The maximum principle in the general problem of optimal control. (Q2755141)

This research monograph presents in 303 pages the latest developments of what may be called ``The Dubovitskij-Milyutin theory of the maximum principle'' for optimal control problems defined by ordinary differential equations with end-points as well as control and phase constraints defined by finite sets of equalities and inequalities. As the author explains in the Preface, the concepts and results in the book are refinements of those obtained jointly in [\textit{A. Ya. Dubovitskij} and \textit{A. A. Milyutin}, ``The theory of the maximum principle'', in ``The methods of the theory of extremal problems in economics'' (Russian), 7-47 (1981; MR 84g:90038)]; moreover, the first step in the very complicated proofs of the main results, is a certain ``local maximum principle'' taken from [\textit{A. Ya. Dubovitskij} and \textit{A. A. Milyutin}, ``Necessary conditions for a weak extremum in the general problem of optimal control'' (Russian), Nauka, Moskva (1971)]. NEWLINENEWLINENEWLINEIn a very succinct manner, the content of the book may be described as follows: in Ch.1 the author presents in the first place the statements of the two optimal control problems to be studied in the book: Problem I, on a fixed time-interval and Problem II, on a variable time-interval; next, after the introduction of the necessary notations and definitions, the author gives the statements of the corresponding variants of the \textit{maximumum principle} termed as MP(I) and MP(II), respectively; finally, a detailed comparison of these results to the earlier forms of the maximum principle (Pontryagin's maximum principle for ``classical problems'', Dubovitskij-Milyutin MP for general problems and problems with phase-constraints) is provided. Ch.2, the largest of the book (82 p.), contains the application of MP(II) on four, rather ``fabricated'' examples, which, however, illustrate several important theoretical aspects of the most important conditions in MP(II). In the last chapters, 3 and 4, the complete (rather technical and very complicated) proofs of MP(I) and, respectively, MP(II) are presented while four of the most important ``auxiliary results'' are proved in Appendices I--IV. NEWLINENEWLINENEWLINEProblem I, formulated on a fixed time-interval \(I=[t_0,t_1]\), consists in the minimization of the functional: NEWLINE\[NEWLINE C(u(.)):=J(x(t_0),x(t_1)) NEWLINE\]NEWLINE subject to constraints of the form: NEWLINE\[NEWLINE F(x(t_0),x(t_1)) \leq 0, \;K(x(t_0),x(t_1))=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE x'(t)=f(x(t),u(t),t), \;u(t)=(u_1(t),u_2(t))\in R^k \times U_2(t), \;U_2(t) \subset R^m,NEWLINE\]NEWLINE NEWLINE\[NEWLINE G(x(t),u(t),t) \leq 0, \;g(x(t),u(t),t)=0 \text{ a.e. }(I) NEWLINE\]NEWLINE while Problem II, formulated on variable time-intervals, consists in the minimization of functionals of the form NEWLINE\[NEWLINE C(u(.)):=J(x(t_0),t_0,x(t_1),t_1) NEWLINE\]NEWLINE in the case the ``end-point constraints'' are of the form: NEWLINE\[NEWLINE F(x(t_0),t_0,x(t_1),t_1) \leq 0, \quad K(x(t_0),t_0,x(t_1),t_1)=0 NEWLINE\]NEWLINE the multifunction \(U_2(.)\) is constant and the other constraints are of the same type as in Problem~I. NEWLINENEWLINENEWLINEThe MP(I), containing necessary conditions for Problem I, states essentially, that if the data satisfy certain (reasonable) hypotheses and if \((x^0(.),u^0(.))\) is an optimal pair for Problem I then for any ``associated multifunction'' \(Z_\theta(.)\) there exists a set of 10 ``multipliers'', \(\pi=(\alpha_0,\alpha,c,p^*,\psi(.),a(.), n(.),b(.),d\nu,s(.))\) that satisfy no less than 12 ``basic conditions'' among which one may identify the four classical ones in the well-known PMP (Pontryagin's Maximum Principle): the adjoint equation (here in a generalized form, involving functions of bounded variations and Radon measures), the maximum condition, the transversality condition and the non-triviality condition. NEWLINENEWLINENEWLINEThe MP(II), for Problem II, derived from MP(I) using an ``extended problem'' in which the time-variable becomes a ``state-variable'', is much more complicated since in addition to the 12 ``basic conditions'', the multipliers in \(\pi\) should satisfy also some 9 additional conditions. NEWLINENEWLINENEWLINEThe term ``Dubovitskij-Milyutin method'' has been adopted in the theory of necessary optimality conditions soon after the publication of their seminal papers [\textit{A. Ya. Dubovitskij} and \textit{A. A. Milyutin}, Dokl. Akad. Nauk SSSR, Mat. 149, 759-762 (1963; Zbl 0133.05501); Zh. Vychisl. Mat. Mat. Fiz. 5, No. 3, 395-453 (1965; Zbl 0158.33504)] which have been used, applied and extended by many authors; in particular, one may note that their method has been much more clearly presented in the book [\textit{I. V. Girsanov}, ``Lectures on mathematical theory of extremum problems'' (1972; Zbl 0234.33504)] and has been significantly extended in some directions by \textit{V. G. Boltyanskij} [Russ. Math. Surv. 30, No. 3, 1-54 (1975; Zbl 0334.49014)]; these and other authors have identified as the starting point of the DM-method a certain ``non-intersection property'' accompanied by a theorem concerning the ``inseparability of a family of convex cones''; a new variant of this result is presented in Appendix I of the present book as ``the three-storey theorem'' which may be of a more general interest in Functional Analysis. A discussion of the DM-non-intersection property and its comparison with other types of intersection properties generating ``multiplier rules'' in optimization may be found also in [\textit{Ş. Mirică}, Optimization 46, No. 2, 135-163 (1999; Zbl 0959.90048)]. NEWLINENEWLINENEWLINEOne may conclude that the present book is the first systematic and (relatively) complete exposition of the ``Dubovitskij-Milyutin theory of the maximum principle in optimal control'' and should be taken as a basic ``reference book'' in the theory of necessary optimality conditions in optimal control.