Stability in measure for uncertain delay differential equations based on new Lipschitz conditions

Abstract

Uncertain delay differential equations (UDDEs) charactered by Liu process can be employed to model an uncertain control system with a delay time. The stability of its solution always be a significant matter. At present, the stability in measure for UDDEs has been proposed and investigated based on the strong Lipschitz condition. In reality, the strong Lipschitz condition is so strictly and hardly applied to judge the stability in measure for UDDEs. For the sake of solving the above issue, the stability in measure based on new Lipschitz condition as a larger scale of applications is verified in this paper. In other words, if it satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz conditions. Conversely, it may not be established. An example is provided to show that it is stable in measure based on the new Lipschitz conditions, but it becomes invalid based on the strong Lipschitz condition. Moreover, a special class of UDDEs is verified to be stable in measure without any limited condition. Besides, some examples are investigated in this paper.