Linear systems in a saturated mode and convergence as gain becomes large of asymptotically stable equilibrium points of neural nets

Abstract

We study solutions of the “linear system in a saturated mode”

$$\begin{array}{*{20}c} {(M)} & {x' \in Tx + c - \partial I_{D^n } x.} \\ \end{array} $$

We show that a trajectory is in a constant face of the cubeD ⁿ on some interval (0,d]. We answer a question about comparing the two systems: (M) and

$$\begin{array}{*{20}c} {(H)} & {\begin{array}{*{20}c} {Cu' = T\upsilon + c - R^{ - 1} u,} & {\upsilon = G(\lambda } \\ \end{array} u)} \\ \end{array} $$

. As λ→∞, limits ofv corresponding to asymptotically stable equilibrium points of (H) are asymptotically stable equilibrium points of (M), and the converse is also true. We study the assumptions to see which are required and which may be weakened.