Lyapunov functions of the mechanical energy type
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. \textit{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.
scientific article; zbMATH DE number 4005666