Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations
The paper which according to its content and size (more than 130 pp) corresponds to a monograph book deals with an investigation of a problem of optimal control by use some of the classical tools of the calculus of variations which are extended and modernized in a proper way. In a previous book [\textit{A. A. Milyutin} and \textit{N. P. Osmolovskii}, Calculus of Variations and Optimal Control, Translations of Mathematical Monographs. 180. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0911.49001)] the simplest problem of the calculus of variations was studied in detail and quadratic necessary and sufficient conditions for broken extremals for the so-called ``Pontryagin minimum'' were obtained. In the same book a problem in terms of optimal control was also posed and later (according to some interpretations) it was named general problem of calculus of variations. The author declares that the presented paper can be considered as a necessary supplement of the above mentioned book (the present paper is devoted to remove a gap which has occurred in the book) but at the same time, the material contained in it is independent and is a complete theory for quadratic conditions of theory of variations and optimal control which was developed by the author in series of papers. The paper is organized as follows. First in Chapter 1 a fragment of the abstract theory of higher-order conditions of Levitin, Milyutin and Osmolovskii is presented. A modification of this theory for smooth problems on the set of sequences determining one or another concept of minimum is given. The main result of the abstract theory is that in a special form necessary and sufficient conditions for optimality are defined. In Chapter 2 on the basis of the results of Chapter 1 the quadratic theory of conditions for Pontryagin minimum in the general problem of calculus of variations is created eliminating one special condition. The decoding of a basic constant for the set of the so-called ``Pontryagin sequences'' is performed. This constant is estimated first from above and after this from below and thus to obtain necessary and sufficient conditions for optimality. Finally in Chapter 3 the quadratic conditions obtained in Chapter 2 are extended also in the case when the above mentioned special condition is not eliminated.
scientific article; zbMATH DE number 2174287