Motion planning problem for a class of hyperbolic PDEs

The purpose of this paper is to investigate the motion planning problem (MPP) for a class of hyperbolic PDEs with a boundary control. Precisely, we aim to construct an input control leading the state output to track a given (sufficiently smooth) target trajectory. First, we establish a solution of the MPP in the form of a boundary state-feedback law. Second, we provide an explicit expression of the whole state of the resulting closed-loop. Third, we furnish an explicit formula of the MP control via the target trajectory and the solution of an appropriate PDE. Besides, the provided control leads to a complete matching of the output with the predefined trajectory at any instant. The control problem is reformulated in the context of operator semigroup theory and solved using the Laplace transform of a suitable input-output function.