Event-Triggered Stabilization and Robust Stability of Stochastic Differential Equations

Influenced by the pioneering work of Ȧström and Bernhardsson (2002), many results on the event-triggered control have been reported for different types of systems, such as ordinary differential equations (ODEs) and stochastic differential equations (SDEs). In the area of SDEs, the event-triggered mechanisms are essentially divided into two categories: (i) a mechanism is designed based on a criterion in terms of the mathematical expectation of the system state; (ii) it is designed based on a trajectory of the state observations. Both mechanisms have their own right. Our paper is concerned with (ii). We will set up a new event-triggered mechanism in category (ii) which differs from those in the existing papers. It is due to the new event-triggered mechanism that we will be able to develop a novel approach which is completely different from those in the existing papers. Our approach is to relate the event-triggered controlled SDE to the corresponding uncertain SDE. Our result shows that if the uncertain SDE is exponentially stable in$p$th moment for some$p >0$, we can then make the event-triggered controlled SDE to be exponentially stable too. It is first time that the event-triggered controlled SDE is related together with the uncertain SDE so that the techniques developed in the area of uncertain SDEs can be applied to the study of the event-triggered controlled SDEs. This is the key contribution in this paper.