New conditions for global stability of neural networks with application to linear and quadratic programming problems.

*(English)* Zbl 0849.68105
Summary: We present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such function may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piecewise linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur’e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS.
In particular, the results are applied to analyze GAS for the class of neural circuits introduced by {\it M. P. Kennedy} and {\it L. O. Chua} [$(*)$ Neural networks for nonlinear programming, ibid. 35, 554-562 (1988)] for solving linear and quadratic programming problems. In this application, the principal result here obtained is that the networks in $(*)$ are GAS also when the constraint amplifiers are dynamical, as it happens in any practial implementation.

##### MSC:

68T05 | Learning and adaptive systems |