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- W1987772339 abstract "New numerical methods are given to find extremal solutions (with corners) in the calculus of variation, where $x(t)$ is an m-vector, $m geqq 1$, for well-defined mixtures of initial value and boundary value problems. These methods are general, accurate, easy to apply and implement, efficient, and stable. Of theoretical interest is the use of sophisticated, quadratic-form methods to establish a priori estimates for the global convergence of the error for the two-point boundary value problem. This error is a maximal pointwise error of at least $O(h^2 )$, and a simple Richardson procedure yields a maximal pointwise error of at least $O(h^4 )$. The results in this paper are suggested by discrete variable methods for ordinary differential equations. Thus, convexity is not required, the Euler–Lagrange equations are satisfied between corners, and the resulting equations are block tridiagonal. The latter property is very important in practical applications. The methods used here are shown to include the case where the functional is quadratic. In this event, solutions are obtained by solving block-tridiagonal, linear equations. It is equally important that these results lead to a new, general theory for the numerical solution of optimal control problems. This theory includes equality and inequality constraints and transversality or minimum-time problems with the accuracy described above. In particular, it is noted that the numerical solutions given by Hestenes [Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966] can be found with the accuracy described in this paper." @default.
- W1987772339 created "2016-06-24" @default.
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- W1987772339 date "1990-04-01" @default.
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- W1987772339 title "Discrete Variable Methods for the m-Dependent Variable, Nonlinear Extremal Problem in the Calculus of Variations" @default.
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- W1987772339 doi "https://doi.org/10.1137/0727029" @default.
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