  New necessary and sufficient conditions are proposed for the stability
investigation of dynamical systems using the flow and the divergence of the
phase vector velocity. The obtained conditions generalize the well-known
results of V.P. Zhukov and A. Rantzer. The relation of Lyapunov methods with
the proposed methods is established. The application of the obtained results to
study the stability of linear systems goes to the problem of matrix inequality
solvability. The new control laws are synthesized for linear and nonlinear
systems. Examples illustrate the applicability of the proposed method and show
the comparison results with some existing ones.
