Evolution semigroups and stability of time-varying systems on Banach spaces

The two main topics addressed are: (i) the relationship between internal, external, and input-output stability, and (ii) stability of time-invariant systems including a new Banach-space formula for the stability radius. With regard to (i), we show that a nonautonomous system is internally stable if and only if it is stabilizable, detectable and input-output stable; the short proof seems to be new even for finite-dimensional autonomous systems. For (ii), new formulas are given: in terms of the coefficients of the system, for the L-p-norm of the input-output operator and for the stability radius of the system.