O(h2 ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations

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The main purpose of this paper is to give numerical algorithms and the error analysis for delay quadratic problems in the calculus of variations. These methods are new, efficient, and accurate and have a global a priori error of O(h2 ), where h is the distance between any two successive node points. We also derive the results for the general, numerical, delay problem, but we focus on proving our results in the simpler quadratic case since the extra technical numerical details have been given previously by the second author. In addition, the authors have previously shown how to reformulate general delay constrained problems in optimal control theory/constrained calculus of variations as unconstrained delay problems. Thus our numerical results and methods will hold for these general constrained problems also. Finally, we note that our algorithm, which solves the stationary condition(s) numerically, avoids the more difficult problems of piecing the solution of second order equations together and requires less smoothness in the solution. Thus we replace difficult second order boundary value problems with the easier task of approximating definite integrals involving first order derivatives.