Abstract
Denoting byS k (ξ k ) the set of solutions of the Cauchy problem\(\dot x \in F_k (t,x),x(0) = \xi _k \), fork∈N∪{∞}, we prove that, under appropriate assumptions, the sequence {S k (ξ k )} k ∈ N converges toS ∞(∈∞) in the Kuratowski sense as well as in the Mosco sense. This result together with some facts from Γ-convergence theory are used to prove a result concerning the existence and the asymptotic behavior of the minima to the optimization problem
withK a compact subset ofR n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buttazzo, G., andDal Maso, G.,Γ-Convergence and Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 38, pp. 385–407, 1982.
De Giorgi, E., andFranzoni, T.,Su un Tipo di Convergenza Variazionale, Atti dell'Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali, Vol. 58, pp. 842–850, 1975.
Piccinini, L. C.,G-Convergence for Ordinary Differential Equations with Peano Phenomenon, Rendiconti del Seminario Matematico dell'Università di Padova, Vol. 58, pp. 65–86, 1977.
Piccinini, L. C.,Linearity and Nonlinearity in the theory of G-Convergence, Recent Advances in Differential Equations, Edited by R. Conti, Academic Press Inc., New York, New York, pp. 337–372, 1981.
Colombo, R. M., Fryszkowski, A., Rzeżuchowski, T., andStaicu, V.,Continuous Selection of Solution Sets of Lipschitzian Differential Inclusions, Funkcialaj Ekvacioj, Vol. 34, pp. 321–331, 1991.
Naselli-Ricceri, O., andRicceri, B.,Differential Inclusions Depending on a Parameter, Bulletin of the Polish Academy of Science, Vol. 37, pp. 665–671, 1989.
Stassinopoulos, G. I., andVinter, R. B.,Continuous Dependence of Solutions of a Differential Inclusion on the Right-Hand Side with Applications to Stability of Optimal Control Problems, SIAM Journal on Control and Optimization, Vol. 17, pp. 432–449, 1979.
Brezis, H.,Analyse Fonctionelle, Théorie et Applications, Masson, Paris, France, 1987.
Dal Maso, G.,An Introduction to Γ-Convergence, SISSA, Trieste, Italy, 1992.
Kuratowski, K.,Topologie, Polish Scientific Publishers, Warszawa, Poland, 1958.
Denkowski, Z.,The Convergence of Generalized Sequences of Sets and Functions in Locally Convex Spaces, Parts 1 and 2, Universitatis Iagellonicae Acta Mathematica, Vol. 22, pp. 59–72, 1981 and Vol. 22, pp., 73–83, 1981.
Denkowski, Z.,On Mosco-Type Characterization of K-Convergence and Lower Semicontinuity of Integral Functionals, Universitatis Jagellonicae Acta Mathematica, Vol. 24, pp. 189–201, 1984.
Himmelberg, C. J., andVan Vleck, F. S.,Lipschitzian Generalized Differential Relations, Rendinconti del Seminario Matematico dell'Università di Padova, Vol. 42, pp. 159–169, 1972.
Filippov, A. F.,Classical Solutions of Differential Equations with Multivalued Right-Hand Side, SIAM Journal on Control and Optimization, Vol. 5, pp. 609–621, 1967.
Cellina, A., andOrnelas, A.,Convexity and Closure of the Solution Set to Differential Inclusions, Bollettino della Unione Matematica Italiana, Vol. 4B, pp. 255–263, 1990.
Aubin, J. P., andCellina, A.,Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.
Author information
Authors and Affiliations
Additional information
Communicated by R. Conti
This work was done in part while the authors were visiting the International School for Advanced Studies (SISSA) in Trieste and was completed while the second author was visiting the Institute for Information Sciences of the Jagellonian University in Krakow. The authors wish to thank Prof. Arrigo Cellina and Prof. Gianni Dal Maso for stimulating discussions.
Rights and permissions
About this article
Cite this article
Denkowski, Z., Staicu, V. Asymptotic behaviour of the minima to a class of optimization problems for differential inclusions. J Optim Theory Appl 81, 21–34 (1994). https://doi.org/10.1007/BF02190311
Issue Date:
DOI: https://doi.org/10.1007/BF02190311