Existence, uniqueness and global exponential stability of a periodic solution for a class of multidirectional associative memory neural network models

Abstract

A multidirectional associative memory (MAM) neural network with periodic coefficients and distributed delays is studied. By constructing a Poincaré mapping, some sufficient conditions are obtained ensuring existence, uniqueness and the global exponential stability of a periodic solution of MAM neural network. The result is new to MAM neural networks. An example is given to illustrate the effectiveness of the result.

Appendix

1.1 Proof of Lemma 1

Proof

Suppose $x(t) = (x_{11}(t), \dots , x_{1n_1}(t), \dots , x_{m1}(t), \dots , x_{mn_m}(t))^T$ is a $T$-periodic solution of MAM (2). Then,

$$\begin{aligned} {\int \limits _0^{ + \infty } {g_{pjki} (s)x_{pj} (t - s)\hbox{d}s} }&= {\sum \limits _{l = 0}^{ + \infty } {\int \limits _{lT}^{(l + 1)T} {g_{pjki} (s)x_{pj} (t - s)\hbox{d}s} } }\\ {}&= {\sum \limits _{l = 0}^{ + \infty } {\int \limits _0^T {g_{pjki} (u + lT)x_{pj} (t - u - lT)\hbox{d}u}.}} \end{aligned}$$

From assumption (H2) and the periodicity of $x(t)$, we obtain

$$\begin{aligned} {\int \limits _0^{ + \infty } {g_{pjki} (s)x_{pj} (t - s)\hbox{d}s} }&= {\int \limits _0^T {\sum \limits _{l = 0}^{ + \infty } {g_{pjki} (u + lT)x_{pj} (t - u)} \hbox{d}u} } \\ {}&= {\int \limits _0^T {G_{pjki} (s)x_{pj} (t - s)\hbox{d}s}.} \end{aligned}$$

Therefore,

$$\begin{aligned} {\mathop{\mathop{\sum}\limits _{p = 1}}\limits_{p \ne k}^m} {\sum \limits _{j = 1}^{n_p } {w_{pjki} (t)f_{pj}\left( {\int \limits _0^{ + \infty } {g_{pjki} (s)x_{pj} \left( {t -s} \right) \hbox{d}s} } \right) } } = {\mathop{\mathop{\sum}\limits_{p = 1}}\limits_{p \ne k}^m} {\sum \limits_{j = 1}^{n_p } {w_{pjki} (t)f_{pj} \left( {\int \limits _0^T{G_{pjki} (s)x_{pj} ( {t - s} )\hbox{d}s} } \right) } }.\end{aligned}$$

It implies that $x(t)$ is a $T$-periodic solution of MAM (4). Now, if $x(t)$ is a $T$-periodic solution of MAM (4), then one can reverse the above sequence of steps and show that $x(t)$ is also a $T$-periodic solution of MAM (2).

1.2 Proof of Lemma 2

Proof

From assumption (H2), we have

$$\begin{aligned} {\int \limits _0^T {G_{pjki} (s)\hbox{d}s} }&= {\int \limits _0^T {\sum \limits _{l = 0}^{ + \infty } {g_{pjki} (s + lT)} \hbox{d}s} = \sum \limits _{l = 0}^{ + \infty } {\int \limits _{lT}^{(l + 1)T} {g_{pjki} (u)\hbox{d}u} } } \\ {}&= {\int \limits _0^{ + \infty } {g_{pjki} (s)\hbox{d}s} = 1,} \end{aligned}$$

and

$$\begin{aligned} {\int \limits _0^T {\exp (\epsilon s) G_{pjki} (s) \hbox{d}s} }&= {\int \limits _0^T {\exp (\epsilon s) \sum \limits _{l = 0}^{ + \infty } {g_{pjki} (s + lT)} \hbox{d}s} = \sum \limits _{l = 0}^{ + \infty } {\int \limits _{lT}^{(l + 1)T} {\exp (\epsilon (u - lT)) g_{pjki} (u) \hbox{d}u} } } \\ {}&< {\sum \limits _{l = 0}^{ + \infty } {\int \limits _{lT}^{(l + 1)T} {\exp (\epsilon u) g_{pjki} (u) \hbox{d}u} } = \int \limits _0^{ + \infty } {\exp (\epsilon u) g_{pjki} (u)\hbox{d}u} } \\ {}&\le {\int \limits _0^{ + \infty } {\exp (u) g_{pjki} (u)\hbox{d}u} < +\infty .} \end{aligned}$$