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arXiv · 2603.20877

On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves

Wim Michiels·Federico Bribiesca-Argomedo·Jean Auriol
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Abstract

The paper addresses the boundary control of a class of hyperbolic PDEs, based on an equivalent representation in terms of an integral-difference equation. The situation is considered where direct compensation of reflection terms induces a fragile closed-loop system, in the sense of lack of strong stability. This is theoretically resolved by adding a low-pass filter to the control law, but the choice of its cut-off frequency is crucial in balancing robustness at high frequencies and performance at low frequencies. First, the maximum stability interval in parameter $T$ is determined, with $T$ the inverse of the filter's cutoff frequency. Next, model mismatch on the PDE parameters is considered and a sufficient stability condition is derived in terms of allowable mismatch and cut-off frequency, satisfied in a region in the combined parameter space with a conic shape around $T=0$. Finally, this qualitative behavior is confirmed by exact stability charts for a special case where all model mismatch is contained into one parameter. It is highlighted that the set of stability crossing curves exhibits a fractal-like structure, which is explained using a limit system with discrete delays.

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Wim Michiels, Federico Bribiesca-Argomedo, Jean Auriol. 2026-03-21. On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves. https://arxiv.org/abs/2603.20877

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