Further Improvement on Delay-Dependent Global Robust Exponential Stability for Delayed Cellular Neural Networks with Time-Varying Delays

Abstract

This paper is concerned with global robust exponential stability for a class of delayed cellular neural networks with time-varying delays. Some new sufficient conditions are presented for the uniqueness of equilibrium point and the global stability of cellular neural networks with time varying delay by constructing Lyapunov functional and using linear matrix inequality and the integral inequality approach. Numerical examples are illustrated to show the effectiveness of the proposed method. From the simulation results, significant improvement over the recent results can be observed.

Appendix A: Proof of Lemma 1

For any positive semi-definite matrices

$$\begin{aligned} \hbox {X}=\left[ {{\begin{array}{l@{\quad }l@{\quad }l} \hbox {X}_{11} &{} \hbox {X}_{12} &{} \hbox {X}_{13} \\ \hbox {X}_{12}^T &{} \hbox {X}_{22} &{} \hbox {X}_{23} \\ \hbox {X}_{13}^T &{} \hbox {X}_{23}^T &{} \hbox {X}_{33} \end{array} }} \right] \ge 0, \end{aligned}$$

(A1)

the following integral inequality holds

$$\begin{aligned}&-\int _{t-h(t)}^t {{\dot{x}}^T (s){X}_{33} \dot{x}(s)ds} \nonumber \\&\quad \le \int _{t-h(t)}^t {\left[ {{ {{x}^T (t)}\quad {{x}^T (t-h(t))}\quad {{\dot{x}}^T (s)} }} \right] } \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {{X}_{11} }&{} {{X}_{12} }&{} {{X}_{13} } \\ {{X}_{12}^T }&{} {{X}_{22} }&{} {{X}_{23} } \\ {{X}_{13}^T }&{} {{X}_{23}^T }&{} 0 \end{array} }} \right] \left[ {{\begin{array}{l} {x(t)} \\ {x(t-h(t))} \\ {\dot{x}(s)} \\ \end{array} }} \right] ds\nonumber \\ \end{aligned}$$

(A2)

Proof

The New-Leibniz formula and integral inequality approach are used to derive the lemma; they are stated below:

$$\begin{aligned}&-\int _{t-h(t)}^t {{\dot{x}}^T (s){X}_{33} \dot{x}(s)ds} \nonumber \\&\quad \le \int _{t-h(t)}^t {\left[ {{ {{x}^T (t)}\quad {{x}^T (t-h(t))}\quad {{\dot{x}}^T (s)} }} \right] } \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {{X}_{11} }&{} {{X}_{12} }&{} {{X}_{13} } \\ {{X}_{12}^T }&{} {{X}_{22} }&{} {{X}_{23} } \\ {{X}_{13}^T }&{} {{X}_{23}^T }&{} 0 \end{array} }} \right] \left[ {{\begin{array}{l} {x(t)} \\ {x(t-h(t))} \\ {\dot{x}(s)} \\ \end{array} }} \right] ds \nonumber \\&\quad \le {x}^T (t)h{X}_{11} x(t)+{x}^T (t)h{X}_{12} x(t-h(t))+{x}^T (t){X}_{13} \int _{t-h(t)}^t {\dot{x}(s)ds} +{x}^T (t-h(t))\mathop {hX}\nolimits _{12}^T x(t) \nonumber \\&\qquad +\,{x}^T (t-h(t))h{X}_{22} x(t-h(t))+{x}^T (t-h(t)){X}_{23} \int _{t-h(t)}^t {\dot{x}(s)ds} \nonumber \\&\qquad +\,\int _{t-h(t)}^t {{\dot{x}}^T (s)ds} {X}_{13}^T x(t)+\int _{t-h(t)}^t {{\dot{x}}^T (s)ds} {X}_{23}^T x(t-h(t))\nonumber \\&\quad ={x}^T (t)[h{X}_{11} +{X}_{13} +{X}_{13}^T ]x(t)+{x}^T (t)[h{X}_{12} -{X}_{13} +{X}_{23}^T ]x(t-h(t)) \nonumber \\&\qquad +\,{x}^T (t-h(t))[h{X}_{12}^T -{X}_{13}^T +{X}_{23} ]x(t)\nonumber \\&\qquad +\,{x}^T (t-h(t))[h{X}_{22} -{X}_{23} -{X}_{23}^T ]x(t-h(t)) \end{aligned}$$

(A3)

$\square$