Implicit Neural Networks as Static Controllers: Certificates and Performance Separation

• URL: http://arxiv.org/abs/2607.11122v1
• Date: Mon, 13 Jul 2026 05:56:49 GMT
• Title: Implicit Neural Networks as Static Controllers: Certificates and Performance Separation
• Abstract summary: Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point equation<n>We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map.<n>For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron--Frobenius and norm conditions for well posedness.<n>We then formulate as a certification synthesis search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas
• Score: 0.8747606955991705
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point {equation}; they include as special cases neural network controllers. We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map, thereby making well-posedness and Lyapunov/IQC analysis mathematically easy to handle. For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron--Frobenius and norm conditions for well posedness, LMI/IQC certificates for exponential stability, and LMIs for discounted infinite-horizon quadratic performance. We then formulate synthesis as a certification-compatible heuristic search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas provide gradients, and the resulting controller is accepted only after independent post-training LMIs or regional admissibility checks are feasible. Finally, we establish constrained-control separation results: for a specific scalar unstable plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. Additional results cover quadratic state-input costs, comparison with linear static output feedback, and computable upper/lower-bound certificates. Numerical examples illustrate the mechanism and the resulting certified performance.

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