Abstract
Abstract. A method for solving the Lagrange problem with phase restrictions for processes described by ordinary differential equations without involvement of the Lagrange principle is supposed. Necessary and sufficient conditions for existence of a solution of the variation problem are obtained, feasible control is found and optimal solution is constructed by narrowing the field of feasible controls. The basis of the proposed method for solving the variation problem is an immersion principle. The essence of the immersion principle is that the original variation problem with the boundary conditions with phase and integral constraints is replaced by equivalent optimal control problem with a free right end of the trajectory. This approach is made possible by finding the general solution of a class of Fredholm integral equations of the first order. The scientific novelty of the results is that: there is no need to introduce additional variables in the form of Lagrange multipliers; proof of the existence of a saddle point of the Lagrange functional; the existence and construction of a solution to the Lagrange problem are solved together.