Variational statements of the problem of controlled motions of a system with elastic elements

Two variational statements of the problem of controlled motions of a linear mechanical system with elastic elements that has a finite number of degrees of freedom are proposed. The first variational statement reduces to nominal minimisation of the non-negative quadratic functional. This functional, the dimension of which is equal to that of the action, comprises an integral residual of constitutive equations of state that define the relations between momenta and velocities of points of the system, and also between elastic forces and relative displacements. The second variational statement, with assignment of certain initial and terminal conditions with respect to time, is related to the Hamilton–Ostrogradskii principle. The variational properties of the two statements of the initial Cauchy problem are shown for the examined type of mechanical system, along with methods for estimating the accuracy of the approximate solutions. An example is given of the numerical calculation of motion of the system and estimates of the accuracy of solution with the chosen control law.