| Further Results on Controllability of Recurrent Neural Networks (1999) | |||||||||||||||
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| This paper studies controllability properties of recurrent neural networks. The new contributions are: (1) an extension of a previous result to a slightly different model, (2) a formulation and proof of a necessary and sufficient condition, and (3) an analysis of a low-dimensional case for which the hypotheses made in previous work do not apply. 1 Introduction This paper deals with controllability properties of what are often called "recurrent neural networks". These constitute a class of nonlinear systems which, although formally analogous to linear systems, exhibit interesting nonlinear characteristics and arise often in applications, see e.g. [3, 4, 5, 7, 9, 10, 11, 14, 15, 18]. A general model of recurrent nets (see e.g. [15]) is as follows. Assume given a Lipschitz map ` : R ! R. The most typical choice is `(x) = tanh x : R ! R : x 7! e x \Gamma e \Gammax e x + e \Gammax ; which is also called the "sigmoid" or "logistic" map. For each positive integer n, we let ~ ` ... | |||||||||||||||
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