Infinite-Dimensional Systems Theory

Abstract

Systems modelled by linear ordinary differential equations are generally written as a set of n first-order differential equations with solution taking values in R. Distributed parameter systems are modelled by partial differential equations but can be written in a similar way. The main difference is that the state-space is no longer R, but an infinite-dimensional Hilbert space, and the matrix A is no longer a matrix, but an operator acting on this infinite-dimensional space. Consequently, systems with partial differential equation models are also called infinite-dimensional systems. In this chapter the familiar state-space framework is extended to distributed parameter systems. Some techniques for establishing well-posedness are described and illustrated with examples. Although there are many similarities, the systems theory for infinite-dimensional systems differs in some important aspects from that for finite-dimensional systems. Some of the differences can be seen by looking at several relatively simple examples. Including a model for the actuator often changes a simple model with an unbounded actuator or sensor to one with bounded control or sensing. This is also described.