Open-loop stabilizability of infinite-dimensional systems

Abstract

In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations$\dot x$=Ax+Bu in an infinite-dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, and by various general definitions of closedloop stabilizability. Here,A is the generator of a strongly continuous semigroup, and we make very few a priori restrictions on the class of controlsu. Our results hinge upon the control operatorB being smoothly left-invertible, which is a very mild restriction when the input space is finite-dimensional. Since open-loop stabilizability is a weak concept, lack of open-loop stability is quites strong. A focus of this paper is to give necessary conditions for open-loop stabilizability, thus identifying classes of systems which are not open-loops stabilizable.

First we give useful frequency domain conditions that are equivalent to our definitions of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. When the input space is finite-dimensional, we give necessary conditions for open-loop stabilizability which involve spectral properties ofA. We show that these results are not true if the conditions onB are weakened. We obtain analogous results for discrete-time systems. We show that, for a class of systems without spectrum determined growth, optimizability is impossible. Finally, we show that a system is open-loop stabilizable with a class of controlu if and only if the system with the sameA but a more boundedB is open-loop stabilizable with a larger class of controls.