  The paper is concerned with the finite-time stabilization of a hybrid PDE-ODE
system describing the motion of an overhead crane with a flexible cable. The
dynamics of the flexible cable is described by the wave equation with a
variable coefficient which is an affine function of the curvilinear abscissa
along the cable. Using several changes of variables, a backstepping
transformation, and a finite-time stable second-order ODE for the dynamics of a
conveniently chosen variable, we prove that a global finite-time stabilization
occurs for the full system constituted of the platform and the cable. The
kernel equations and the finite-time stable ODE are numerically solved in order
to compute the nonlinear feedback law, and numerical simulations validating our
finite-time stabilization approach are presented.
