Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusiński's operational calculus. The method is illustrated through an application to a model of a Timoshenko beam, which is clamped on a rotating disk and carries a load at its free end.

In this paper we consider a free boundary problem for a nonlinearparabolic partial differential equation. In particular, we areconcerned with the inverse problem, which means we know thebehavior of the free boundary a priori and would like a solution,e.g. a convergent series, in order to determine what thetrajectories of the system should be for steady-state tosteady-state boundary control. In this paper we combine twoissues: the free boundary (Stefan) problem with a quadraticnonlinearity. We prove convergence of a series solution and give adetailed parametric study on the series radius of convergence.Moreover, we prove that the parametrization can indeed can be usedfor motion planning purposes; computation of the open loop motionplanning is straightforward. Simulation results are given and weprove some important properties about the solution. Namely, a weakmaximum principle is derived for the dynamics, stating that themaximum is on the boundary. Also, we prove asymptotic positivenessof the solution, a physical requirement over the entire domain, asthe transient time from one steady-state to another gets large.