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Invariant imbedding and a class of variational problems

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Abstract

Consider the minimization of an integral

$$I = \int_a^T {[\frac{1}{2}\dot w^2 + F(w,y)]} dy$$

withw(a)=c andw(T)=free. An initial-value problem for the optimizing function is derived directly from the variational problem. It is shown that the solution of the initial-value problem satisfies the usual Euler equation. The Bellman-Hamilton-Jacobi partial differential equation is also treated.

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References

  1. Bellman, R., andKalaba, R.,On the Fundamental Equations of Invariant Imbedding I, Proceedings of the National Academy of Science, Vol. 47, No. 3, 1961.

  2. Kagiwada, H., andKalaba, R.,An Initial Value Method Suitable for the Computation of Certain Fredholm Resolvents, Journal of Mathematical and Physical Sciences, Vol. 1, Nos. 1 and 2, 1967.

  3. Kagiwada, H., Kalaba, R., andSchumitzky, A.,An Initial Value Method for Fredholm Integral Equations, Journal of Mathematical Analysis and Applications, Vol. 19, No. 1, 1967.

  4. Kagiwada, H., Kalaba, R., Schumitzky, A., andSridhar, R.,Cauchy and Fredholm Methods for Euler Equations, Journal of Optimization Theory and Applications, Vol. 2, No. 3, 1968.

  5. Bellman, R., Kagiwada, H., andKalaba, R.,Invariant Imbedding and Nonvariational Principles in Analytical Mechanics, International Journal of Nonlinear Mechanics, Vol. 1, No. 1, 1966.

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Authors and Affiliations

  1. The RAND Corporation, Santa Monica, California

    J. Casti (Mathematician), R. Kalaba & B. Vereeke

Authors
  1. J. Casti
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  2. R. Kalaba
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  3. B. Vereeke
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Casti, J., Kalaba, R. & Vereeke, B. Invariant imbedding and a class of variational problems. J Optim Theory Appl 3, 81–88 (1969). https://doi.org/10.1007/BF00932458

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  • DOI: https://doi.org/10.1007/BF00932458

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