Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks

Zuowei Cai; Jianhua Huang; Lihong Huang

doi:10.3934/dcdsb.2017181

Article Contents

2017, Volume 22, Issue 9: 3591-3614. Doi: 10.3934/dcdsb.2017181

This issue Previous Article Expansivity implies existence of Hölder continuous Lyapunov function Next Article Governing equations for Probability densities of stochastic differential equations with discrete time delays

Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks

a.
College of Science, National University of Defense Technology, Changsha, Hunan 410073, China
b.
Department of Information Technology, Hunan Women's University, Changsha, Hunan 410002, China
c.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114 China

* Corresponding author: Jianhua Huang
* Corresponding author: Jianhua Huang

Received: August 31, 2016

Revised: May 31, 2017

Published: October 2017

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)

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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 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.

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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

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Figure 3. Discontinuous neuron activation functions of Example 2

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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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