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New proof and some generalizations of the minimum principle in optimal control

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Abstract

An optimal control problem is reduced to the finite-dimensional problem of minimizing the terminal payoff over the intersection of the target set with the reachable set. The pointwise Pontryagin minimum principle is derived from two simple preliminary results: the first states that the intersection of two inseparable derived cones at a common point of two given sets is contained in the quasitangent cone (hence, in the contingent cone) to their intersection; the second identifies a derived cone to the reachable set. The standard variants of the minimum principle are easily generalized to problems defined by non-differentiable terminal payoffs on arbitrary target sets.

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References

  1. 1.

    Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, Wiley, New York, New York, 1966.

  2. 2.

    Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMischenko, E. F.,The Mathematical Theory of Optimal Processes, Wiley, New York, New York, 1962.

  3. 3.

    Berkovitz, L. D.,Optimal Control Theory, Springer, New York, New York, 1974.

  4. 4.

    Cesari, L.,Optimization: Theory and Applications, Springer, New York, New York, 1983.

  5. 5.

    Fleming, W. H., andRishel, R. W.,Deterministic and Stochastic Optimal Control, Springer, New York, New York, 1975.

  6. 6.

    Dubovitskii, A. Ya., andMilyiutin, A. A.,Extremum Problems in the Presence of Constraints, Zhurnal Vychislitelnyi Matematiki i Matematicheskii Fiziki, Vol. 5, No. 3, pp. 395–453, 1965 (in Russian).

  7. 7.

    Boltyanskii, V. G.,The Method of Tents in the Theory of Extremal Problems, Russian Mathematical Surveys, Vol. 30, No. 3, pp. 1–54, 1975.

  8. 8.

    Girsanov, I. V.,Lectures on Mathematical Theory of Extremum Problems, Springer, New York, New York, 1972.

  9. 9.

    Mirică, Ş., andMirică, I.,Distinct Types of Intrinsic Tangent Cones to Arbitrary Subsets of Euclidean Spaces, Analele Universităţii Bucureşti, Seria Matematică, Vol. 38, No. 1, pp. 21–39, 1989.

  10. 10.

    Bazaraa, M. S., andShetty, C. M.,Foundations of Optimization, Springer, New York, New York, 1976.

  11. 11.

    Kurzweil, J.,Ordinary Differential Equations, Elseiver, Amsterdam, Holland, 1986.

  12. 12.

    Mirică, Ş.,A Short Easy Proof of Pontryagin's Minimum Principle, Preprint 50/1988, Increst, Institutul de Matematică, Bucureşti, Romania, 1988.

  13. 13.

    Penot, J. P.,Calcul Sous-Différentiel et Optimisation, Journal of Functional Analysis, Vol. 27, No. 2, pp. 248–276, 1978.

  14. 14.

    Mirică, Ş., Staicu, V., Angelescu, N.,Fréchet Semidifferentials of Real-Valued Functions, Preprint 8706, Istituto Matematico Ulisse Dini, Firenze, Italia, 1987.

  15. 15.

    Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley, New York, New York, 1983.

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Communicated by L. D. Berkovitz

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MiricĂ, S. New proof and some generalizations of the minimum principle in optimal control. J Optim Theory Appl 74, 487–508 (1992). https://doi.org/10.1007/BF00940323

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Key Words

  • Optimal control problems
  • minimum principle
  • derived cones
  • quasitangent cones
  • contingent cones
  • superdifferentials
  • generalized gradients