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 p rovide a framework to study the stability of general linear control systems . These include autonomous and nonautonomous 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-varyi ng systems. In particular, the complex stability radius may be expressed ex plicitly in terms of the generator of an (evolution) semigroup. Examples ar e given to show that classical formulas for the stability radius of an auto nomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are proven for Banach-space systems. In addi tion, it is shown that the theory of evolution semigroups allows for a stra ightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonauto nomous Banach-space systems. In particular, for the nonautonomous setting, internal stability is shown to be equivalent to input-output stability for stabilizable and detectable systems. For the autonomous setting, an explici t formula for the norm of input-output operator is given.