## Abstract and Applied Analysis

### Bounded Motions of the Dynamical Systems Described by Differential Inclusions

#### Abstract

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 617936, 9 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745566

Digital Object Identifier
doi:10.1155/2009/617936

Mathematical Reviews number (MathSciNet)
MR2516008

Zentralblatt MATH identifier
1175.37088

#### Citation

Ege, Nihal; Guseinov, Khalik G. Bounded Motions of the Dynamical Systems Described by Differential Inclusions. Abstr. Appl. Anal. 2009 (2009), Article ID 617936, 9 pages. doi:10.1155/2009/617936. https://projecteuclid.org/euclid.aaa/1268745566

#### References

• M. A. Aizerman and E. S. Pyatnitskii, Foundations of a theory of discontinuous systems. I,'' Automatic Remote Control, vol. 35, no. 7, pp. 1066--1079, 1974.
• A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988.
• N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer Series in Soviet Mathematics, Springer, New York, NY, USA, 1988.
• N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Nauka, Moscow, Russia, 1974.
• A. I. Subbotin, Minimax Inequalities and Hamilton-Jacobi Equations, Nauka, Moscow, Russia, 1991.
• J.-W. Chen, J.-F. Huang, and L. Y. Lo, Viable control for uncertain nonlinear dynamical systems described by differential inclusions,'' Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 41--53, 2006.
• J.-P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, vol. 264 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1984.
• F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998.
• K. Deimling, Multivalued Differential Equations, vol. 1 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 1992.
• Kh. G. Guseinov, A. I. Subbotin, and V. N. Ushakov, Derivatives for multivalued mappings with applications to game-theoretical problems of control,'' Problems of Control and Information Theory, vol. 14, no. 3, pp. 155--167, 1985.
• E. Roxin, Stability in general control systems,'' Journal of Differential Equations, vol. 1, no. 2, pp. 115--150, 1965.
• B. N. Pchenitchny, $\varepsilon$-strategies in differential games,'' in Topics in Differential Games, pp. 45--99, North-Holland, Amsterdam, The Netherlands, 1973.
• V. I. Blagodatskikh and A. F. Filippov, Differential inclusions and optimal control,'' Proceedings of the Steklov Institute of Mathematics, vol. 169, pp. 194--252, 1985.
• I. P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, Russia, 3rd edition, 1974.
• H. Brézis and F. E. Browder, A general principle on ordered sets in nonlinear functional analysis,'' Advances in Mathematics, vol. 21, no. 3, pp. 355--364, 1976.
• C. Ursescu, Evolutors and invariance,'' Set-Valued Analysis, vol. 11, no. 1, pp. 69--89, 2003.
• Kh. G. Guseinov and N. Ege, On the properties of positionally weakly invariant sets with respect to control systems described by differential inclusions,'' Differential Equations, vol. 43, no. 3, pp. 299--310, 2007.