Abstract
We discuss a class of linear control problems in a Hilbert space setting. The aim is to show that these control problems fit in a particular class of evolutionary equations such that the discussion of well-posedness becomes easily accessible. Furthermore, we study the notion of conservativity. For this purpose we require additional regularity properties of the solution operator in order to allow point-wise evaluations of the solution. We exemplify our findings by a system with unbounded control and observation operators.
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Notes
- 1.
In this case
is a continuous linear operator.
- 2.
Note that in the fourth equality \(\langle \cdot | \cdot \rangle_{H_{0} (\sqrt{A_{0}}+\mathrm {i})}\) is used not as the inner product in \(H_{0} (\sqrt{A_{0}}+\mathrm {i})\) but as its continuous extension to the duality pairing between \(H_{-1} (\sqrt{A_{0}}+\mathrm {i})\) and \(H_{1} (\sqrt{A_{0}}+\mathrm {i})\). This will be utilized throughout without explicit mention.
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Picard, R., Trostorff, S., Waurick, M. (2013). A Note on a Class of Conservative, Well-Posed Linear Control Systems. In: Reissig, M., Ruzhansky, M. (eds) Progress in Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 44. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00125-8_12
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DOI: https://doi.org/10.1007/978-3-319-00125-8_12
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