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A Bayesian Approach to Feedback Control for Hyperbolic Balance Laws

Markus Bambach, Shaoshuai Chu, Michael Herty, Yunong Lin

TL;DR

The paper addresses stabilizing boundary control for one-dimensional hyperbolic balance laws in the form $\mathbf{U}_t + \mathbf{F}(\mathbf{U})_x = \mathbf{S}(\mathbf{U})$ by introducing a Bayesian framework that evolves a posterior over boundary gains $\kappa$ using Lyapunov decay as the data. The method is discretization-agnostic and nonintrusive, relying on a first-order LLF scheme (extensible to second order) and Bayes’ rule with a damping mechanism to concentrate probability on stabilizing parameters. Across linear and nonlinear deterministic cases, stochastic variants, nonconservative systems, and a two-parameter LPBF example, the approach recovers known stability domains and remains robust to prior choices and indicator choices. The work demonstrates a practical route to identify stabilizing boundary controls in complex, possibly nonlinear or stochastic settings, with potential extensions to networks and multi-parameter controls. The contribution is significant for computational control of hyperbolic systems, providing a data-driven, solver-compatible tool to delineate stability ranges without requiring bespoke Lyapunov functionals for each problem.

Abstract

We propose a Bayesian framework for feedback boundary control for hyperbolic balance laws. The method propagates a probability distribution over feedback parameters by using Lyapunov decay estimates as a likelihood. In the linear setting, the framework recovers the available analytical results, while simultaneously extending them to nonlinear regimes where such results are not readily accessible. We first validate the method using the first-order local Lax-Friedrichs (LLF) discretizations on linear models -- the decoupled wave system and the linearized Saint-Venant equations -- recovering the known stability intervals and mixed boundary couplings reported in the control literature. We then consider nonlinear and stochastic settings, including the nonlinear Saint-Venant system and Burgers equation with random initial data, as well as a nonconservative perturbation with source terms, and demonstrate that the computed stability domains remain accurate and robust with respect to the choice of indicator and the initial prior. We further show that the methodology carries over to a second-order semi-discrete LLF scheme and to a model with two interacting control parameters for the temperature field development in laser powder bed fusion with feedback power regulation. These numerical experiments confirm consistency with available theory on benchmark cases and highlight the practicality of the proposed, discretization-agnostic feedback selection procedure.

A Bayesian Approach to Feedback Control for Hyperbolic Balance Laws

TL;DR

The paper addresses stabilizing boundary control for one-dimensional hyperbolic balance laws in the form by introducing a Bayesian framework that evolves a posterior over boundary gains using Lyapunov decay as the data. The method is discretization-agnostic and nonintrusive, relying on a first-order LLF scheme (extensible to second order) and Bayes’ rule with a damping mechanism to concentrate probability on stabilizing parameters. Across linear and nonlinear deterministic cases, stochastic variants, nonconservative systems, and a two-parameter LPBF example, the approach recovers known stability domains and remains robust to prior choices and indicator choices. The work demonstrates a practical route to identify stabilizing boundary controls in complex, possibly nonlinear or stochastic settings, with potential extensions to networks and multi-parameter controls. The contribution is significant for computational control of hyperbolic systems, providing a data-driven, solver-compatible tool to delineate stability ranges without requiring bespoke Lyapunov functionals for each problem.

Abstract

We propose a Bayesian framework for feedback boundary control for hyperbolic balance laws. The method propagates a probability distribution over feedback parameters by using Lyapunov decay estimates as a likelihood. In the linear setting, the framework recovers the available analytical results, while simultaneously extending them to nonlinear regimes where such results are not readily accessible. We first validate the method using the first-order local Lax-Friedrichs (LLF) discretizations on linear models -- the decoupled wave system and the linearized Saint-Venant equations -- recovering the known stability intervals and mixed boundary couplings reported in the control literature. We then consider nonlinear and stochastic settings, including the nonlinear Saint-Venant system and Burgers equation with random initial data, as well as a nonconservative perturbation with source terms, and demonstrate that the computed stability domains remain accurate and robust with respect to the choice of indicator and the initial prior. We further show that the methodology carries over to a second-order semi-discrete LLF scheme and to a model with two interacting control parameters for the temperature field development in laser powder bed fusion with feedback power regulation. These numerical experiments confirm consistency with available theory on benchmark cases and highlight the practicality of the proposed, discretization-agnostic feedback selection procedure.
Paper Structure (18 sections, 2 theorems, 72 equations, 18 figures, 1 algorithm)

This paper contains 18 sections, 2 theorems, 72 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A Bayesian Boundary Feedback Method
  3. First-Order Local Lax-Friedrichs Scheme
  4. Evolution of the Probability Distribution
  5. Numerical Results
  6. Linear Deterministic Examples
  7. Nonlinear Deterministic Examples
  8. Example 4---Burgers Equation.
  9. Stochastic Examples
  10. Nonconservative System
  11. Second-Order Discretization
  12. Feedback Control of Two Parameters
  13. Conclusions
  14. Funding.
  15. Conflicts of interest.
  16. ...and 3 more sections

Key Result

Proposition 2.3

Assume that the deterministic system 1.1 with boundary conditions 2.1–2.2 satisfies the hypotheses of coron2007controlbanda2013numerical and that the numerical solution is computed by the first–order LLF scheme of §sec2.1. Then there exist constants $\nu>0$ and $\widetilde{C}\ge 0$, depending only o and, in addition, the exponential stabilization inequality in the discrete $L^2$–norm An analogous

Figures (18)

  • Figure 3.1: Example 1: Initial (left) and final probability distributions for Cases I (top row) and II (bottom row).
  • Figure 3.2: Example 1: Final probability distributions for Cases I (left) and II (right).
  • Figure 3.3: Example 2: Initial (left) and final probability distributions for Cases I (top row) and II (bottom row) for the boundary conditions (\ref{['3.3a']}).
  • Figure 3.4: Example 2: Final probability distributions for Cases I (left) and II (right) for the boundary conditions (\ref{['3.4a']}).
  • Figure 3.5: Example 3: Final probability distributions for Cases I (left) and II (right).
  • ...and 13 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Definition 2.1: Exponential Stabilization
  • Definition 2.2: Lyapunov–Type Indicators
  • Proposition 2.3: Discrete Exponential Stabilization for the LLF Scheme
  • Definition 2.4: Probability Distributions of Feedback Parameters
  • Definition 2.5: Bayes’ Formula
  • Definition 2.6: Bayesian Update Operator
  • Proposition 2.7: Positivity and Normalization
  • Remark 2.2
  • Remark 3.1