Abnormality in the theory of necessary optimality conditions

The problems of existence, non-uniqueness, abnormality of solutions arising from the using of necessary optimality conditions are discussed in terms of the extension principle and sufficient optimality conditions. Simple examples are used.
It is shown that these problems are often generated not by the sense of problem, but by the method of solving and these problems can disappear by using another method.
The study is based on the extension principle in the abstract extremum problem proposed by V.F. Krotov. This method was further developed in the works of V.I. Gurman, M.M. Khrustalev and A.I. Moskalenko.
Particular attention is paid to the phenomenon of abnormality that occurs in the using of the classical scheme of the formation of the Lagrange function in the finite-dimensional extremal problems and in optimal control problems.
Lagrange multipliers are constants in the classical Lagrange method for the finite-dimensional problems. In this case, the problem of conditional extremum is completely reduced to the problem of unconditional extremum only in special cases described in the Kuhn–Tucker theorem.
With regard to the optimal control problems, functional analog of classical constant Lagrange multipliers is used in the calculus of variations and in the Pontryagin maximum principle. In the Pontryagin maximum principle it is a vector of conjugate variables. The problem of abnormality is presented here in full. An example of a class of optimal control problems, where any permissible process is an abnormal Pontryagin extremal, is considered.
The extension principle allows you to use the Lagrange multipliers, which are the functions depended on the optimized vector in the finite-dimensional case, and depended on the system state in the optimal control problem. Using this principle, you can get the optimal conditions where problem of abnormality doesn't occur.