Problems of stability with respect to part of the variables

The Lyapunov-Malkin theorem on stability and (simultaneously) exponential asymptotic stability with respect to part of the variables in the linear approximation in critical cases (in Lyapunov's sense) has served as a point of departure for various previous results. These results are strengthened by relaxing all additional assumptions (other than continuity) regarding the coefficients of the linear part of the non-linear system under consideration. The result is extended to the problem of polystability with respect to part of the variables. In addition, a method for narrowing down the admissible domain of variation of "uncontrollable" variables is worked out as applied to problems of asymptotic stability with respect to part of the variables. Examples are considered. (C) 2000 Elsevier Science Ltd. All rights reserved.