Exponential stability in the Lagrange sense for Clifford-valued recurrent neural networks with time delays

G Rajchakit,C P Lim,P Hammachukiattikul,R Sriraman,N Boonsatit,P Agarwal

doi:10.1186/s13662-021-03415-8

Abstract

This paper considers the Clifford-valued recurrent neural network (RNN) models, as an augmentation of real-valued, complex-valued, and quaternion-valued neural network models, and investigates their global exponential stability in the Lagrange sense. In order to address the issue of non-commutative multiplication with respect to Clifford numbers, we divide the original n-dimensional Clifford-valued RNN model into 2^{m}n real-valued models. On the basis of Lyapunov stability theory and some analytical techniques, several sufficient conditions are obtained for the considered Clifford-valued RNN models to achieve global exponential stability according to the Lagrange sense. Two examples are presented to illustrate the applicability of the main results, along with a discussion on the implications.

Highlights

1 Introduction Recurrent neural network (RNN) models have been successfully employed for solving problems of optimization, control, associative memory, and signal and image processing
Motivated by the above discussions, we investigate the exponential stability in the Lagrange sense for Clifford-valued RNNs with time delays in this paper
Proof First of all, we prove that NN model (6) is uniformly stable in the Lagrange sense

Summary

Introduction

Recurrent neural network (RNN) models have been successfully employed for solving problems of optimization, control, associative memory, and signal and image processing. As an example, when employing NN models for associative memory or pattern recognition, multiple equilibrium points are often required In these cases, NN models are no longer globally stable in the Lyapunov sense. Our main objective of this paper is to derive new sufficient conditions to ensure the Lagrange exponential stability for Clifford-valued RNNs. The main contributions of this work are as follows: (1) This is the first paper to study such a problem for global exponential stability. (2) By using the Lyapunov functional and some analytical methods, new sufficient conditions for ascertaining the exponential stability of the considered Clifford-valued RNN models in the Lagrange sense are derived. This is achieved by decomposing the n-dimensional Clifford-valued RNN model into 2mn-dimensional real-valued RNN models.

Notations

Clifford algebra The Clifford real algebra over Rm is defined as

Main results

Exponential stability

Conclusions