Exact Controllability and Boundary Stabilization of A One-Dimensional Vibrating String With a Point Mass in its Interior

Abstract

The dynamics of hybrid systems have attracted the attention of control theorists in the recent past, the control and stabilization of these systems is of paramount importance. The complexity of the structure has increased over time. There has been need to investigate the dynamics of such complex systems as to establish their optimal operational requirements. Earlier researches on the control and stabilization of these systems have been done with homogeneous system models. These earlier works have been criticized as being unable to address the ever-growing complexities of such systems. This research project has examined the problems of boundary control and stabilization for a one- dimensional vibrating string with an interior point mass. We have investigated what happens to the singularities in waves as they cross a point mass. In the case of an interior point mass for a vibrating string with the point mass placed at the point x=0, for example, with the L2- Dirichlet control at the left end, we have established the most reachable spaces from position of the point mass both to the left and to the right of the point mass. We have studied the problem of a one-dimensional wave equation with an interior point mass and have established a way of regulating the boundary vibrations of this system in a way that the vibration at a time t=T coincide with a function fixed in advance. This project has given an appropriate way of choosing a control u from the admissible set of controls U for a system with the state y so that the system is exactly controllable.