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Extremum-Seeking Boundary Control for Schrödinger-Type PDEs

Paulo Henrique Foganholo Biazetto, Gustavo Artur de Andrade, Tiago Roux Oliveira, Miroslav Krstic

TL;DR

The paper tackles real-time optimization of a boundary-controlled Schrödinger-type PDE with an unknown quadratic performance map. It advances a novel ES framework in which complex-valued states are mapped to a real $2\times2$ representation, enabling Newton-based Hessian inverse estimation, while a two-step backstepping strategy compensates for actuation dynamics. The main contributions are (i) first application of ES to complex-valued PDEs, (ii) a constructive backstepping-based actuation compensation, and (iii) an averaging-based analysis proving convergence to a neighborhood of the extremum with local exponential stability of the Schrödinger dynamics. Numerical results validate the method and illustrate convergence behavior under the proposed ES law. This work broadens data-driven adaptive optimization for dispersive PDEs and offers a practically implementable boundary-control approach for quantum- and wave-like systems.

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.

Extremum-Seeking Boundary Control for Schrödinger-Type PDEs

TL;DR

The paper tackles real-time optimization of a boundary-controlled Schrödinger-type PDE with an unknown quadratic performance map. It advances a novel ES framework in which complex-valued states are mapped to a real representation, enabling Newton-based Hessian inverse estimation, while a two-step backstepping strategy compensates for actuation dynamics. The main contributions are (i) first application of ES to complex-valued PDEs, (ii) a constructive backstepping-based actuation compensation, and (iii) an averaging-based analysis proving convergence to a neighborhood of the extremum with local exponential stability of the Schrödinger dynamics. Numerical results validate the method and illustrate convergence behavior under the proposed ES law. This work broadens data-driven adaptive optimization for dispersive PDEs and offers a practically implementable boundary-control approach for quantum- and wave-like systems.

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.

Paper Structure

This paper contains 16 sections, 4 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Schrodinger Mathematical Model
  4. Control Problem
  5. Extremum Seeking Boundary Control Design
  6. Demodulation Signals
  7. Additive Probing Signal
  8. Estimation Errors and PDE-Error Dynamics
  9. Schrodinger Compensation via Backstepping Boundary Control
  10. Target System
  11. Backstepping transformation
  12. Invertibility of the Transformations
  13. Implementable extremum seeking control law
  14. Stability and convergence analysis
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Lemma 1

The solution of problem eq:trajectoryPDE--eq:trajectorybc03 is where

Figures (4)

  • Figure 1: Block diagram of the ESC system applied to the Schrödinger problem.
  • Figure 2: The closed-loop response of the Schrödinger PDE with ES compensating controller.
  • Figure 3: The response of the real part of the Schrödinger equation evolving in time and space.
  • Figure 4: The response of the imaginary part of the Schrödinger equation evolving in time and space.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1