Generalized Lyapunov-Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. (English) Zbl 1371.34107
Authors’ abstract: In this paper, a general class of nonlinear dynamical systems described by retarded differential equations with discontinuous right-hand sides is considered. Under the extended Filippov-framework, we investigate some basic stability problems for retarded differential inclusions (RDI) with given initial conditions by using the generalized Lyapunov-Razumikhin method. Comparing with the previous work, the main results given in this paper show that the Lyapunov-Razumikhin function is allowed to have an indefinite or positive definite derivative for almost everywhere along the solution trajectories of RDI. However, in most of the existing literature, the derivative (if it exists) of Lyapunov-Razumikhin function is required to be negative or semi-negative definite for almost everywhere. In addition, the Lyapunov-Razumikhin function in this paper is allowed to be non-smooth. To deal with the stability, we also drop the specific condition that the Lyapunov-Razumikhin function should have an infinitesimal upper limit. Finally, we apply the proposed Razumikhin techniques to handle the stability or stabilization control of discontinuous time- delayed neural networks. Meanwhile, we present two examples to demonstrate the effectiveness of the developed method.
Reviewer: Andrej V. Plotnikov (Odessa)
MSC:
| 34K09 | Functional-differential inclusions |
| 34K20 | Stability theory of functional-differential equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
Keywords:
retarded differential inclusions (RDI); Filippov solution; stability; Razumikhin techniques; almost indefinite Lyapunov-like functionReferences:
| [1] | J. P. Aubin, <em>Set-Valued Analysis</em>,, MA: Birkhauser (1990) |
| [2] | J. P. Aubin, <em>Differential Inclusions</em>,, Springer-Verlag (1984) · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4 |
| [3] | A. Baciotti, Stability and stabilization of discontinuous systems and non-smooth Lyapunov function,, ESAIM Control Optim. Calc. Var., 4, 361 (1999) · Zbl 0927.34034 · doi:10.1051/cocv:1999113 |
| [4] | M. Benchohra, Existence results for functional differential inclusions,, Electronic Journal of Differential Equations, 2001, 1 (2001) · Zbl 0980.34062 |
| [5] | M. di Bernardo, <em>Piecewise-smooth Dynamical Systems, Theory and Applications</em>,, Springer-Verlag (2008) · Zbl 1146.37003 |
| [6] | V. I. Blagodat-skik, Differential inclusions and optimal control,, Proc. Steklov Inst. Math., 4, 199 (1986) · Zbl 0608.49027 |
| [7] | Z. W. Cai, Finite-time stabilization of delayed memristive neural networks: Discontinuous state-feedback and adaptive control approach,, IEEE Trans. Neural Netw. Learn. Syst., PP, 1 (2017) · doi:10.1109/TNNLS.2017.2651023 |
| [8] | Z. W. Cai, Novel adaptive control and state-feedback control strategies to finite-time stabilization of discontinuous delayed networks,, IEEE Trans. Syst., PP, 1 (2017) · doi:10.1109/TSMC.2017.2657784 |
| [9] | Z. W. Cai, On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions,, Neural Networks, 33, 97 (2012) · Zbl 1269.34076 · doi:10.1016/j.neunet.2012.04.009 |
| [10] | E. K. P. Chong, An analysis of a class of neural networks for solving linear programming problems,, IEEE Trans. Autom. Control, 44, 1995 (1999) · Zbl 0957.90087 · doi:10.1109/9.802909 |
| [11] | F. H. Clarke, <em>Optimization and Nonsmooth Analysis</em>,, Wiley (1983) · Zbl 0582.49001 |
| [12] | F. H. Clarke, <em>Nonsmooth Analysis and Control Theory</em>,, Springer-Verlag (1998) · Zbl 1047.49500 |
| [13] | J. Cortés, Discontinuous dynamical systems,, IEEE Control Syst. Mag., 28, 36 (2008) · Zbl 1395.34023 · doi:10.1109/MCS.2008.919306 |
| [14] | A. F. Filippov, <em>Differential Equations with Discontinuous Right-hand Side, in: Mathematics and Its Applications (Soviet Series)</em>,, Kluwer Academic (1988) · Zbl 0138.32204 · doi:10.1007/978-94-015-7793-9 |
| [15] | M. Forti, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations,, Physica D, 214, 88 (2006) · Zbl 1103.34044 · doi:10.1016/j.physd.2005.12.006 |
| [16] | M. Forti, Generalized neural network for nonsmooth nonlinear programming problems,, IEEE Trans. Circuits Syst. I, 51, 1741 (2004) · Zbl 1374.90356 · doi:10.1109/TCSI.2004.834493 |
| [17] | Z. Y. Guo, Generalized Lyapunov method for discontinuous systems,, Nonlinear Anal., 71, 3083 (2009) · Zbl 1182.34008 · doi:10.1016/j.na.2009.01.220 |
