Abstract
Consider minimizing the integral
where
ForT sufficiently small, it is shown that
where the functionx, viewed as a function ofT, is a solution of the Cauchy problem
and the auxiliary functionr satisfies the Riccati system
In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.
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Kagiwada, H., Kalaba, R. Optimal trajectories for quadratic variational processes via invariant imbedding. J Optim Theory Appl 4, 99–108 (1969). https://doi.org/10.1007/BF00927415
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DOI: https://doi.org/10.1007/BF00927415