Exponential Stability of Positive Conformable BAM Neural Networks with Communication Delays (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we consider a class of nonlinear differential equations with delays described by conformable fractional derivative. This typeof differential equations can be used to describe dynamics of various practical models including biological and artificial neural networks with heterogeneous time-varying delays. By novel comparison techniques via fractionaldifferential and integral inequalities, we prove under assumptions involvingthe order-preserving property of nonlinear vector fields that, with nonnegativeinitial states and inputs, the system state trajectories are always nonnegative for all time. This feature, called positivity, induces a special character,namely the monotonicity of the system. We then derive tractable conditionsin terms of linear programming and prove, by utilizing the Brouwer’s fixedpoint theorem and comparisons induced by the monotonicity, that the systempossesses a unique positive equilibrium point which attracts exponentially allstate trajectories. An application to the exponential stability of fractional linear time-delay systems is also discussed. Numerical examples with simulationsare given to illustrate the theoretical results.
Keywords: Conformable derivative; time-delay systems; BAM neural networks; positive equilibrium; M-matrix.