The criterion for solvability of a linear Noether’s boundary value problem for a system of dynamical equations on a time scale

Abstract

On a time scale, the linear Noether’s boundary value problem for a system of second-order dynamical equations is considered. This boundary-value problem is considered in the case when the operator of the linear part is irreversible, that is, the number of boundary conditions of the problem and the order of the operator system are different. To establish the solvability conditions of the boundary-value problem under consideration, the apparatus of the theory of pseudo-inverse matrices is used. A connection is established between the condition of solvability of a dynamical system and the condition of solvability of an algebraic system of equations. That is, using the theory of pseudo-inverse matrices, the condition for solvability of a dynamical system of equations is established, which reduces the considered boundary value problem. In this case, the condition of solvability of a dynamical system of equations follows from the condition of solvability of the corresponding algebraic system of equations. A set of solutions of the boundary value problem under consideration is found. Also, partial cases of the boundary value problem are given, when the number of boundary conditions is greater than the number of unknowns of the system of dynamic equations and vice versa. For each of these cases, the solvability conditions of the boundary-value problem under consideration are found and its solutions are found. An example is provided illustrating the application of the results obtained.