A motion planning problem for anti-damped wave equation

Abstract

 
A motion planning strategy is developed for a boundary-controlled system governed by a one-dimensional wave equation subject to spatially distributed anti-damping and boundary actuation. The goal is to construct a boundary input that enforces the tracking of a prescribed trajectory at the system output. Our proposed methodology relies on a Volterra-type integral transformation of backstepping nature, which converts the original dynamics into a suitably chosen auxiliary system. The transformed system is analyzed within a semigroup framework, which enables the use of Laplace transform techniques to derive explicit representations of the system trajectory and the associated control input. The state and control of the original system are subsequently recovered through the inverse transformation, yielding a constructive solution to the motion planning problem expressed in terms of the prescribed trajectory and the solutions of the associated kernel partial differential equations.