Delay sensitivity of quadratic controllers. A singular perturbation approach

We study the variations of the quadratic performance associated to a linear differential system of retarded type for small values of the delays. From an interpretation of delays as singular perturbations of abstract evolution operators, we revisit the usual theory of representation and optimal control of retarded systems. This leads to a new parameterization of associated Riccati operators for which insight is gained in the dependence on the delays. This explicit parameterization of Riccati operators by the delays enables us to prove differentiability at zero for performance viewed as a function of the delays, in the LQ-optimal or H_∞ sub-optimal control. The gradient is explicitly computed in terms of the non-negative solution of the finite dimensional Riccati equation associated to the non-delay control problem.