Vibrations of sympathetic pendulums are studied. The set of initial conditions for beats and energy transfer has been found, and the corresponding motions have been constructed with help of the reversible mechanical systems theory.

We consider the problem of control the triple integrator with dual phase constraints and bounded control. The quasi-time-optimal control algorithms for finite-time stabilization of a double and triple integrators with phase constraints are developed. We construct attraction domains of the origin for appropriate closed-loop systems. The theoretical results are supported by numerical simulations.

A new approach was proposed to analyze the stability of the linear continuous stationary dynamic systems. It is based on the decomposition of a square H2 norm of the transfer function of the dynamic system into parts corresponding either to particular eigenvalues of the system matrix, or to pairwise combinations of these eigenvalues. The spectral decompositions of a square H2 norm of the transfer function with multiple poles were obtained using the residues of the transfer function and their derivatives. Exact analytical expressions for calculation of the quadratic forms of the corresponding expansions were derived for an arbitrary location of the eigenvalues in the left half-plane. The obtained decompositions allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic variation of the system energy. We propose the energy criterion for estimation of the system stability margins that uses an evaluation of energy accumulated in a group of weakly stable system modes. This approach is illustrated by calculating the energy of a band-pass filter.