Download PDF - Singular perturbations for systems of differential inclusionsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Singular perturbations for systems of differential inclusions

Authors 1

Affiliations

  1. Département de Mathématiques, Université François Rabelais, Parc de Grandmont, F-37200 Tours, France

Abstract

We study a system of two differential inclusions such that there is a singular perturbation in the second one. We state new convergence results of solutions under assumptions concerning contingent derivative of the perturbed inclusion. These results state that there exists at least one family of solutions which converges to some solution of the reduced system. We extend this result to perturbed systems with state constraints.

Bibliography

  1. J.-P. Aubin, Viability Theory, Birkhäuser. Boston, Basel, 1992.
  2. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1991.
  3. P. Binding, Singularly perturbed optimal control problems.I. Convergence, SIAM J. Control Optim. 14 (1976), 591-612.
  4. A. L. Dontchev and I. I. Slavov, Singular perturbation in a class of nonlinear differential inclusions, Proceedings IFIP Conference, Leipzig, 1989, Lecture Notes in Inform. Sci. 143, Springer, Berlin, 1990, 273-280.
  5. A. L. Dontchev and V. M. Veliov, Singular perturbations in linear control systems with weakly coupled stable and unstable fast subsystems, J. Math. Anal. Appl. 110 (1985), 1-130.
  6. A. L. Dontchev and V. M. Veliov, Continuity of a family of trajectories of linear control systems with respect to singular perturbations, Soviet Math. Dokl. 35 (1987), 283-286.
  7. N. Dunford and J. Schwartz, Linear Operators, Part I, Wiley, New York.
  8. A. F. Filippov, On some problems of optimal control theory, Vestnik Moskov. Univ. Mat. 1958 (2), 25-32 (in Russian); English transl.: SIAM J. Control 1 (1962), 76-84.
  9. T. F. Filippova and A. B. Kurzhanskiĭ, Methods of singular perturbations for differential inclusions, Dokl. Akad. Nauk SSSR 321 (1991), 454-460 (in Russian).
  10. H. Frankowska, S. Plaskacz and T. Rzeżuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation, J. Differential Equations 116 (1995), 265-305.
  11. P. V. Kokotović, Applications of singular perturbation techniques to control problems, SIAM Rev. 26 (1984), 501-550.
  12. R. O'Malley, Introduction to Singular Perturbation, Academic Press, 1974.
  13. A. N. Tikhonov, A. B. Vassilieva and A. G. Sveshnikov, Differential Equations, Springer, 1985.
  14. H. Tuan, Asymptotical solution of differential systems with multivalued right-hand side, Ph.D. Thesis, University of Odessa, 1990, in Russian.
  15. M. Quincampoix, Contribution à l'étude des perturbations singulières pour les systèmes contrôlés et les inclusions différentielles, C. R. Acad. Sci. Paris Sér. I 316 (1993), 133-138.
  16. M. Quincampoix, Singular perturbations for control systems and for differential inclusions, in: Cahiers Mathématiques de la Décision, Université Paris-Dauphine, 1994.
  17. V. M. Veliov, Differential inclusions with stable subinclusions, Nonlinear Anal. 23 (1994), 1027-1038.
Banach Center Publications
1995/32/1
Export to PBN, Opens in new tab
Pages:
341-348
Main language of publication
English
Published
1995
Exact and natural sciences