Structure-preserving Koopman predictive control for memristive neural dynamics: input-exact and commutator-defect lifting

Abstract

Finite-dimensional Koopman MPC for nonlinear controlled systems requires care when a learned LTI lift is used as a finite-horizon surrogate. We recast the memristive Hindmarsh–Rose benchmark through an input-exact Lie-lifting certificate. Because the stimulation vector field is g=e1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=e_1$$\end{document}, the augmented polynomial dictionary is closed under Lg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_g$$\end{document}; the pure stimulation flow is represented exactly by a nilpotent matrix exponential. Moreover, the drift–input commutator cascade terminates after three input commutators, so the controlled Koopman error can be written as a finite shifted-drift defect rather than an uncontrolled truncation heuristic. The resulting theory supports a bilinear, stimulation-aligned surrogate and places the affine EDMDc-MPC implementation in a conservative finite-horizon deterministic setting. Paired comparisons with Hermite and SINDy polynomial baselines, controlled-pulse prediction, measurement-noise stress tests, and affine-versus-bilinear Lie-MPC evaluations show that the bilinear Lie model gives the lowest controlled-prediction error, closed-loop RMSE, and control energy in the deterministic benchmark.