Motion Planning Problem For A 1-D Wave Equation

This paper addressed the motion planning problem (MPP) for a 1-D wave equation with boundary control. Our approach utilizes advanced control methods, such as semigroup theory and the Laplace transform, to determine a control that ensures the state output accurately follows a predefined target trajectory (sufficiently smooth) at every moment. Initially, we derive a boundary state-feedback law to solve the MPP. Subsequently, we present a detailed expression for the whole state of the resulting closed-loop system, providing a detailed formula for the MP control, expressed in terms of the target trajectory.