Admissible observation operators for linear semigroups
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Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=C T t x,t≧0, should be inL p and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=C T t x, even if the trajectory ofx is not in the domain ofC.
KeywordsBanach Space Continuous Semigroup Distribute Parameter System Graph Norm Lebesgue Point
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