Extremum-Seeking Boundary Control for Schr\"odinger-Type PDEs

TL;DR

This paper develops an extremum-seeking boundary control approach for Schrödinger-type PDEs, combining boundary actuation, real-valued functional mapping, and backstepping to achieve local stability near the extremum.

Contribution

It introduces a novel extremum-seeking control method tailored for Schrödinger-type PDEs with boundary actuation, using a real-valued quadratic functional and a two-step backstepping approach.

Findings

Proves local exponential stability near the extremum.

Establishes an isomorphism for PDE to real-valued space conversion.

Demonstrates effectiveness through a numerical example.

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.