Extremum-Seeking Boundary Control for Schr\"odinger-Type PDEs
Abstract
This paper addresses the design and analysis of an extremum-seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by complex-valued partial differential equations (PDEs) of Schrodinger type. The system is actuated at one boundary, and the map input is defined as a real-valued quadratic functional corresponding to the squared norm of the complex state at the uncontrolled boundary. An isomorphism between the complex Hilbert space and its two-dimensional real-valued representation is established to enable the use of the standard multivariable Newton-based ES method. To compensate for the PDE actuation dynamics, a boundary control strategy based on a two-step backstepping procedure is employed. With a perturbation-based estimate of the Hessian inverse, the local exponential stability to a small neighborhood of the unknown extremum point is proved. A numerical example illustrates the effectiveness of the proposed extremum-seeking methodology.
Cite
@article{arxiv.2511.11994,
title = {Extremum-Seeking Boundary Control for Schr\"odinger-Type PDEs},
author = {Paulo Henrique Foganholo Biazetto and Gustavo Artur de Andrade and Tiago Roux Oliveira and Miroslav Krstic},
journal= {arXiv preprint arXiv:2511.11994},
year = {2025}
}