Existence and Uniqueness Theorem on Uncertain Nonlinear Switching Systems with Time Delay

This paper considers an uncertain nonlinear switching system with time delay, which is denoted as a series of uncertain delay differential equations. Previously, there were few published results on such kinds of uncertain switching systems. To fill this void, the internal property of the solutions is thoroughly explored for uncertain switching systems with time delay in state. Under the linear growth condition and the Lipschitz condition, existence and uniqueness with respect to the solutions are derived almost surely in the form of a judgement theorem. The theorem is strictly verified by applying uncertainty theory and the contraction mapping principle. In the end, the validity of above theoretical results is illustrated through a microbial symbiosis model.