Gao Yin, Jia Lifen
School of Mathematics, Renmin University of China, Beijing, 100872 China.
School of Management and Engineering, Capital University of Economics and Business, Beijing, 100070 China.
J Ambient Intell Humaniz Comput. 2022 Apr 2:1-15. doi: 10.1007/s12652-022-03826-9.
Stability in distribution for uncertain delay differential equations based on the strong Lipschitz condition only involving the current state has been successfully investigated. In reality, the uncertain delay differential equation is not only relate to the current state, but also relate to the past state, so it is very hard to obtain the strong Lipschitz condition. In this paper, the new Lipschitz condition concerning the current state and the past state is provided, if the uncertain delay differential equation satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz condition, conversely, it may not be established. By means of the new Lipschitz condition, a sufficient theorem for the uncertain delay differential equation being stable in distribution is proved. Meanwhile, a class of uncertain delay differential equation is certified to be stable in distribution without any limited condition. Besides, the effectiveness of the above sufficient theorem is verified by two numerical examples.
仅基于涉及当前状态的强利普希茨条件，对不确定延迟微分方程的分布稳定性进行了成功研究。实际上，不确定延迟微分方程不仅与当前状态有关，还与过去状态有关，因此很难得到强利普希茨条件。本文给出了关于当前状态和过去状态的新利普希茨条件，若不确定延迟微分方程满足强利普希茨条件，则它一定满足新利普希茨条件，反之则不一定成立。借助新利普希茨条件，证明了不确定延迟微分方程在分布上稳定的一个充分定理。同时，证明了一类不确定延迟微分方程在无任何限制条件下在分布上是稳定的。此外，通过两个数值例子验证了上述充分定理的有效性。
