On periodic oscillations for a class of feedback control systems in Hilbert spaces

Nguyen Van Loi

Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 33 (2013), p. 205-219 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper, by using the topological degree theory for multivalued maps and the method of guiding functions in Hilbert spaces we deal with the existence of periodic oscillations for a class of feedback control systems in Hilbert spaces.

Détail

Publié le : 2013-01-01

Zbl 1300.34148

EUDML-ID : urn:eudml:doc:270450

Bibliographie

[000] [1] R. Bader and W. Kryszewski, Fixed-point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR's, Set-Valued Anal. 2 (3) (1994), 459-480. doi: 10.1007/BF01026835 | Zbl 0846.55001

[001] [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. | Zbl 0328.47035

[002] [3] Yu.G. Borisovich, B.D. Gel'man, A.D. Myshkis and V.V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential inclusions, Second edition, Librokom, Moscow, 2011 (in Russian).

[003] [4] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Topological methods in the theory of fixed points of multivalued mappings, (Russian) Uspekhi Mat. Nauk 35 (1980), 59-126. English translation: Russian Math. Surveys 35 (1980), 65-143. doi: 10.1070/RM1980v035n01ABEH001548

[004] [5] K. Borsuk, Theory of Retracts. Monografie Matematyczne, 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. | Zbl 0153.52905

[005] [6] I. Ekeland and R. Temam, Convex Analysis and Variation Problems, North Holland, Amsterdam, 1979. | Zbl 0546.90057

[006] [7] R.E. Gaines and J.L. Mawhin, Coincidence degree and nonlinear differential equations, Lecture Notes in Mathematics, no. 568, Springer-Verlag, Berlin-New York, 1977. | Zbl 0339.47031

[007] [8] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, 2nd edition. Topological Fixed Point Theory and Its Applications, 4. Springer, Dordrecht, 2006. | Zbl 1107.55001

[008] [9] L. Górniewicz, A. Granas and W. Kryszewski, On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts, J. Math. Anal. Appl. 161 (2) (1991), 457-473. doi: 10.1016/0022-247X(91)90345-Z | Zbl 0757.54019

[009] [10] Ph. Hartman, Ordinary Differential Equations, Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA], Classics in Applied Mathematics, 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.

[010] [11] D.M. Hyman, On decreasing sequences of compact absolute retracts, Fund Math. 64 (1969), 91-97. | Zbl 0174.25804

[011] [12] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications 7, Walter de Gruyter, Berlin-New York, 2001. doi: 10.1515/9783110870893 | Zbl 0988.34001

[012] [13] N.V. Loi, Method of guiding functions for differential inclusions in a Hilbert space, Differ. Uravn. 46 (10) (2010), 1433-1443 (in Russian); English tranl.: Differ. Equat. 46 (10) (2010), 1438-1447. doi: 10.1134/S0012266110100071

[013] [14] N.V. Loi, V. Obukhovskii and P. Zecca, Non-smooth guiding functions and periodic solutions of functional differential inclusions with infinite delay in Hilbert spaces, Fixed Point Theory 13 (2) (2012), 565-582. | Zbl 1287.34057

[014] [15] A.D. Myshkis, Generalizations of the theorem on a stationary point of a dynamical system inside a closed trajectory, (Russian) Math. Sb. 34 (1954), 525-540.

[015] [16] V. Obukhovski, P. Zecca, N.V. Loi and S. Kornev, Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes in Math. 2076, Springer, Berlin, 2013. doi: 10.1007/978-3-642-37070-0 | Zbl 1282.34003

[016] [17] L. Schwartz, Cours d'Analyse 1, Second edition, Hermann, Paris, 1981.