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

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 $B^{*}P\left(t\right)$ 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.