Exponential stabilization and motion planning of an overhead crane system

Two control problems for an overhead crane system are investigated: exponential stabilization and motion planning. The considered crane system is modelled by a hyperbolic PDE containing spatially varying terms. These terms influence the spectrum location of the open-loop system and therefore affect the analysis of the control problems under investigation. The first problem is solved in-semigroups context using an appropriate Lyapunov functional. The suggested stabilizer is explicitly constructed by means of a collocated-type controller of an auxiliary system combined with a term including the kernel of a suitable Volterre transformation. The second problem is solved using the Laplace transform of an adequate input-output function. Further, the explicit expressions of the motion planner and the corresponding full-state are provided. The two pursued methods are systematic and offer an intrinsic insight about the mechanism of the controller's construction.