Admissible observation operators for linear semigroups. (English) Zbl 0696.47040
The author treats the well-posedness of the following observation problem: \(y(t)=CT(t)x\), \(t\geq 0\), where T(t), \(t\geq 0\) is a \(C_ 0\)- linear semigroup in a Banach space X, \(x\in X\), and C is an unbounded densely defined operator from X to another Banach space Y. The observation operator C is admissible if \(y(t)\in L^ p(0,r;Y)\) for every \(r>0\) and y(t) depends continuously on the initial state x. the author develops a pointwise interpretation of the admissibility problem for x such that T(t)x is not in the domain of C. An analogy between admissible control operators and admissible observation operators is discussed. The results are applicable to control theory for linear partial differential equations and several examples are given for illustration.
Reviewer: G.F.Webb
MSC:
| 47D03 | Groups and semigroups of linear operators |
| 49J27 | Existence theories for problems in abstract spaces |
Keywords:
well-posedness; observation problem; \(C_ 0\)-linear semigroup in a Banach space; unbounded densely defined operator; pointwise interpretation of the admissibility problem; admissible control operators; admissible observation operatorsReferences:
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