Regularity of solutions of retarded equations and application to sensitivity of linear quadratic controllers to small delays

We analyze the behaviour of the solution of a linear differential equation of retarded type when the delays vary, for L^p initial conditions, given some p∈[1,∞]. First, we state continuity and differentiability results for the solution viewed as function of the delays. Those regularity results are applied to analyze the small delay sensitivity for quadratic performance index associated to stabilization with a finite dimensional a priori feedback.