On input-output persistency and the interconnection of positive nonlinear systems

What the math gives to ML

The paper provides a nonstandard input-output boundedness mechanism for positive feedback interconnections: a plant with persistent-input/persistent-output (PIPO) behavior amplifies sustained large inputs, while a controller with persistent-input/transient-output (PITO) behavior attenuates its output when the plant output remains large. Under forward completeness and a one-sided affine bound on controller-output growth, these complementary properties imply boundedness of the interconnected control signal even without requiring a bounded controller state. A transferable neural-network construction is a positive continuous-depth recurrent or state-space module whose recurrent gain is regulated by a PITO feedback state, preventing runaway activations while retaining a measurable threshold-and-decay signature.