Stability and boundedness of continuous- and discrete-time systems

is stable. For infinite-dimensional systems the situation is more complicated. Although the eigenvalues still possess the same property as in the finite dimensional situation, they don’t longer rule the stability. Since many years it is an open question whether the stability of (1.1) and (1.2) are equivalent. For Banach spaces it is not hard to find a counter example, and so we concentrate on the situation where the state space is a Hilbert space. For Hilbert spaces there are some positive results, such as for dissipative operator and for normal operators. The results of Crouzeix et. al. [1] and Palencia [5] imply that if A is the infinitesimal generator of the holomorphic semigroup T (t) which is sectorially bounded, i.e.,