A Bayesian Approach to Feedback Control for Hyperbolic Balance Laws

TL;DR

This paper introduces a Bayesian framework for feedback boundary control of hyperbolic balance laws, effectively handling linear, nonlinear, and stochastic systems, and demonstrating robustness and practical applicability across various models.

Contribution

It extends Bayesian feedback control methods to nonlinear and stochastic hyperbolic systems, providing a discretization-agnostic approach validated through multiple complex models.

Findings

Accurately recovers stability intervals for linear models.

Maintains robustness in nonlinear and stochastic settings.

Applicable to second-order schemes and multi-parameter control.

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.