Limit Set Dichotomy and Convergence of Cooperative Piecewise Linear Neural Networks

Abstract

This paper considers a class of nonsymmetric cooperative neural networks (NNs) where the neurons are fully interconnected and the neuron activations are modeled by piecewise linear (PL) functions. The solution semiflow generated by cooperative PLNNs is monotone but, due to the horizontal segments in the neuron activations, is not eventually strongly monotone (ESM). The main result in this paper is that it is possible to prove a peculiar form of the LIMIT SET DICHOTOMY for this class of cooperative PLNNs. Such a form is slightly weaker than the standard form valid for ESM semiflows, but this notwithstanding it permits to establish a result on convergence analogous to that valid for ESM semiflows. Namely, for almost every choice of the initial conditions, each solution of a fully interconnected cooperative PLNN converges toward an equilibrium point, depending on the initial conditions, as + . From a methodological viewpoint, this paper extends some basic techniques and tools valid for ESM semiflows, in order that they can be applied to the monotone semiflows generated by the considered class of cooperative PLNNs.Index Terms-Convergence, cooperative neural networks, dynamical systems, limit set dichotomy, monotone and eventually strongly monotone semiflows.

On the Kamke–Müller conditions, monotonicity and continuity for bi-modal piecewise-smooth systems

“…As A and B are Metzler, the orthant R n + is invariant under (13). For this reason, the definitions of monotonicity and stability for (13) are understood with respect to the state space R n + . For (13) the mapping φ is given by φ(x) = c T x.…”

“…The origin is a globally exponentially stable equilibrium of (13) if there exist positive constants K > 0, α > 0 such that…”

“…Consider the system (13) and assume that B = A + bc T for some b ∈ R n . If there exists v ∈ D A with Av 0 or v ∈ D B with Bv 0, then the origin is a globally exponentially stable equilibrium of (13).…”

“…Piecewise systems whose local dynamics are monotone provide a potential modelling framework for these scenarios. It should also be noted that related results for piecewise linear Neural Networks have recently appeared in [13]. The structure of the paper is as follows.…”

“…The proof for v ∈ D B with Bv 0 is identical. Corollary 5.2 below provides a sufficient condition for the origin to be a globally exponentially stable equilibrium of (13). We first recall the relevant definition.…”