A uniformly differentiable approximation scheme for delay systems using splines

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of the semigroups on the Hilbert space Rn ?? L2. The approximating semigroups preserve (uniformly) the logarithmic sectorial property (i.e., the differentiability) of the solution semigroup. We prove the convergence of the scheme both in W1,2, Rn ?? L2, and using the uniform differentiability we are able to establish error estimates and obtain the quadratic convergence for a class of initial data. We also apply the scheme for computing the feedback solutions to the linear quadratic regulator problem. The uniform differentiability is again the essential basis for obtaining a complete theory for convergence of the Riccati solutions.