Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays

Volume 18, Issue 3, pp 328--345 Publication Date: June 27, 2018       Article History


R. P. Agarwal - Department of Mathematics, Texas A\(\&\)M University-Kingsville, Kingsville, TX 78363, USA \(\&\) Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA
S. Hristova - University of Plovdiv Paisii Hilendarski, Plovdiv, Bulgaria
Donal O'Regan - School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland


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.