Considered here is a higher-order generalization of the classical Boussinesq system which models the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal or the propagation of long-crested waves on large lakes and oceans. Our aim is to investigate the controllability properties of this nonlinear model in terms of the values of the parameters involved in the system. We give general conditions which ensure both the well-posedness and the local exact controllability of the nonlinear problem in some well chosen Sobolev spaces.
In this paper we are concerned with a Boussinesq system for smallamplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.
<p style='text-indent:20px;'>The paper deals with the internal controllability and stabilizability of a family of Boussinesq systems introduced by J. L. Bona, M. Chen and J.-C. Saut to describe two-way propagation of small amplitude gravity waves on the surface of water in a canal. By applying the moment method, we first obtain the exact controllability of the linearized system in suitable Hilbert spaces, which implies its exponential stabilization by suitable feedback. Then, by using a contraction mapping principle, we establish the local exact controllability and exponential stabilization for the original nonlinear system.</p>
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