Necessary conditions of the second order in optimal control problems

The author studies finite-dimensional minima of an optimal control problem. An admissible process (x_{0},u_{0}) is a finite-dimensional minimum for a control problem if for any finite-dimensional space Wsubset L_{infty }^{m}[t_{1},t_{2}] containing u_{0} the process (x_{0},u_{0}) is a local minimum for the control problem with the additional constraint uin W. Second-order necessary optimality conditions, without a normality assumption, are derived.