Abstract
A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert spaceR n ×L 2. The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g., among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spacesR n ×L 2 andH 1. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.
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Work done by K. Ito was supported by AFOSR under Contract No. F-49620-86-C-0111, by NASA under Grant No. NAG-1-517, and by NSF under Grant No. UINT-8521208. Work done by F. Kappel was supported by AFOSR under Grant No. 84-0398 and by FWF(Austria) under Grants S3206 and P6005.
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Ito, K., Kappel, F. A uniformly differentiable approximation scheme for delay systems using splines. Appl Math Optim 23, 217–262 (1991). https://doi.org/10.1007/BF01442400
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DOI: https://doi.org/10.1007/BF01442400