Dalembert-Lagrange principle as a basis for control synthesis using reduction to the isoperemetric problem

Dynamic systems satisfying the Dalembert-Lagrange principle are considered. The problem of control synthesis is posed as an inverse problem of dynamics, the solution of which is conjugated with a known problem. Its solution can be based on the reduction of the Lagrangian problem to an isoperimetric problem, which requires a search for the conditions of the minimum of the corresponding extended target functional. This leads to a variational inequality. On its basis it is established that the synthesis of controls can be based on obtaining the conditions of optimality using the constraints following from the differential and integral variational principles. For this purpose, it is necessary to use constraints in the form of the Dalembert-Lagrange equation, Hamilton-Ostrogradsky action integral or Appel's acceleration energy when constructing the convolution of these expressions with the target functional. Aim of work – synthesis of controls of electromechanical systems using the reduction of the Lagrangian problem to the isoperimetric problem. The considered problem of control synthesis is solved using the maximum principle of L.S. Pontryagin and the results of the Lagrange problem reduction to the isoperimetric problem. The correspondence of the obtained solutions is analytically established. This is additionally confirmed by the results of mathematical modeling given in the phase portrait. It is shown that the application of transversality conditions constraining the boundary value problem allows to synthesize a new quasi-optimal control law in the form that takes into account the terminal conditions. However, at the final section, the control trajectories differ in the moments of time of sign change, and the calculations demonstrate an acceptable in practice loss in the quality index of the synthesized terminal control to the optimal one. The practical significance of the obtained results is determined by the fact that the synthesis of controls using the reduction of the Lagrange problem to the isoperimetric problem does not require the solution of the known two-point boundary value problem of the maximum principle of L.S. Pontryagin. This greatly simplifies the solution in the case of a problem of large dimension. In addition, in a number of practical applications the closeness of the solution to the optimal one is sufficient.