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Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays. (English) Zbl 1427.34104

Summary: A reaction diffusion equation with a Caputo fractional derivative in time and with time-varying delays is considered. Stability properties of the solutions are studied via the direct Lyapunov method and arbitrary Lyapunov functions (usually quadratic Lyapunov functions are used). In this paper, we give a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of reaction-diffusion fractional neural network with variable coefficients and time-varying delays. We show the quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability are obtained and we illustrate our theory on a particular nonlinear Caputo reaction-diffusion fractional neural network with time dependent delays.

MSC:

34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
35K57 Reaction-diffusion equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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