Boundary control, quiet boundaries, super-stability and super-instability

This paper deals with the boundary feedback control of a bar undergoing axial vibrations. The feedback force is applied to one end of the bar and it is proportional to its velocity. The problem is of fundamental interest in control theory, structural dynamics, and the development of 'quiet boundaries' in fields like earthquake engineering and computational mechanics. The system is not self-adjoint and exhibits a variety of interesting behaviors which are explained through a combination of several inter-twined strands of thought using mathematics, physical interpretations, and numerical simulations. Besides providing a rigorous mathematical solution to the problem, the paper explains the physical origin of super-stable behavior, introduces the new concept of super-unstable behavior, and points out that these regimes of behavior are intricately connected with the continuum model that is used. It is shown that any finite-dimensional approximation of the system, no matter how finely discretized, can not qualitatively depict superstable behavior.