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.

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We suggest a method for solving a boundary value problem for ordinary differential equations with boundary conditions in the presence of state and integral constraints. The method is based on the embedding principle, which permits one to reduce the original boundary value problem to a special optimal control problem with the use of the general solution of a Fredholm integral equation of the first kind. STATEMENT OF THE PROBLEMConsider the boundary value probleṁwith the boundary conditionsthe phase constraintsand the integral constraintsHere A(t) and B(t) are given n × n and n × m matrices, respectively, with piecewise continuous entries, μ(t), t ∈ I, is a given n-dimensional vector function with piecewise continuous components, f (x, t) is an m-dimensional vector function that is jointly continuous with respect to the variables (x, t) ∈ R n × I and satisfies the conditionsand S is a given convex closed set. The function F (x, t) = (F 1 (x, t), . . . , F r (x, t)), t ∈ I, is an r-dimensional jointly continuous vector function, and γ(t) = (γ 1 (t), . . . , γ r (t)) and δ(t) = (δ 1 (t), . . . , δ r (t)), t ∈ I, are given continuous functions.149

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