Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach

In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are provided for these general systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systemsComment: Also at http://www.math.missouri.edu/~stephen/preprint

Some remarks on adaptive stabilization of infinite-dimensional systems

It is the purpose of this note to show that a first-order adaptive controller stabilizes a large class of infinite-dimensional systems described by strongly continous semigroups. It is assumed that the plant is minimum-phase and has invertible high-frequency gain. Knowledge of the sign of the high-frequency gain is not required