| Popis: |
This thesis concerns the well-posedness and controllability of certain dispersive partial differential equations. The first part focuses on an initial boundary value problem (IBVP) for the Korteweg-de Vries equation posed on a bounded interval with specific nonhomogeneous boundary conditions; this problem was introduced by T. Colin and J.-M. Ghidaglia. The IBVP is shown to be well-posed on the L2-based Sobolev spaces Hs(0,L) for s= 0. Moreover, the existence of global solutions is proved for small data, and these solutions decay exponentially when the boundary data decay exponentially. This result is improved by showing that the IBVP is locally well-posed in Hs(0,L) for s>-3/4. Both results are proved using regularity properties and suitable techniques developed for KdV equations.The boundary controllability of the Korteweg-de Vries equation on a finite interval is also treated in this thesis. The cases of one and two controls are analyzed, and controllability is initially shown for the linear control system. The controllability of the system for one control is guaranteed only if the length of the interval does not belong to a specific set, but no conditions are needed for the case of two controls. These results are then extended to nonlinear cases by using the contraction mapping theorem. Finally, a well-posedness property for the IBVP of the Boussinesq equation posed on the half-line is shown in Hs(R+) for s=0, thanks to recent work on the Schrodinger equation and the regularity properties of its solutions. |
