## Exact controllability of the Boussinesq equation on a bounded domain.(English)Zbl 1161.93302

Summary: The purpose of this article is to study the exact boundary controllability of the classical Boussinesq equation $y_{tt}-y_{xx}+y_{xxxx}+(y^2)_{xx}=0.$ The control is applied to the first spatial derivative and then to the second spatial derivative both at the right endpoint. The exact controllability of the linearized problem is essentially proved by using the Hilbert uniqueness method. From this result, we deduce the exact boundary controllability for the nonlinear Boussinesq equations. The main improvements compared to existing results in the literature are the absence of restrictions on controllability time, the use of more regular spaces and the extension of the exact boundary controllability to the nonlinear Boussinesq equation.

### MSC:

 93B05 Controllability 35Q53 KdV equations (Korteweg-de Vries equations) 76D55 Flow control and optimization for incompressible viscous fluids 93C20 Control/observation systems governed by partial differential equations