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Control of the surface of a fluid by a wavemaker. (English) Zbl 1094.93014

This is a lengthy article by an expert on control of the Korteweg-de Vries equation, which is here presented in the commonly accepted form: \[ y_t+y_x+yy_x +y_{xxxx}=0, \quad 0<t<T, \quad x\in(0,L), \quad (y-1/6y^2+ y_{xx})_{x=0}=h.\tag{1} \] The wave is generated in a canal by a moving vertical wall device called the wave maker, with all waves moving in a fixed direction (call it from left to right). If the channel is long and only two-way waves of small amplitude are considered, the widely accepted model is the Boussinesq system: \[ h_t+(uh)_x=0,\quad u_t+uu_x+gh_x+ 1/3 h_0h_{xt}=0.\tag{2} \] The author asserts that no control is applied from the right end of the channel. The existence and uniqueness of \(L^2\cap H^1\) solutions of problem (1) is the main theorem of this article. The technique follows a similar proof of null controllability for the Burgers equation. The second most important result is the lack of global controllability. Excluding effects of waves from right to left (which is very reasonable), the author observes that in that case large solutions of the KdV equations behave just like solutions to the much simpler Hopf system \(y_t+yy_x=0\) (ignoring effects of \(y_x+y_{xxxx})\) when a negative wave cannot be generated by the left end control. This article contains important estimates, much clever manipulation, and several minor results, such as smoothness of the trajectory, and an interesting application of the Schauder fixed-point theorem. An interesting exercise of conversion from Eulerian to Lagrangian coordinates shows some possible benefits of the Lagrangian viewpoint.

MSC:

93C20 Control/observation systems governed by partial differential equations
35B37 PDE in connection with control problems (MSC2000)
35Q53 KdV equations (Korteweg-de Vries equations)
93B05 Controllability
49J20 Existence theories for optimal control problems involving partial differential equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B75 Flow control and optimization for incompressible inviscid fluids
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