Control Time for Gravity-Capillary Waves on Water

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.…”

Asymptotic behaviour of a linearized water waves system in a rectangle

“…Different with the control introduced in the system (1.4), Alazard discussed in [6] the stabilization of the nonlinear water waves system in a rectangle where the external pressure as the control signal acts on a part of the free surface, by absorbing the waves coming from the left. For the problem in a cubic domain, in an irregular domain and the case of the water waves with surface tension, please refer to Reid [7] and [8], Craig et al [9], Alazard et al [10] and [11]. Recently, for u ∈ L 2 loc [0, ∞), we established in our paper [1] the well-posedness of the system (1.4), and further showed that it can be recast as a well-posed linear control system (for this concept, please refer to [1], Weiss [12] or Tucsnak and Weiss [13]).…”

Boundary observability of gravity water waves

“…where we used the Poincaré inequality in the first line. Now, since η ≥ −h/2 by assumption, using the identity (16) we see that the left hand side of (19) is larger than the sum of the last two terms in the right hand side of (19) and of…”

“…which, by assumption (17), is larger than H. ii) (Unique continuation) If m = 0 then (19) implies that H = 0 and hence η = 0 and φ = 0.…”