Stable, but not fast
A mathematical analysis of a damped, linearized periodic Whitham-Boussinesq system finds stabilization in a precise but limited sense. For initial data in the domain of the damped generator, the operator governing time evolution, the state norm declines at order 1/t and the energy at order 1/t^2. When damping is genuinely localized in the first component, the paper identifies both exponents as optimal among polynomial rates, or rates described by powers of time.
Here, strong stability means that the damped evolution converges to zero for every initial state in the energy space. The analysis also states that the damped generator has no spectrum on the imaginary axis. That result does not by itself provide a uniform decay rate or establish exponential stability.
A model built around feedback
The work studies a mathematical system rather than an empirical sample. It fixes the periodic domain T = R/(2pi Z) and a parameter beta > 0, then examines Fourier modes and function-space states of the system. The paper states that no new data were generated.
The damping is specified by the feedback rules f = -a h and g = -b P_beta u for the two components. In the operator framework, both the conservative and damped dynamics are well posed. The damped evolution operator, called a semigroup, is contractive, and its energy derivative is the negative of the two squared damping norms.
To handle high frequencies, the proof changes variables and reduces the problem to the scalar operator L_beta = T_beta partial_x. After multiplication by beta, it becomes |D| plus a smoothing Fourier multiplier R_beta. That reduction supplies the route to the paper's resolvent estimates. The resolvent is the frequency-domain operator used to track how large the response can be at a chosen frequency.
The spectral picture
Without damping, the spectrum consists of zero and paired imaginary frequencies, written as plus or minus i omega_n. At high frequency, those frequencies grow like the square root of the mode number. That sublinear growth is the baseline for the damping analysis.
After damping is added, the generator has no spectrum on the imaginary axis, and its contraction semigroup converges strongly to zero for every energy-space initial state. The paper therefore establishes strong stability, while stopping short of a uniform rate for all initial states.
Why the open region matters
The sharpness result depends on a geometric condition. There must be a nonempty open region in which the first-component damping vanishes. Under that genuine localization, the authors construct high-frequency quasimodes, approximate modes used to probe the system, and obtain an unbounded resolvent on the imaginary axis. Along a sequence of frequencies, the lower bound grows linearly.
The upper estimate is also linear at high frequency, and it does not require the additional genuine-localization condition. Where the open undamped region exists, the upper and lower estimates therefore agree in order. That is the mathematical reason the polynomial exponents are described as optimal.
The rates come with a domain restriction. The paper states them for initial data in D(Ad), the domain of the damped generator, not for arbitrary energy-space data. Strong stability itself is stated for every state in that energy space, but the broader statement does not carry a uniform decay rate.
A result with a clear boundary
The work remains a statement about the linearized periodic model. It is not an empirical measurement, and the optimality claim does not cover damping that acts everywhere on the torus or lacks an open zero region.
The supplied document is an arXiv version 1 preprint dated 25 Aug 2026. Its acknowledgements report partial support for R. de A. Capistrano-Filho from CAPES/COFECUB, CNPq and PROPG (UFPE), with CAPES funding the article processing charge. The paper declares no conflicts of interest.
Paper data and sources
Original title: Optimal Polynomial Stabilization of the Linearized Periodic Whitham--Boussinesq System
Authors: Roberto de A. Capistrano Filho, William Artiles Roqueta
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text