Estimating the Ultimate Bound and Positively Invariant Set for a Class of Hopfield Networks

In this paper, we investigate the ultimate bound and positively invariant set for a class of Hopfield neural networks (HNNs) based on the Lyapunov stability criterion and Lagrange multiplier method. It is shown that a hyperelliptic estimate of the ultimate bound and positively invariant set for the HNNs can be calculated by solving a linear matrix inequality (LMI). Furthermore, the global stability of the unique equilibrium and the instability region of the HNNs are analyzed, respectively. Finally, the most accurate estimate of the ultimate bound and positively invariant set can be derived by solving the corresponding optimization problems involving the LMI constraints. Some numerical examples are given to illustrate the effectiveness of the proposed results.