Flatness-based trajectory planning for the linear Kuramoto-Sivashinsky equation with a single boundary input
Abstract
We study the null controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval using a single Dirichlet boundary control via the flatness method. The usual flatness representation for this fourth-order equation naturally involves two boundary traces, which serve as flat outputs. When two suitable boundary controls are available, these outputs can be chosen independently. With only one control, however, the remaining homogeneous boundary condition imposes a compatibility relation between the two flat outputs. The main novelty of this work is the construction of a one-output flat parametrization that automatically incorporates this relation. It is obtained through a weighted Bézout theorem for the entire functions associated with the boundary conditions. The required growth estimates for the Bézout coefficients are established using Hörmander's weighted ∂ estimate. Combining this parametrization with the regularizing effect of the free dynamics and a Gevrey cutoff, we construct the control and the corresponding trajectory as convergent flat series. We prove null controllability at any positive time by means of a boundary control whenever the anti-diffusion parameter lies outside a countable critical set determined by the length of the interval. In particular, the control has Gevrey regularity in time, while the state has Gevrey regularity in time and space away from the initial time.For critical parameter values, we exhibit an unobservable eigenmode and prove that null controllability under the considered boundary control fails. To the best of our knowledge, this is the first flatness-based analysis of the Kuramoto-Sivashinsky equation that successfully handles the critical-set phenomenon.
Related
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- Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay
- Global Well-Posedness of the Kuramoto--Sivashinsky Equation in One and Two Dimensions under a Codimension-One Constraint
- Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion
- Amortized Generative Modeling of Invariant Measures across A Family of Kuramoto-Sivashinsky PDEs
BibTeX
@misc{majumdar2026flatnessbased,
title = {{Flatness-based trajectory planning for the linear Kuramoto-Sivashinsky equation with a single boundary input}},
author = {Subrata Majumdar},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05765772v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Flatness-based trajectory planning for the linear Kuramoto-Sivashinsky equation with a single boundary input
AU - Subrata Majumdar
PY - 2026
DA - 2026/09/27
PB - HAL
UR - https://hal.science/hal-05765772v2
AB - We study the null controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval using a single Dirichlet boundary control via the flatness method. The usual flatness representation for this fourth-order equation naturally involves two boundary traces, which serve as flat outputs. When two suitable boundary controls are available, these outputs can be chosen independently. With only one control, however, the remaining homogeneous boundary condition imposes a compatibility relation between the two flat outputs. The main novelty of this work is the construction of a one-output flat parametrization that automatically incorporates this relation. It is obtained through a weighted Bézout theorem for the entire functions associated with the boundary conditions. The required growth estimates for the Bézout coefficients are established using Hörmander's weighted ∂ estimate. Combining this parametrization with the regularizing effect of the free dynamics and a Gevrey cutoff, we construct the control and the corresponding trajectory as convergent flat series. We prove null controllability at any positive time by means of a boundary control whenever the anti-diffusion parameter lies outside a countable critical set determined by the length of the interval. In particular, the control has Gevrey regularity in time, while the state has Gevrey regularity in time and space away from the initial time.For critical parameter values, we exhibit an unobservable eigenmode and prove that null controllability under the considered boundary control fails. To the best of our knowledge, this is the first flatness-based analysis of the Kuramoto-Sivashinsky equation that successfully handles the critical-set phenomenon.
ER -