Periodic orbit analysis for the delayed Filippov system

Cai, Zuowei; Huang, Jianhua; Huang, Lihong

doi:10.1090/proc/13883

Periodic orbit analysis for the delayed Filippov system
HTML articles powered by AMS MathViewer

by Zuowei Cai, Jianhua Huang and Lihong Huang PDF

Proc. Amer. Math. Soc. 146 (2018), 4667-4682 Request permission

Abstract:

In this paper, a general class of the delayed differential equation with a discontinuous right-hand side is considered. Under the extended Filippov differential inclusions framework, some new criteria are obtained to guarantee the existence of a periodic solution by employing Kakutani’s fixed point theorem of set-valued maps and matrix theory. Then, we apply these criteria to the time-delayed neural networks with discontinuous neuron activations. Our analysis method and theoretical results are of great significance in the design of time-delayed neural network circuits with discontinuous neuron activation under a periodic environment.

References

Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, Electron. J. Differential Equations (2001), No. 41, 8. MR 1836809
Abraham Berman and Robert J. Plemmons, Nonnegative matrices in the mathematical sciences, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544666
Francesco S. de Blasi, Lech Górniewicz, and Giulio Pianigiani, Topological degree and periodic solutions of differential inclusions, Nonlinear Anal. 37 (1999), no. 2, Ser. A: Theory Methods, 217–243. MR 1689752, DOI 10.1016/S0362-546X(98)00044-3
Abdelkader Boucherif and Christopher C. Tisdell, Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth, Appl. Math. Comput. 204 (2008), no. 1, 441–449. MR 2458382, DOI 10.1016/j.amc.2008.07.001
Jorge Cortés, Discontinuous dynamical systems: a tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Syst. Mag. 28 (2008), no. 3, 36–73. MR 2413715, DOI 10.1109/MCS.2008.919306
B. C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications, Comput. Math. Appl. 51 (2006), no. 3-4, 589–604. MR 2207444, DOI 10.1016/j.camwa.2005.07.017
A. F. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR 1028776, DOI 10.1007/978-94-015-7793-9
M. Forti, M. Grazzini, P. Nistri, and L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D 214 (2006), no. 1, 88–99. MR 2200797, DOI 10.1016/j.physd.2005.12.006
Georges Haddad, Monotone viable trajectories for functional-differential inclusions, J. Differential Equations 42 (1981), no. 1, 1–24. MR 633828, DOI 10.1016/0022-0396(81)90031-0
Georges Haddad, Topological properties of the sets of solutions for functional-differential inclusions, Nonlinear Anal. 5 (1981), no. 12, 1349–1366. MR 646220, DOI 10.1016/0362-546X(81)90111-5
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
H. Hermes, Discontinuous vector fields and feedback control, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P.R., 1965) Academic Press, New York, 1967, pp. 155–165. MR 0222424
Shi Huang Hong, Existence results for functional differential inclusions with infinite delay, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 773–780. MR 2219689, DOI 10.1007/s10114-005-0600-y
Shihuang Hong and Li Wang, Existence of solutions for integral inclusions, J. Math. Anal. Appl. 317 (2006), no. 2, 429–441. MR 2208929, DOI 10.1016/j.jmaa.2006.01.057
Shouchuan Hu, Dimitrios A. Kandilakis, and Nikolaos S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc. 127 (1999), no. 1, 89–94. MR 1451808, DOI 10.1090/S0002-9939-99-04338-5
Shou Chuan Hu and Nikolaos S. Papageorgiou, On the existence of periodic solutions for nonconvex-valued differential inclusions in $\mathbf R^N$, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3043–3050. MR 1301503, DOI 10.1090/S0002-9939-1995-1301503-6
Guocheng Li and Xiaoping Xue, On the existence of periodic solutions for differential inclusions, J. Math. Anal. Appl. 276 (2002), no. 1, 168–183. MR 1944344, DOI 10.1016/S0022-247X(02)00397-9
Yong Li and Zheng Hua Lin, Periodic solutions of differential inclusions, Nonlinear Anal. 24 (1995), no. 5, 631–641. MR 1319074, DOI 10.1016/0362-546X(94)00111-T
J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein. MR 0481301
A. Lasota and Z. Opial, An application of the Kakutani—Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 781–786 (English, with Russian summary). MR 196178
Vasile Lupulescu, Existence of solutions for nonconvex functional differential inclusions, Electron. J. Differential Equations (2004), No. 141, 6. MR 2108912
Donal O’Regan, Integral inclusions of upper semi-continuous or lower semi-continuous type, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2391–2399. MR 1342037, DOI 10.1090/S0002-9939-96-03456-9
Brad E. Paden and Shankar S. Sastry, A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators, IEEE Trans. Circuits and Systems 34 (1987), no. 1, 73–82. MR 871547, DOI 10.1109/TCS.1987.1086038
N. S. Papageorgiou, Periodic solutions of nonconvex differential inclusions, Appl. Math. Lett. 6 (1993), no. 5, 99–101. MR 1349674, DOI 10.1016/0893-9659(93)90110-9
D. Turkoglu and I. Altun, A fixed point theorem for multi-valued mappings and its applications to integral inclusions, Appl. Math. Lett. 20 (2007), no. 5, 563–570. MR 2303994, DOI 10.1016/j.aml.2006.07.002
Sitian Qin and Xiaoping Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math. Anal. Appl. 424 (2015), no. 2, 988–1005. MR 3292713, DOI 10.1016/j.jmaa.2014.11.057
Vadim I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automatic Control AC-22 (1977), no. 2, 212–222. MR 0484664, DOI 10.1109/tac.1977.1101446
Xiaoping Xue and Jinfeng Yu, Periodic solutions for semi-linear evolution inclusions, J. Math. Anal. Appl. 331 (2007), no. 2, 1246–1262. MR 2313711, DOI 10.1016/j.jmaa.2006.09.056
Pietro Zecca and Pier Luigi Zezza, Nonlinear boundary value problems in Banach spaces for multivalue differential equations on a noncompact interval, Nonlinear Anal. 3 (1979), no. 3, 347–352. MR 532895, DOI 10.1016/0362-546X(79)90024-5