Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay
Abstract
Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto--Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor--backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.
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BibTeX
@misc{cisse2026predictorbased,
title = {{Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay}},
author = {Amadou Cissé},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00048},
eprint = {2610.00048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00048},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay
AU - Amadou Cissé
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00048
UR - https://arxiv.org/abs/2610.00048
AB - Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto--Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor--backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.
ER -