Particular solution of infinite-dimensional linear systems with applications to trajectory planning of boundary control systems

Abstract

This paper considers a general class of infinite-dimensional linear control systems described by either a state-space (SCS) or boundary control (BCS) system formulation. A key objective of the paper is to accomplish trajectory planning of a BCS through a ‘stable’ dynamic inversion without resorting to discretization. To this end, the paper first formulates the particular solution of an infinite-dimensional SCS within a Sobolev space together with a set of necessary and sufficient conditions for its existence and an explicit formula for computing it. The resulting solution, which may be noncausal, is further utilized to explicitly compute the bounded control input needed for output tracking of a BCS without requiring its inverse to be minimum phase or even to possess a \(C_{0}\)-semigroup. The key results of the paper are illustrated on a flexible beam and a one-dimensional heat conduction system.