Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors

Abstract: This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error $\epsilon$. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.