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Abnormality in constrained optimal control: optimality conditions

Abstract

This research project deals with the theory of abnormal problems and its application in Optimal Control. By saying "abnormal problem" we understand an optimization problem with constraints (first let it be a finite-dimensional problem) in which, however, the conventional regularity assumption (the so-called Lyusternik condition) at the point of minimum does not hold true. This implies that the derivative of the constraint function degenerates at the point of minimum. Then, the minimizer turns out to be a singular point of the constraint map. Such minimizers are called abnormal. The interest of this research lies in various classes of optimality conditions for abnormal minimizers, and, in the context of Optimal Control Theory, in the investigation of issues of non-controllability arising in various engineering applications. Note that, for abnormal optimization problems, the classic Lagrange multiplier rule is no longer informative since it is trivially met. Moreover, as we know, the classic second-order necessary conditions do not hold for abnormal problems. Per se, "abnormality" means that the local linear approximation to a problem has a structure quite different from that of the original nonlinear problem. This can be explained by the following fundamental fact: the statement of the classic inverse function theorem does not hold in the proximity of abnormal points. Therefore, the question of inverse function theorems in the proximity of abnormal (singular) points of a smooth map (the question, which is interesting in itself) becomes extremely crucial for the present theory and plays a major role in derivation of extremum conditions for abnormal minimizers. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ARUTYUNOV, ARAM; DE OLIVEIRA, VALERIANO ANTUNES; PEREIRA, FERNANDO LOBO; ZHUKOVSKIY, EVGENIY; ZHUKOVSKIY, SERGEY. On the solvability of implicit differential inclusions. APPLICABLE ANALYSIS, v. 94, n. 1, SI, p. 129-143, JAN 2 2015. Web of Science Citations: 7.

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