[Paper Review] Control of three dimensional water waves

Hui Zhu|arXiv (Cornell University)|Dec 17, 2017

Stability and Controllability of Differential Equations45 references3 citations

TL;DR

This paper establishes the exact controllability of three-dimensional water waves with surface tension under localized exterior pressures, proving that any sufficiently small and regular initial and final wave states can be connected within arbitrarily short time. The proof combines an iterative scheme for reducing the quasi-linear system to linear problems and a semiclassical approach for high frequencies, with a uniqueness-compactness argument for low frequencies, under the geometric control condition on the control region.

ABSTRACT

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard, Baldi, and Han-Kwan for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.

Motivation & Objective

• To establish exact controllability for three-dimensional water waves with surface tension on the torus.
• To extend the 2D controllability result of Alazard, Baldi, and Han-Kwan to higher dimensions.
• To address the lack of Ingham's inequality in higher dimensions by distinguishing high and low frequency regimes.
• To prove controllability within arbitrarily short time for small, regular data under the geometric control condition.
• To ensure conservation of zero-frequency modes by imposing mean-zero constraints on the wave height.

Proposed method

• The proof uses an iterative scheme to reduce the nonlinear quasi-linear water wave system to a sequence of linear control problems.
• For high frequencies, a semiclassical approach inspired by Lebeau and Burq–Zworski is applied to handle the linearized equations.
• For low frequencies, the uniqueness-compactness method of Bardos–Lebeau–Rauch is employed to establish controllability.
• The control is localized in space and time, with the exterior pressure $ P_{\mathrm{ext}} $ supported in a region $ \omega \subset \mathbb{T}^d $ satisfying the geometric control condition.
• The analysis relies on the Zakharov/Craig–Sulem formulation of the water wave system, involving the Dirichlet–Neumann operator and surface tension effects.
• Energy estimates and regularity bounds are derived via paradifferential calculus and operator theory in Sobolev spaces.

Experimental results

Research questions

• RQ1 Can three-dimensional water waves with surface tension be exactly controlled using localized exterior pressures within arbitrarily short time?
• RQ2 Is the geometric control condition both sufficient and necessary for controllability in 3D water wave systems?
• RQ3 How can the absence of Ingham’s inequality in higher dimensions be overcome in the control theory of quasi-linear PDEs?
• RQ4 What role does the mean curvature term $ H(\eta) $ play in the controllability of the nonlinear water wave system?
• RQ5 Can the 2D controllability result by Alazard, Baldi, and Han-Kwan be extended to 3D using different analytical techniques?

Key findings

• Exact controllability holds for three-dimensional water waves with surface tension within arbitrarily short time for sufficiently small and regular initial and final data.
• The geometric control condition is necessary and sufficient for controllability, ensuring every geodesic in $ \mathbb{T}^d $ eventually enters the control region $ \omega $.
• The result extends to infinite depth with the same proof, though finite depth is considered for simplicity.
• The high-frequency regime is controlled via a semiclassical method, while the low-frequency regime relies on the uniqueness-compactness argument.
• The control pressure $ P_{\mathrm{ext}} $ is shown to exist in $ C([0,T], H^s(\mathbb{T}^d)) $, compactly supported in $ \omega $, and real-valued.
• The solution $ (\eta, \psi) $ exists uniquely in $ C([0,T], H^{s+1/2} \times H^s) $, matching the initial and final states.