Quadratic order conditions of a local minimum for abnormal extremals Original Research Article

It is well known in the theory of extremal problems that the abnormal case, i.e. the case when equality constraints are degenerate at the examined point, is a difficult subject to obtain higher order conditions of a local minimum. Especially it is true for necessary conditions. The matter is that “standard” necessary conditions, relevant to the general case, are always trivially fulfilled in the abnormal case and do not provide any information about the presence or absence of a local minimum at the given point. Here we present a method of treatment extremal problems with degenerate equality constraints, originally proposed By A.A. Milyutin. It consists of the passing from the given problem to another one, in which the equality constraints are nondegenerate. Application of this method and of its refinement allows one to obtain informative quadratic order necessary conditions for local minima in some classes of problems.