Improved Integral Inequality Based on Free Matrices and Its Application to Stability Analysis of Delayed Neural Networks via Matrix-valued Cubic Polynomial Inequality

Abstract

This paper deals with the stability analysis of delayed neural networks (DNNs). Firstly, an improved integral inequality based on free matrices (I3BFM) is newly proposed. The I3BFM gives the upper bound of the integral quadratic form including the state-related vector stacked by the state, its derivative, and its integration by introducing some free matrices. The upper bound of the I3BFM fully reflects the information not only on each term in the state-related vector but also on the cross-terms between the three terms. Secondly, the double integral Lyapunov-Krasovskii functional (LKF) of the integral quadratic form including the state-related vector is established to utilize the proposed I3BFM in the stability analysis of DNN. From the derivative of the double integral LKF, a matrix-valued cubic polynomial on a time-varying delay is generated in the process of stability analysis. Therefore, a matrix-valued cubic polynomial inequality is applied to make the matrix-valued cubic polynomial numerically tractable. Finally, the relaxed stability criteria utilizing the I3BFM and the double integral inequality are derived, and the superiority of the derived stability criteria is demonstrated by two well-known numerical examples.