Application of vibrational control to linear-quadratic control problems

The application of the theory of vibrational control to linear-quadratic control problems is developed. The solution of the matrix Riccati differential equation (RDE) and the optimal trajectory are found approximately as power series of a small parameter. The minimizing effect of vibrations on the maximal solution of RDE and the cost functional is studied. The region of attraction of the maximal solution of RDE for the case of Hamiltonian matrix with imaginary axis eigenvalues is found. Special attention is paid to the application of vibrations to the linear-quadratic problem of stabilization with respect to a part of variables and transfer of the other variables to a given position. A problem of vibrational stabilization and optimal control of a carriage with an inverted pendulum is solved as an example.