Absztrakt: | When examining the properties of the stability and asymptotic
behaviour of a system a Lyapunov function is often used as the
total mechanical energy of the system. By analogy with the division
of the energy into kinetic and potential energy, it is proposed
below to construct a Lyapunov function in the form of the sum
of two subsidiary scalar functions, such that its derivative
on account of the system is estimated using some kind of function
of these subsidiary functions. We examine the case when the derivative
of the Lyapunov function can also take positive values, and the
equation of comparison the emerges from the estimate of the Lyapunov
function does not permit a separation of variables. {\it V. V.
Rumjancev}'s theorem [ibid. 35, 105-110 (1971; Zbl 0265.34069).]
on the asymptotic stability with respect to the velocities of
the equilibrium position of a dissipative mechanical system is
generalized on the basis of the results obtained. |