Lyapunov functions for nonlinear nonautonomous systems: Their construction and application

The apparatus of Lyapunov functions developed in [Ij for investigating the stability "in the small" of unperturbed motion of mechanical systems and applied in [2] for constructing mechanical systems having asymptotically stable motion "as a whole" is extended to a wider class containing both mechanical and nonmechanical systems. The family of Lyapunov functions is also applied to refine the quality of transition processes of nonmechanical systems through an optimal control. The feasibility of relay control for systems is examined. Examples are given to illustrate the use of the procedure to stabilize the motion of a material point of variable mass in a Newton field and control the approach with a satellite under large initial deviations.