Some Controllability and Stabilization Problems of Surface Waves on Water with Surface tension


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Virginia Tech


The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation

wt + wxxx - wxxxxx - nj=1 ajwjwx = 0, w(x, 0) = w0(x)

posed on a periodic domain x ∈ [0, 2π] with boundary conditions wix(0, t) = wix(2π, t), i = 0, 2, 3, 4 and an L2-stabilizing feedback control law wx(2π, t) = αwx(0, t) + (1 - α)wxxx(0; t) where n is a fixed positive integer, aj, j = 1, 2, ... n, α are real constants, and |α| < 1. It is shown that for w0(x) ∈ H1α(0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T]; H1α(0, 2π)) with a conserved volume of w, [w] = ∫0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w0(x)]/(2π) as t → + ∞. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case.



Kawahara Equation, Contraction Mapping Principle, Boundary Control, Hydrodynamics