Stability of input-output maps and their minimal realizations in state-linear, state-affine, LPV, and linear switched systems

What the math gives to ML

The paper's transferable asset is an input-output notion of stability for input-dependent linear state updates: uniform forgetting of past inputs is equivalent to exponential decay and, for minimal realizations, determines the decay rate of the hidden state. This suggests designing recurrent or state-space neural layers around a joint contraction condition over all input-conditioned transition matrices, rather than checking stability for one fixed matrix or one training trajectory. A practical adaptation is to regularize or parameterize the transition family with a common quadratic Lyapunov metric and to monitor finite-horizon joint-spectral-radius estimates. The main expected benefit is prevention of exploding hidden states and long-horizon sensitivity while retaining input-dependent dynamics.