Implicit Neural Networks as Static Controllers: Certificates and Performance Separation

What the math gives to ML

The paper offers a strong transfer mechanism: represent a neural controller as a trainable linear interconnection closed through an implicit activation map, making well-posedness, stability, and quadratic performance certifiable. The most directly useful neural-network adaptation is a certified implicit layer whose fixed point is guaranteed unique by a contraction or Perron-Frobenius condition, while gradients are computed by solving the corresponding implicit linear system. A second transfer is to train feedback networks subject to explicit well-posedness constraints and accept them only when an independent Lyapunov/IQC or discounted-cost certificate is feasible. These mechanisms predict sharp failure boundaries in spectral gain and closed-loop stability rather than relying only on benchmark improvements.