Stability properties of linear systems in phase-variable form

At least two types of stability may be defined for linear, finite-dimensional, dynamic systems: Lyapunov stability, reflecting the internal stability of the system, and bounded-input, bounded-output stability, reflecting the external stability of the system. Although connections between these two types of stability are well understood for time-invariant systems, this is not the case for time-varying systems, as witnessed by past controversy and present interest in the topic. A connection between the two types of stability is established for systems which may be represented in a special canonical form; the extent to which systems can be put into this canonical form has been discussed by others and is reviewed at the end of the paper.