# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 M. P. Kennedy and L. O. Chua [$\left(*\right)$ 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 $\left(*\right)$ are GAS also when the constraint amplifiers are dynamical, as it happens in any practial implementation.

##### MSC:
 68T05 Learning and adaptive systems
##### Keywords:
global asymptotic stability; Lyapunov functions