On the Observability and Controllability of Leaky-ReLU Networks

Abstract

This paper studies minimum-node observability and controllability of Leaky rectified linear unit (Leaky-ReLU) networks under degree constraints. The objective is to characterize how many state nodes must be measured or directly actuated to determine the initial state from a finite output sequence or to steer the network between arbitrary states within a finite horizon. For observability, a graph-theoretic analysis yields a class-wide upper bound on the minimum number of observation nodes. We construct a family of networks attaining this bound, thereby determining the exact worst-case minimum number of observation nodes. We also construct networks that are observable from a single node over a finite horizon, establishing the exact best-case value of one. By establishing an observability--controllability duality under the corresponding degree constraints, we obtain analogous exact best- and worst-case results for the minimum number of control nodes. A comparison with ReLU networks shows how replacing the zero negative slope with a nonzero slope changes the observation-node requirement. More generally, the observability arguments require only injectivity of the activation function, whereas the controllability results extend to bijective activation functions.