Singular perturbations for systems of differential inclusions

Marc Quincampoix

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 341-348
  • ISSN: 0137-6934

Abstract

top
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.

How to cite

top

Quincampoix, Marc. "Singular perturbations for systems of differential inclusions." Banach Center Publications 32.1 (1995): 341-348. <http://eudml.org/doc/262848>.

@article{Quincampoix1995,
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.},
author = {Quincampoix, Marc},
journal = {Banach Center Publications},
keywords = {system of two differential inclusions; singular perturbation; convergence; contingent derivative; perturbed inclusion},
language = {eng},
number = {1},
pages = {341-348},
title = {Singular perturbations for systems of differential inclusions},
url = {http://eudml.org/doc/262848},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Quincampoix, Marc
TI - Singular perturbations for systems of differential inclusions
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 341
EP - 348
AB - 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.
LA - eng
KW - system of two differential inclusions; singular perturbation; convergence; contingent derivative; perturbed inclusion
UR - http://eudml.org/doc/262848
ER -

References

top
  1. [1] J.-P. Aubin, Viability Theory, Birkhäuser. Boston, Basel, 1992. 
  2. [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1991. 
  3. [3] P. Binding, Singularly perturbed optimal control problems.I. Convergence, SIAM J. Control Optim. 14 (1976), 591-612. Zbl0329.49004
  4. [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. [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. Zbl0576.93031
  6. [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. [7] N. Dunford and J. Schwartz, Linear Operators, Part I, Wiley, New York. 
  8. [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. [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. [10] H. Frankowska, S. Plaskacz and T. Rzeżuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation, J. Differential Equations 116 (1995), 265-305. Zbl0836.34016
  11. [11] P. V. Kokotović, Applications of singular perturbation techniques to control problems, SIAM Rev. 26 (1984), 501-550. Zbl0548.93001
  12. [12] R. O'Malley, Introduction to Singular Perturbation, Academic Press, 1974. 
  13. [13] A. N. Tikhonov, A. B. Vassilieva and A. G. Sveshnikov, Differential Equations, Springer, 1985. 
  14. [14] H. Tuan, Asymptotical solution of differential systems with multivalued right-hand side, Ph.D. Thesis, University of Odessa, 1990, in Russian. 
  15. [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. Zbl0769.93055
  16. [16] M. Quincampoix, Singular perturbations for control systems and for differential inclusions, in: Cahiers Mathématiques de la Décision, Université Paris-Dauphine, 1994. 
  17. [17] V. M. Veliov, Differential inclusions with stable subinclusions, Nonlinear Anal. 23 (1994), 1027-1038. Zbl0816.34011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.