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**An abstract setting for differential Riccati equations in optimal control problems for hyperbolic/Petrowski-type P. D. E. s with boundary control and slightly smoothing observation.**
*(English)*
Zbl 0940.49005

Author’s summary: “We study, by the variational method, the differential Riccati equation which arises in the theory of quadratic optimal control problems for ‘abstract hyperbolic’ equations (which encompass hyperbolic and Petrowski-type partial differential equations (P.D.E.) with boundary control). We markedly relax, at the abstract level, the original assumption of smoothing required for the observation operator by the direct method of G. Da Prato, I. Lasiecka and the author [\((*)\) J. Differ. Equations 64, 26-47 (1986; Zbl 0601.49003)]. This is achieved by imposing additional higher level regularity requirements on the dynamics, which, however, are always satisfied by the class of hyperbolic and Petrowski-type mixed P.D.E. problems which we seek to cover. To appreciate the additional level of generality, and related technical difficulties associated with it, it suffices to point out that in the present treatment – unlike in [\((*)\)] – the gain operator \(B^*P(t)\) is no longer bounded between the state space \(Y\) and the control space \(U\). The abstract theory is illustrated by its application to a Kirchhoff equation with one boundary control. This requires establishing new higher level interior and boundary regularity results”.

Reviewer: Kunihiko Ichikawa (Tokyo)

### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

49N10 | Linear-quadratic optimal control problems |

93C20 | Control/observation systems governed by partial differential equations |

93D15 | Stabilization of systems by feedback |