LMI optimization approach on stability for delayed neural networks of neutral-type.

*(English)*Zbl 1157.34056The authors investigate the global asymptotic stability of delayed cellular neural network described by equations of neutral type

\[ \dot{u}_i(t)=-a_iu_i(t)+\sum_{j=1}^nw_{ij}^1\overline{f}_j(u_j(t))+ \sum_{j=1}^nw_{ij}^2\overline{g}_j(u_j(t-h))+b_i, \;\;i=1,\dots,n, \] where \(u_i(t)\) corresponds to the state of the \(i\)-th neuron, \(\overline{f}_j\) and \(\overline{g}_j\) denotes the activation functions, \(w_{ij}^1\) and \(w_{ij}^2\) are constant connection weights, \(b_i\) is the external input on the \(i\)-th neuron, \(a_i>0\). The activation fuctions satisfy a usual condition of the form

\[ 0\leq\frac{\overline{f}_j(\xi_1)-\overline{f}_j(\xi_2)}{\xi_1-\xi_2}\leq l_j \;\text{for any }\xi_1\neq\xi_2\text{ in }\mathbb{R}. \] A theorem on delay-dependent conditions for global asymptotic stability is obtained in the form of linear matrix inequality. For the proof of the result an appropriate Lyapunov functional is constructed as a sum of quadratic forms and integrals of them.

The advantage of the proposed approach is that the result can be used efficiently through existing numerical convex optimization algorithm for solving the linear matrix inequality. An illustrative numerical example is presented.

\[ \dot{u}_i(t)=-a_iu_i(t)+\sum_{j=1}^nw_{ij}^1\overline{f}_j(u_j(t))+ \sum_{j=1}^nw_{ij}^2\overline{g}_j(u_j(t-h))+b_i, \;\;i=1,\dots,n, \] where \(u_i(t)\) corresponds to the state of the \(i\)-th neuron, \(\overline{f}_j\) and \(\overline{g}_j\) denotes the activation functions, \(w_{ij}^1\) and \(w_{ij}^2\) are constant connection weights, \(b_i\) is the external input on the \(i\)-th neuron, \(a_i>0\). The activation fuctions satisfy a usual condition of the form

\[ 0\leq\frac{\overline{f}_j(\xi_1)-\overline{f}_j(\xi_2)}{\xi_1-\xi_2}\leq l_j \;\text{for any }\xi_1\neq\xi_2\text{ in }\mathbb{R}. \] A theorem on delay-dependent conditions for global asymptotic stability is obtained in the form of linear matrix inequality. For the proof of the result an appropriate Lyapunov functional is constructed as a sum of quadratic forms and integrals of them.

The advantage of the proposed approach is that the result can be used efficiently through existing numerical convex optimization algorithm for solving the linear matrix inequality. An illustrative numerical example is presented.

Reviewer: Oleg Anashkin (Simferopol)

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K40 | Neutral functional-differential equations |

92B20 | Neural networks for/in biological studies, artificial life and related topics |

##### Software:

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\textit{J. H. Park} et al., Appl. Math. Comput. 196, No. 1, 236--244 (2008; Zbl 1157.34056)

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