On quadratic optimization in distributed parameter systems

Systems described by parabolic partial differential equations are formulated as ordinary differential equations in a Hilbert space. Quadratic cost criteria are then formulated as inner products on this Hilbert space. Existence of an optimal control is proved both in the case where the system operator is "coercive" and in the case where the system operator is the infinitesimal generator of a semigroup of operators. The optimal control is given by a linear state feedback law in which the feedback operator is shown to be the bounded positive self-adjoint solution of a nonlinear operator equation of the Riccati type.