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A Study on Abnormality in Variational and Optimal Control Problems

Author

Listed:
  • C. Bruni

    (Università di Roma – La Sapienza)

  • D. Iacoviello

    (Università di Roma – La Sapienza)

Abstract

In this paper, a variational problem is considered with differential equality constraints over a variable interval. It is stressed that the abnormality is a local character of the admissible set; consequently, a definition of regularity related to the constraints characterizing the admissible set is given. Then, for the local minimum necessary conditions, a compact form equivalent to the well-known Euler equation and transversality condition is given. By exploiting this result and the previous definition of regularity, it is proved that nonregularity is a necessary and sufficient condition for an admissible solution to be an abnormal extremal. Then, a necessary and sufficient condition is given for an abnormal extremal to be weakly abnormal. The analysis of the abnormality is completed by considering the particular case of affine constraints over a fixed interval: in this case, the abnormality turns out to have a global character, so that it is possible to define an abnormal problem or a normal problem. The last section is devoted to the study of an optimal control problem characterized by differential constraints corresponding to the dynamics of a controlled process. The above general results are particularized to this problem, yielding a necessary and sufficient condition for an admissible solution to be an abnormal extremal. From this, a previously known result is recovered concerning the linearized system controllability as a sufficient condition to exclude the abnormality.

Suggested Citation

  • C. Bruni & D. Iacoviello, 2004. "A Study on Abnormality in Variational and Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 41-64, April.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:1:d:10.1023_b:jota.0000026130.86491.22
    DOI: 10.1023/B:JOTA.0000026130.86491.22
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    References listed on IDEAS

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    1. Hadley, G. & Kemp, M. C., 1971. "Variational Methods in Economics," Elsevier Monographs, Elsevier, edition 1, number 9780720436013 edited by Bliss, C. J..
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