The paper describes novel necessary and sufficient conditions for stability study of nonautonomous dynamical systems. The necessary conditions establish the existence of a surface through which the flow of the vector field has a given sign. The relation between the proposed necessary conditions with the integral and differential forms of continuity equations is shown. The sufficient condition establishes uniform stability and uniform asymptotic stability of a system equilibrium point using some properties of the vector field divergence. The proposed sufficient conditions are applied to design the state feedback control laws. The control law is found as a solution of a partial differential inequality, whereas the control law based on Lyapunov method is a solution of algebraic inequality. The examples illustrate the efficiency of the proposed method compared with some existing ones.

本文描述了非自治动力系统稳定性研究的新颖必要条件和充分条件。必要条件确定了表面的存在，矢量场通过该表面具有给定的符号。显示了所提出的必要条件与连续性方程的积分形式和微分形式之间的关系。充分条件利用矢量场散度的某些性质建立了系统平衡点的一致稳定性和一致渐近稳定性。提出的充分条件可用于设计状态反馈控制律。发现控制律是偏微分不等式的解，而基于Lyapunov方法的控制律是代数不等式的解。