Admissible extremals in the problems of variational calculus with constraints in the form of a system of linear inhomogeneous differential equations of the first order with rectangular matrices (Q2663190)

The paper is about the determination of necessary and sufficient conditions for the existence of smooth admissible extremals of the functional, where the state variable is subject to an ordinary differential equation (ODE) constraint. When the equations for constraints are regarded as independent, the classical method can be used for the solution of the problems of variational calculus with the help of Lagrange multipliers. Because this restriction is absent, to check their independence, the ODE system is reduced to the generalized canonical form. Some equations are excluded after preliminary verification of solvability of the system. Necessary and sufficient conditions are established with constraints in the form of a system of linear inhomogeneous differential equations of the first order with rectangular matrices and different types of boundary conditions.