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

  • * Corresponding author: Jianhua Huang

    * Corresponding author: Jianhua Huang 

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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  • 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.

    Mathematics Subject Classification: Primary:34D20, 34K09;Secondary:34K20.


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  • Figure 1.  Discontinuous neuron activation functions of Example 1

    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 3.  Discontinuous neuron activation functions of Example 2

    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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