• Wang, Zengyun ;
  • Huang, Lihong ;
  • Zuo, Yi ;
  • Zhang, Lingling
  • Published : 2010.01.31


This paper concerns the problem of global robust stability of a time-delay discontinuous system with a positive-defined connection matrix under polytopic-type uncertainty. In order to give the stability condition, we firstly address the existence of solution and equilibrium point based on the properties of M-matrix, Lyapunov-like approach and the theories of differential equations with discontinuous right-hand side as introduced by Filippov. Second, we give the delay-independent and delay-dependent stability condition in terms of linear matrix inequalities (LMIs), and based on Lyapunov function and the properties of the convex sets. One numerical example demonstrate the validity of the proposed criteria.


global robust stability;delayed neural network;delay-independent condition;delay-dependent condition;linear matrix inequality;discontinuous neuron activation


  1. J. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.
  2. G. Avitabile, M. Forti, S. Manetti, and M. Marini, On a class of nonsymmetrical neural networks with application to ADC, IEEE Trans. Circuits Syst. 38 (1991), 202–209.
  3. S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.
  4. A. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988.
  5. M. Forti, M. Grazzini, P. Nistri, and L. Pancioni, Generalized lyapunov approach for convergence of neural networks with discontinuous or non-lipschitz activations, Physica D: Nonlinear Phenomena 214 (2006), 88–99.
  6. M. Forti, S. Manetti, and M. Marini, A condition for global convergence of a class of symmetric neural networks, IEEE Trans. Circuits Syst. I 39 (1992), 480–483.
  7. M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I 50 (2003), no. 11, 1421–1435.
  8. M. Forti, P. Nistri, and D. Papini, Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Trans. Neural Netw. 16 (2005), no. 6, 1449–1463.
  9. S. Grossberg, Nonlinear neural networks: Principles, mechanisms and architectures, Neural Netw. 1 (1988), 17–61.
  10. Y. He, M. Wu, and J. She, Parameter-dependent Lyapunov functional for stability of the time-delay system with polytopic-type uncertainty, IEEE Trans. Automatic Control 49 (2004), 828–832.
  11. M. Hirsch, Convergent activation dynamics in continuous time networks, Neural Netw. 2 (1989), no. 5, 331–349.
  12. S. Hu and J. Wang, Global stability of a class of continuous-time recurrent neural networks, IEEE Trans. Circuits Systems I Fund. Theory Appl. 49 (2002), no. 9, 1334–1347.
  13. X. Liang and J. Wang, A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints, IEEE Trans. Neural Netw. 11 (2000), no. 6, 1251–1262.
  14. C. Lim, Y. Park, and S. Moon, Robust saturation controller for linear time-invariant system with structured real parameter uncertainties, J. Sound Vibra. 294 (2006), no. 1-2, 1–14.
  15. W. Lu and T. Chen, Dynamical behaviors of delayed neural network systems with discontinuous activation functions, Neural Computation 18 (2006), 683–708.
  16. Y. Moon, P. Park, W. Kwon, and Y. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control 74 (2001), no. 14, 1447–1455.
  17. S. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269. Springer-Verlag London, Ltd., London, 2001.
  18. D. Papini and V. Taddei, Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations, Phys. Lett. A 343 (2005), 117–128.
  19. P. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automat. Control 44 (1999), no. 4, 876–877.

Cited by

  1. Long Time Behavior for a System of Differential Equations with Non-Lipschitzian Nonlinearities vol.2014, 2014,
  2. H∞ control for neural networks with discontinuous activations and nonlinear external disturbance vol.352, pp.8, 2015,