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- W2017034164 abstract "This article concerns the interrelation between the existence of a bounded solution Π to the operator equation Π S = A Π + 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 C 0 -semigroups T A ( t ) and T S ( 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 T A ( t ) , z ( t ) asymptotically inherits the dynamics of the inputs if there exists Π ∈ L ( W , Z ) such that Π S = A Π + BQ in D ( S ) . If T A ( t ) and T S ( t ) are bounded, then z ( t ) is bounded and uniformly continuous provided that Π S = A Π + BQ in D ( S ) . For the converse we show that if z ( t ) asymptotically inherits the dynamics of the inputs and if T S ( t ) is a suitable C 0 -group, then T A ( t ) is strongly stable and there exists Π ∈ L ( W , Z ) such that Π S = A Π + BQ in a subspace of D ( S ) . We also discuss why inputs u ( t ) frequently completely determine the asymptotic properties of z ( t ) if T A ( t ) is exponentially stable. As an application, we consider almost periodic inputs u ( t ) in Sobolev spaces H ( U , f n , ω n ) ." @default.
- W2017034164 created "2016-06-24" @default.
- W2017034164 creator A5031675737 @default.
- W2017034164 date "2005-11-01" @default.
- W2017034164 modified "2023-09-27" @default.
- W2017034164 title "Function generators, the operator equation <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:mi mathvariant=normal>Π</mml:mi><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mi mathvariant=normal>Π</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant=italic>BQ</mml:mi></mml:math> and inherited dynamics in inhomogenous Cauchy problems" @default.
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- W2017034164 doi "https://doi.org/10.1016/j.jde.2005.07.018" @default.
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