Neural-Spectral and Newton-HUM Methods for Reconstructing Controls of Wave Equations

Abstract

We develop a framework for constructing and rigorously validating controls for semilinear wave equations with localized interior control. We analyze the continuous energy-space problem and certify exact control for finite-dimensional Galerkin systems. Under explicit smallness and neighborhood conditions, our a posteriori theorem proves the existence of an exact Galerkin control near a computed candidate. It combines a validated terminal-state error, a bounded right inverse of the derivative of the control-to-terminal-state map, and a Lipschitz bound for that derivative. Under additional uniform right-inverse and neighborhood conditions, we also show that Newton correction converges with quadratic decay of terminal-state errors. We use residual estimates and interval arithmetic to enclose the nonlinear trajectory and its linearized controllability Gramian. For the specified eight-mode tanh problem, we rigorously bound a source-response operator to prove contraction of the exact source-frozen iteration. We compare deterministic controls based on the Hilbert Uniqueness Method with neural approximations of temporal modal coefficients. Two Newton corrections reduce the computed terminal-state errors of all six tested neural candidates below $10^{-8}$. For the three certified candidates, we obtain smaller bounds on the distance to an exact Galerkin steering control by refining the validation mesh while keeping the computed controls fixed.

arXiv commentNewton-HUM method is added to the study