Optimality conditions

The book is devoted to optimality conditions for degenerate (the so-called abnormal) problems, both in the abstract and optimal control settings. par In Chapter 1, the author considers the problem f_0(x)tomin,; F(x)in C, where X is a vector space equipped with the so-called finite topology, and C is a closed polyhedral cone in Y ={bf R}^k. A point x_0 is called abnormal if exists yneq 0 such that pm yin N_C(F(x_0)),; F'(x_0)^*y=0. The Lagrange multiplier rule is not informative for such points, and the "standard" second-order necessary condition holds trivially: max_{lambdainLambda},d^2L(x_0,lambda)[h]^2ge 0 on the cone of critical variations, where Lambda is the set of all normalized tuples of Lagrange multipliers lambda=(lambda_0,y). The author suggests a refinement of this condition, substituting Lambda by the set Lambda_k of those lambdainLambda for which the corresponding second variation d^2L(x_0,lambda)[h]^2ge 0 on a subspace Pi(lambda) of codimension le k. The proof is based on the penalty function method. Under the Lyusternik condition, this new second-order condition turns into the usual one. par Next, the author studies the relation between this second-order necessary condition and the known sufficient conditions, for minimality in the finite topology and for the local minimality in a Banach space X. A notion of 2-normal mapping Fcolon Xto Y at a point x_0 with respect to a cone Csubset Y is introduced. In the case when 0notin{rm co},Lambda_k and F is 2-normal, it is shown that the above necessary condition turns into a sufficient one by an arbitrarily small perturbation of the problem. (In [E. S. Levitin, A. A. Milyutin and N. P. Osmolovskiĭ, Uspekhi Mat. Nauk {bf 33} (1978), no.~6(204), 85--148, 272; [msn] MR0526013 (80f:49001) [/msn]], this property was called s-necessity.) par Considering the problem f_0(x)tomin,; F(x)=0 with thrice smooth functions, the author obtains second-order necessary conditions using a method based on an extended Lagrange function, proposed by E. R. Avakov [Zh. Vychisl. Mat. i Mat. Fiz. {bf 25} (1985), no.~5, 680--693, 798; [msn] MR0796117 (86j:49056) [/msn]; Trudy Mat. Inst. Steklov. {bf 185} (1988), 3--29; [msn] MR0979298 (90a:49025) [/msn]]. To this aim, the notion of a 2-regular mapping F at a point x_0 in the direction h is introduced. The tangent cone to the level set of F at an abnormal point is described. Then, higher-order necessary and sufficient conditions in terms of the extended Lagrange function are considered. par Chapter 2 is devoted to a general optimal control problem with state and mixed constraints, terminal equality and inequality constraints, and with an inclusion constraint for some of the control components. This statement is mostly due to A. P. Afanasʹev et al. [{it A necessary condition in optimal control} (Russian), "Nauka", Moscow, 1990; [msn] MR1116118 (92i:49001) [/msn]], V. V. Dikusar and A. A. Milyutin [{it Qualitative and numerical methods in the maximum principle} (Russian), "Nauka", Moscow, 1989; [msn] MR1072997 (91h:49001) [/msn]], and A. Ya. Dubovitskiĭ and Milyutin [in {it Methods of the theory of extremal problems in economics}, 7--47, "Nauka", Moscow, 1981; [msn] MR0694700 (84g:90038) [/msn]]. The author introduces a notion of regularity for the endpoint, mixed and state constraints, and also a notion of compatibility of the state constraints with the endpoint constraints. Using finite-dimensional approximations and the penalty function methods, he proves the Pontryagin Maximum Principle. Special attention is given to guaranteeing its nontriviality in the case when the endpoints of the examined trajectory lie on the boundary of the state constraint. This is proved under the assumption of controllability at the endpoints with respect to the state constraints. Then, relaxations and perturbations of optimal control problems are considered. par In Chapter 3, the author studies the integral quadratic form of the classical calculus of variations, whose Legendre coefficient vanishes at some points tau as |t-tau|^alpha,; alpha>0. He proves necessary and sufficient conditions for the nonnegativity of this form in terms of focal and "breakdown" points, and gives estimates for its index. par Chapter 4 is a study of mappings in a neighborhood of an abnormal point. Here, an implicit function theorem and an inverse function theorem are proposed for 2-normal mappings. In the normal case, these results transform into the classical theorems. It is also proved that, in a neighborhood of an abnormal point, the zero set of a 2-regular mapping is locally diffeomorphic to the zero set of its second differential. A criterion for strong 2-regularity of a quadratic mapping is given. par In each chapter, illustrative examples are provided. par {The Russian original has been reviewed [[msn] MR1469734 (99c:49001) [/msn]].}