Abstract
In this chapter the existence of a solution to an operator Riccati equation related to the linear regulator problem in separable Hilbert spaces is shown first. Then a formula for the optimal solution on an arbitrary finite time interval is given. The existence of an optimal solution on the infinite time interval is discussed as well. Some applications to the stabilizability problem are also included. Regulator problem for delay systems is treated as well.
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The direct proof of the existence of the solution \(P(\cdot )\) to the Riccati equation (17.9) based on Lemma 17.3 is due to G. Da Prato [24]. Theorem 17.4 is taken from the author’s paper J. Zabczyk [106]. More information on the subject can be found in the monographs by R.F. Curtain and A.J. Pritchard [22] and I. Lasiecka and R. Triggiani [62]. A specific application to delay systems can be found in R.F. Curtain and H. Zwart [23, pp. 288–298].
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Zabczyk, J. (2020). Linear regulators in Hilbert spaces. In: Mathematical Control Theory. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-44778-6_17
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DOI: https://doi.org/10.1007/978-3-030-44778-6_17
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-44776-2
Online ISBN: 978-3-030-44778-6
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