Expansivity implies existence of Hölder continuous Lyapunov function
Governing equations for Probability densities of stochastic differential equations with discrete time delays
November 2017, 22(9): 3591-3614. doi: 10.3934/dcdsb.2017181
Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks
Zuowei Cai a,b, , Jianhua Huang a,, and Lihong Huang c,
College of Science, National University of Defense Technology, Changsha, Hunan 410073, China
Department of Information Technology, Hunan Women's University, Changsha, Hunan 410002, China
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114 China
* Corresponding author: Jianhua Huang
Received September 2016 Revised June 2017 Published July 2017
Fund Project: The first author is supported by NSF of China(No.11626100), Natural Science Foundation of Hunan Province(No.2016JJ3078) and Scientific Research Youth Project of Hunan Provincial Education Department(No.16B133)
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 a 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.
Keywords: Retarded differential inclusions(RDI), Filippov solution, stability, Razumikhin techniques, almost indefinite Lyapunov-like function.
Mathematics Subject Classification: Primary:34D20, 34K09;Secondary:34K20.
Citation: Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181
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Figure 1. Discontinuous neuron activation functions of Example 1
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Figure 2. Time-domain behaviors of the state variables $x_{1}(t)$ and $x_{2}(t)$ for system (50) under the switching state-feedback controller (53) in Example 1
Figure 4. Time-domain behaviors of the state variables $x_{1}(t)$ and $x_{2}(t)$ for system (50) without external input in Example 2
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Zuowei Cai Jianhua Huang Lihong Huang