Viability of the Solution with a Perturbation and Feedback Control

Maxim V. Balashov

doi:10.68381/jca31018

Abstract

We consider a dynamic system with a perturbation in the right side. The perturbation is given by some geometrical restriction. We consider the question: find a feedback control that stabilizes a solution of the system required to be viable in a given compact convex subset? We construct minimal, in a certain sense, continuous feedback control which stabilizes the system. Some examples are considered.

Author information

Contact details are reproduced from the original publication and may be historical.

Maxim V. Balashov

V. A. Trapeznikov Institute of Control Sciences, Moscow, Russia

balashov73@mail.ru

Keywords

Viability theory
Hausdorff distance
supporting function
dynamic system
geometric difference

Mathematics Subject Classification

49J52, 49J53, 52A20; 34D20

References

J.-P. Aubin: Viability Theory, Birkhäuser, Boston (1991).Google Scholar
J.-P. Aubin, A. Cellina: Differential Inclusions, Springer, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4 Google Scholar
J.-P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley, New York (1984).Google Scholar
J.-P. Aubin, G. D. Prato: The viability theorem for stochastic differential inclusions, Stoch. Analysis Appl. 16/1 (1998) 1–15.https://doi.org/10.1080/07362999808809512 Google Scholar
O. Carja, T. Donchev, M. Rafaqat, R. Ahmed: Viability of fractional differential inclusions, Appl. Math. Letters 38 (2014) 48–51.https://doi.org/10.1016/j.aml.2014.06.012 Google Scholar
J. B. Krawczyk, A. Pharo: Viability theory: an applied mathematics tool for achieving dynamic systems’ sustainability, Math. Appl. 41/1 (2013) 97–126.https://doi.org/10.14708/ma.v41i1.409 Google Scholar
V. M. Veliov: Sufficient conditions for viability under imperfect measurement, Set-Valued Analysis 1 (1993) 305–317.https://doi.org/10.1007/BF01027640 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.

[1] J.-P. Aubin: Viability Theory, Birkhäuser, Boston (1991).Google Scholar

[2] J.-P. Aubin, A. Cellina: Differential Inclusions, Springer, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4 Google Scholar

[3] J.-P. Aubin, I. Ekeland: Applied Nonlinear Analysis, Wiley, New York (1984).Google Scholar

[4] J.-P. Aubin, G. D. Prato: The viability theorem for stochastic differential inclusions, Stoch. Analysis Appl. 16/1 (1998) 1–15.https://doi.org/10.1080/07362999808809512 Google Scholar

[5] O. Carja, T. Donchev, M. Rafaqat, R. Ahmed: Viability of fractional differential inclusions, Appl. Math. Letters 38 (2014) 48–51.https://doi.org/10.1016/j.aml.2014.06.012 Google Scholar

[6] J. B. Krawczyk, A. Pharo: Viability theory: an applied mathematics tool for achieving dynamic systems’ sustainability, Math. Appl. 41/1 (2013) 97–126.https://doi.org/10.14708/ma.v41i1.409 Google Scholar

[7] V. M. Veliov: Sufficient conditions for viability under imperfect measurement, Set-Valued Analysis 1 (1993) 305–317.https://doi.org/10.1007/BF01027640 Google Scholar