Control of PDE Systems with Delays

We consider a special class of control systems governed by infinite dimensional neutral functional differential equations of the formwhere the linear operator A0 : D(A0) ⊆ Z → Z generates a C0-semigroup on a product space Z of the form Z = T × Rn where T is a Hilbert space. We assume A1 : Z → Z,D : Z → Z and B : Rm → Z are bounded linear operators. We show how such systems arise naturally when one includes delays in an actuated system or when one is concerned with sensitivities with respect to delays. Well-posedness of these systems are presented and numerical approximations are discussed. We apply these results to boundary control of PDE systems and illustrate the ideas with a numerical example.