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.

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