| [18] | Z. Y. Guo, Attractivity analysis of memristor-based cellular neural networks with time-varying delays,, IEEE Trans. Neural Netw. Learn. Syst., 25, 704 (2014) · doi:10.1109/TNNLS.2013.2280556 |
| [19] | G. Haddad, Topological properties of the sets of solutions for functional differential inclusions,, Nonlinear Anal., 5, 1349 (1981) · Zbl 0496.34041 · doi:10.1016/0362-546X(81)90111-5 |
| [20] | G. Haddad, Monotone viable trajectories for functional differential inclusions,, J. Differential Equations, 42, 1 (1981) · Zbl 0472.34043 · doi:10.1016/0022-0396(81)90031-0 |
| [21] | J. K. Hale, <em>Theory of Functional Differential Equations</em>,, Springer-Verlag (1977) · Zbl 0352.34001 |
| [22] | H. Hermes, Discontinuous vector fields and feedback control,, in: Differential Equations and Dynamical Systems, 155 (1967) · Zbl 0183.15905 |
| [23] | S. H. Hong, Existence results for functional differential inclusions with infinite delay,, Acta Math. Sin., 22, 773 (2006) · Zbl 1111.34059 · doi:10.1007/s10114-005-0600-y |
| [24] | L. H. Huang, <em>Theory and Applications of Differential Equations with Discontinuous Right-hand Sides</em> (in Chinese),, Science Press (2011) |
| [25] | M. P. Kennedy, Neural networks for nonlinear programming,, IEEE Trans. Circuits Syst. I, 35, 554 (1988) · doi:10.1109/31.1783 |
| [26] | N. N. Krasovskii, <em>Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay</em>,, CA: Stanford Univ. Press (1963) · Zbl 0109.06001 |
| [27] | K. Z. Liu, Stability theorems for delay differential inclusions,, IEEE Trans. Autom. Control, 61, 3215 (2016) · Zbl 1359.34062 · doi:10.1109/TAC.2015.2507782 |
| [28] | K. Z. Liu, Razumikhin-type theorems for hybrid system with memory,, Automatica, 71, 72 (2016) · Zbl 1343.93073 · doi:10.1016/j.automatica.2016.04.038 |
| [29] | X. Y. Liu, Nonsmooth finite-time synchronization of switched coupled neural networks,, IEEE Trans. Cybern., 46, 2360 (2016) · doi:10.1109/TCYB.2015.2477366 |
| [30] | A. C. J. Luo, <em>Discontinuous Dynamical Systems on Time-varying Domains</em>,, Higher Education Press (2009) · Zbl 1213.34001 · doi:10.1007/978-3-642-00253-3 |
| [31] | V. Lupulescu, Existence of solutions for nonconvex functional differential inclusions,, Electronic Journal of Differential Equations, 2004, 1 (2004) · Zbl 1075.34055 |
| [32] | B. E. Paden, A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulator,, IEEE Trans. Circuits Syst., 34, 73 (1987) · Zbl 0632.34005 · doi:10.1109/TCS.1987.1086038 |
| [33] | T. T. Su, Finite-time synchronization of competitive neural networks with mixed delays,, Discrete & Continuous Dynamical Systems-Series B, 21, 3655 (2016) · Zbl 1357.34126 · doi:10.3934/dcdsb.2016115 |
| [34] | A. Surkov, On the stability of functional-differential inclusions with the use of invariantly differentiable Lyapunov functionals,, Differential Equations, 43, 1079 (2007) · Zbl 1187.34084 · doi:10.1134/S001226610708006X |
| [35] | V. I. Utkin, Variable structure systems with sliding modes,, IEEE Trans. Automat. Contr., 22, 212 (1977) · Zbl 0382.93036 |
| [36] | K. N. Wang, Stability analysis of differential inclusions in banach space with application to nonlinear systems with time delays,, IEEE Trans. Circuits Syst. I, 43, 617 (1996) · doi:10.1109/81.526677 |
| [37] | L. Wang, Finite-time stabilizability and instabilizability of delayed memristive neural networks with nonlinear discontinuous controller,, IEEE Trans. Neural Netw. Learn. Syst., 26, 2914 (2015) · doi:10.1109/TNNLS.2015.2460239 |
| [38] | S. P. Wen, Circuit design and exponential stabilization of memristive neural networks,, Neural Networks, 63, 48 (2015) · Zbl 1323.93065 · doi:10.1016/j.neunet.2014.10.011 |
| [39] | A. L. Wu, Synchronization control of a class of memristor-based recurrent neural networks,, Information Sciences, 183, 106 (2012) · Zbl 1243.93049 · doi:10.1016/j.ins.2011.07.044 |
| [40] | C. J. Xu, Exponential stability of almost periodic solutions for memristor-based neural networks with distributed leakage delays,, Neural Comput., 28, 2726 (2016) · Zbl 1482.34199 · doi:10.1162/NECO_a_00895 |
| [41] | X. S. Yang, Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays,, IEEE Trans. Fuzzy Syst., 23, 2302 (2015) · doi:10.1109/TFUZZ.2015.2417973 |
| [42] | X. S. Yang, Synchronization of delayed memristive neural networks: Robust analysis approach,, IEEE Trans. Cybern., 46, 3377 (2016) · doi:10.1109/TCYB.2015.2505903 |
| [43] | B. Zhou, Razumikhin and Krasovskii stability theorems for time-varying time-delay systems,, Automatica, 71, 281 (2016) · Zbl 1343.93061 · doi:10.1016/j.automatica.2016.04.048 |
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.
