Open Loop Control of Water Waves in an Irregular Domain

Stabilization of the Water-Wave Equations with Surface Tension

“…However, one cannot easily adapt these studies to the water-wave system (2.8) since the latter system is quasi-linear (instead of semi-linear) and since it is a pseudo-differential system, involving the Dirichlet-Neumann operator which is nonlocal and also depends nonlinearly on the unknown. The first results about the possible applications of control theory to the water-wave equations are due to Reid and Russell [30] and Reid [28,29], who studied the linearized equations at the origin (see also Miller [27], Lissy [26] and Biccari [7] for other control results about dispersive equations involving a fractional Laplacian). Alazard, Baldi and Han-Kwan proved in [3] the first controllability result for the nonlinear water-wave equations with surface tension, namely a controllability result in arbitrarily small time, under a smallness assumption on the size of the data.…”

Estimation of velocity potential of water waves using a Luenberger-like observer

“…In this section, we propose a finite difference scheme for solving the LWWE (4) and observer equation (5). Then we demonstrate the effectiveness of the observer with an example.…”

“…In applications, these estimated states can be used to design state feedback control laws [3]. An open-loop control for water waves in a finite interval was studied in [4,5]. These studies reported that water waves are stabilizable over an infinite time interval.…”

Boundary observability of gravity water waves

“…x -norm of the trace ψ. Hence √ H does not control the L 2 -norm of ∂ x ψ and this why we need the assumption for ψ in (18). Now, notice that this assumption for ∂ x ψ holds at t = 0 with A = K √ N if the Fourier transform of ψ(0) is supported in [−N, N ] (as in (15)).…”