Stability Results for Classes of Distributed Parameter Networks and Systems

A number of stability results for general systems of linear and nonlinear partial differential equations that describe distributed parameter networks and systems are obtained by means of Liapunov´s direct method. Conditions for the asymptotic stability of the null solution of linear time-invariant distributed parameter systems are given. This result is then extended to a class of distributed parameter systems consisting of a linear system and a zero-memory nonlinear elment described by a set of Lurie-like partial differential equations. Bounded input-bounded output stability for the above two classes of systems is established by extension of a result of Goldwyn, Chao, and Chang for lumped parameter systems. Finally, it is shown how Malkin´s theorem on the stability under persistent disturbances can be extended to provide a solution to the dynamic range problem for systems described by a general set of nonlinear partial differential equations.