  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 systems
