A Study on Abnormality in Variational and Optimal Control Problems |
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Authors: | Bruni C Iacoviello D |
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Institution: | (1) Dipartimento di Informatica e Sistemistica Antonio Ruberti, Università di Roma – La Sapienza, Rome, Italy;(2) Dipartimento di Informatica e Sistemistica Antonio Ruberti, Università di Roma – La Sapienza, Rome, Italy |
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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. |
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Keywords: | Calculus of variations optimal control problems abnormal extremals |
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