On the controllability of a Boussinesq system for two-way propagation of dispersive waves

On the controllability of a model system for long waves in nonlinear dispersive media

“…Notice that a fixed point of Γ gives a controlled solution of system (1)- (3). Therefore, if we show that Γ is a contraction in some ball B(0, R) of the space C([0, T]; V), the result is proved.…”

“…This section is devoted do analyse the well-posedness and the controllability properties of the full system (1)- (3). In what concerns the well-posedness, we consider a more general system:…”

“…From our point of view, the choice of the periodic boundary conditions (2) is very convenient, since it allows an explicit spectral analysis of the linear problem. Other types of boundary conditions have been considered in similar contexts (see, for instance [3], where Dirichlet type conditions are imposed) but, to our knowledge, the controllability problem has not been addressed in the infinite domain setting.…”

Long-time asymptotics of a linear higher-order water wave model

“…The parameters appearing in (2) and (3) satisfy the conditions a + b + c + d = 1 3 , γ 1 + γ 2 = 1 6 , γ = 1 24 (5 − 18γ 1 ), and δ 2 − δ 1 = 19 360 − 1 6 γ 1 , where δ 1 > 0 and γ 1 > 0. Equation (1) describes the unidirectional propagation of water waves and is known as the fifth-order KdV-BBM type equation.…”

“…Controllability results for system (4) are studied in [1] when a control localized in the interior of the domain is considered and acts on only one equation. Some negative results of controllability are studied in [2,23] when the KdV terms are dropped in the systems above, i.e. (4) with a = a 1 = c = c 1 = 0 and (5) with a = c = 0.…”