OPEN-LOOP STABILIZABILITY OF INFINITE-DIMENSIONAL SYSTEMS

In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations (x) over dot = Ax Bu in an infinite-dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, an d by various general definitions of closed-loop stabilizability. Here, A is the generator of a strongly continuous semigroup, and we make ve ry few a priori restrictions on the class of controls u. Our results h inge upon the control operator B 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 quite strong. A focus of this paper is to give necessary conditions fur open-loop stabilizability, thus identifying classes of systems which are not open-loop stabilizable. First we give useful fr equency domain conditions that are equivalent to our definitions of op en-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 invo lve spectral properties of A. We show that these results are not true if the conditions on B are weakened. We obtain analogous results for d iscrete-time systems. We show that, for a class of systems without spe ctrum determined growth, optimizability is impossible. Finally, we sho w that a system is open-loop stabilizable with a class of controls u i f and only if the system with the same A but a more bounded B is open- loop stabilizable with a larger class of controls.