Control processes

On asymptotic stability of mechanical systems with nonstationary leading parameter under dissipative forces

A. Yu. Aleksandrov^a, A. A. Kosov^b, A. V. Platonov<sup>a

a</sup> St. Petersburg State University, Faculty of Applied Mathematics and Control Processes
^b Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: Mechanical systems described by Lagrange differential equations of the second kind with nonstationary evolution of dissipative forces resulting in their domination are considered. In case of nonapplicability of known for nonstationary linearizations classical criteria, the theorems on asymptotic stability of the equilibrium position by the linear approximation are proved. The classes of nonstationary mechanical systems are determined, such that the asymptotic stability of their equilibrium is not exponential, however it is preserved for arbitrary perturbation whose order of smallness is higher than one. Furthermore, systems with essentially nonlinear dissipative forces are investigated. It is assumed that dissipative forces are determined by the homogeneous Rayleigh function, or depend on generalized coordinates. For such systems the conditions of asymptotic stability of the equilibrium position under the nonstationary domination of dissipative forces are obtained as well. It is proved that in the comparison with the case of linear dissipative forces the overdamping arises under higher velocities of evolution for essentially nonlinear ones.

Keywords: mechanical systems, dissipative forces, stability, Lyapunov functions, nonstationary parameter.

UDC: 531.36


Accepted: February 28, 2012