×

A system of fractional-order interval projection neural networks. (English) Zbl 1329.34013

Summary: A system of fractional-order interval projection neural networks is introduced and investigated. Under some suitable assumptions, the existence and uniqueness of the equilibrium point of this type of interval projection neural networks is proved. Moreover, \(\alpha\)-exponential stability of this type of neural networks is obtained. Also, in the last section, we give two numerical examples to illustrate our results.

MSC:

34A08 Fractional ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34D20 Stability of solutions to ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wu, Z. B.; Zou, Y. Z., Global fraction-order projective dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 19, 2811-2819 (2014) · Zbl 1510.34024
[2] Wu, X. K.; Wu, Z. B.; Zou, Y. Z., Existence, uniqueness and stability for a class of interval projective dynamical systems, Commun. Appl. Nonlinear Anal., 20, 81-94 (2013) · Zbl 1295.34022
[3] Ding, K.; Huang, N. J., A new interval projection neural networks for solving interval quadratic program, Chaos Solitons Fractals, 35, 718-725 (2008) · Zbl 1137.90020
[4] Cojocaru, M. G., Projected dynamical systems on Hilbert spaces (2002), Queen’s University: Queen’s University Canada, (Ph.D. thesis)
[5] Dupuis, P.; Nagurney, A., Dynamical systems and variational inequalities, Ann. Oper. Res., 44, 19-42 (1993) · Zbl 0785.93044
[6] Friesz, T. L.; Bernstein, D. H.; Mehta, N. J.; Tobin, R. L.; Ganjlizadeh, S., Day-to-Day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42, 1120-1136 (1994) · Zbl 0823.90037
[7] Friesz, T. L.; Suo, Z. G.; Bernstein, D. H., A dynamic disequilibrium interregional commodity flow model, Transp. Res. B, 42, 467-483 (1998)
[8] Kinderlehrer, D.; Stampcchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic: Academic New York
[9] Wu, H. Q.; Shi, R.; Qin, L. J.; Tao, F.; He, L. J., A nonlinear projection neural network for solving interval quadratic programming problems and its stability analysis, Math. Probl. Eng., 2010, 13 (2010), Artical ID 403749 · Zbl 1195.90072
[10] Xia, Y. S.; Vincent, T. L., On the stability of global projected dynamical systems, J. Optim. Theory Appl., 106, 129-150 (2000) · Zbl 0971.37013
[11] Xia, Y. S., Further results on global convergence and stability of global projected dynamical systems, J. Optim. Theory Appl., 122, 627-649 (2004) · Zbl 1082.34043
[12] Zhang, D.; Nagurney, A., On the stability of projected dynamical systems, J. Optim. Theory Appl., 85, 97-124 (1995) · Zbl 0837.93063
[13] Zou, Y. Z.; Li, X.; Huang, N. J.; Sun, C. Y., Global dynamical systems involving generalized \(f\)-projection operators and set-valued perturbation in Banach spaces, J. Appl. Math., 2012, 12 (2012), Article ID 682465 · Zbl 1305.47041
[14] Zou, Y. Z.; Sun, C. Y., Equilibrium points for two related projective dynamical systems, Commun. Appl. Nonlinear Anal., 19, 109-117 (2012)
[15] Wu, Z. B.; Zou, Y. Z., Stability analysis of two related projective dynamical systems in Hilbert spaces, Nonlinear Anal. Forum, 19, 37-51 (2014) · Zbl 1309.49011
[16] Zou, Y. Z.; Wu, X. K.; Zhang, W. B.; Sun, C. Y., An iterative method for a class of generalized global dynamical system involving fuzzy mappings in Hilbert spaces, Lecture Notes in Comput. Sci., 7666, 44-51 (2012)
[17] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier · Zbl 1092.45003
[18] Lu, J. G.; Chen, Y. Q., Robust stability and stabilization of fractional-order interval systems with the fractional order \(\alpha \): the \(0 < \alpha < 1\) case, IEEE Trans. Automat. Control, 55, 152-158 (2010) · Zbl 1368.93506
[19] Li, C. P.; Zhang, F. R., A survey on the stability of fractional differential equations, Eur. Phys. J. Spec. Top., 193, 27-47 (2011)
[20] Ozalp, N.; Koca, I., A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Differential Equations, 2012, 189 (2012) · Zbl 1377.91140
[21] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Siego · Zbl 0918.34010
[22] Torvik, P. J.; Bagley, R. L., On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, 294-298 (1984) · Zbl 1203.74022
[23] Skaar, S. B.; Michel, A. N.; Miller, R. K., Stability of viscoelastic control systems, IEEE Trans. Automat. Control, 33, 348-357 (1988) · Zbl 0641.93051
[24] Yu, J.; Hu, C.; Jian, H. J., \( \alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks, Neural Netw., 35, 82-87 (2012) · Zbl 1258.34118
[25] Li, Y.; Chen, Y. Q.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: lyapunov direct method and generalized Mittag-leffler stability, Comput. Math. Appl., 59, 1810-1821 (2010) · Zbl 1189.34015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.