Quadratic Control of Evolution Equations with Delays in Control

The quadratic cost problem of evolution equations with delays in control is considered. A semigroup model which involves no explicit delays in control, but contains an unbounded control operator is introduced. With the aid of a family of approximating systems, it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Three examples are given to illustrate the theory. The filtering problem of evolution equations with observation delays is also solved through the duality relation.