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 [math] .
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.