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The paper describes a novel method for studying the stability ofnonautonomous dynamical systems. This method based on the flow and divergenceof the vector field with coupling to the method of Lyapunov functions. Thenecessary and sufficient stability conditions are formulated. It is shown thatthe necessary stability conditions are related to the integral and differentialforms of continuity equations with the sources (the flux is directed inward)located in the equilibrium points of the dynamical system. The sufficientstability conditions are applied to design the state feedback control laws. Theproposed control law is found as a solution of partial differential inequality,whereas the control law based on Lyapunov technique is found from the solutionof algebraic inequality. The examples illustrate the effectiveness of theproposed method compared with some existing ones.

Divergence Conditions for Investigation and Control of Nonautonomous Dynamical Systems