Abstract and Applied Analysis
Volume 1 (1996), Issue 4, Pages 435-484
doi:10.1155/S1085337596000243
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
Department of Applied Mathematics, University of Virginia, Thornton Hall, Charlottesville 22903, VA, USA
Received 6 December 1996
Copyright © 1996 R. Triggiani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 of the observation operator by the
direct method of [D-L-T.1]. 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 associate with it, it suffices to point out that in the
present treatment—unlike in [D-L-T.1]—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 Kirchoff equation with one boundary control. This requires establishing new higher level interior and boundary regularity results.