principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail.
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References
Benamara, M., Points extremaux, multi-applications et fonctionelles integrales. Thèse de 3ème cycle. Université de Grenoble, Grenoble, 1976.
Bressan, A. andColombo, G.,Extensions and selections of maps with decomposable values. Studia Math90 (1988), 69–86.
DeBlasi, F. andPianigiani, G.,Nonconvex valued differential inclusions in Banach spaces. J. Math. Anal. Appl.157 (1991), 469–494.
Fryszkowski, A.,Continuous selections for a class of nonconvex multivalued maps. Studia Math.76 (1983), 163–174.
Giles, R.,Convex analysis with application to differentiation of convex functions. Pitman, Boston, 1982.
Holmes, R.,Geometric functional analysis and its applications. Springer, New York, 1975.
Klein, E. andThompson, A.,Theory of correspondences. Wiley, New York, 1984.
Nagy, E.,A theorem on compact embedding for functions with values in an infinite dimensional Hilbert space. Ann. Univ. Sci. Budapest. Sect. Math.23 (1980), 243–245.
Papageorgiou, N. S.,On the theory of Banach space valued multifunctions 1. Integration and conditional expectation. J. Multivariate Anal.17 (1985), 185–206.
Papageorgiou, N. S.,On measurable multifunctions with applications to random multivalued equations. Math. Japonica32 (1987), 437–464.
Papageorgiou, N. S.,Continuous dependence results for a class of evolution inclusions. Proc. Edinburgh Math. Soc. (2)35 (1992), 139–158.
Tolstonogov, A.,Extreme continuous selectors of multivalued maps and the bang-bang principle for evolution inclusions. Soviet Math. Dokl.317 (1991), 1–8.
Tolstonogov, A.,Extreme continuous selectors of multivalued maps and their applications. SISSA, Preprint 72M. Trieste, Italy, 1991.
Wagner, D.,Survey of measurable selection theorems. SIAM J. Control Optim.15 (1977), 859–903.
Zeidler, E.,Nonlinear functional analysis, III. Springer, Berlin, 1990.