Semistability theory for spatially distributed systems

The concepts of semistability and exponential semistability are well-developed for finite-dimensional systems with nonisolated equilibrium points, where asymptotic or exponential stability is not possible. Definitions of semistability and exponential semistability have recently been formulated for networks with time-delays. This paper further extends the semistability theory to continuous and discrete spatially distributed systems. This requires the definition of the notions of exact and approximate semicontrollability and semiobservability, and discrete approximate semicontrollability and semiobservability. Also introduced is the property of weak semistability. Necessary and sufficient conditions are given for exponential semistability and weak semistability, and sufficient conditions are given for semistability.