Construction and application of a family of Lyapunov functions for nonlinear nonautonomous systems

Summary (translated from the Russian): "Earlier [Vest. Ross. Univ. Druzhby Narodov Ser. Prikl. Mat. i Informat. {bf 1995}, no. 1, 9--12; per bibl.] we constructed a family of Lyapunov functions to study the stability `in the small' of the unperturbed motion of mechanical systems. We used this approach in another paper [Vest. Ross. Univ. Druzhby Narodov Ser. Prikl. Mat. i Informat. {bf 1998}, no. 1, 16--21; per bibl.] to construct mechanical systems that have asymptotically stable programmed motion `in the large'. In the present paper we extend these results to a wider class of material systems, both mechanical and nonmechanical. In addition, the family of Lyapunov functions constructed here is used to improve the quality of the transient of nonmechanical systems by means of optimal control. We discuss the possibility of the relay control of a system. We give examples of the use of the procedure proposed in the problem of the stabilization of motion of a point of variable mass in a Newtonian gravitational field and in the control of the approach to a satellite under large initial deviations."