Function generators, the operator equation Î S=AÎ +BQ and inherited dynamics in inhomogenous Cauchy problems

This article concerns the interrelation between the existence of a bounded solution Π to the operator equation ΠS=AΠ+BQ in D(S) and the asymptotic behaviour of the mild solutions z(t) of the abstract Cauchy problem z˙(t)=Az(t)+Bu(t), t⩾0, in a Banach space Z. Here B and Q are bounded, whereas A and S generate C0-semigroups TA(t) and TS(t) on Z and W (W is a Banach space), respectively. Banach space valued inputs u(t)∈U are generated by linear dynamical systems. We define asymptotically inherited dynamics of z(t) and show that for strongly stable semigroups TA(t), z(t) asymptotically inherits the dynamics of the inputs if there exists Π∈L(W,Z) such that ΠS=AΠ+BQ in D(S). If TA(t) and TS(t) are bounded, then z(t) is bounded and uniformly continuous provided that ΠS=AΠ+BQ in D(S). For the converse we show that if z(t) asymptotically inherits the dynamics of the inputs and if TS(t) is a suitable C0-group, then TA(t) is strongly stable and there exists Π∈L(W,Z) such that ΠS=AΠ+BQ in a subspace of D(S). We also discuss why inputs u(t) frequently completely determine the asymptotic properties of z(t) if TA(t) is exponentially stable. As an application, we consider almost periodic inputs u(t) in Sobolev spaces H(U,fn,ωn).