Stabilisation of super-linear hybrid stochastic differential equation by discrete-time trajectory-based event-triggered controls

Liu et al. (2025) have recently established their new theory on how to design an event-triggered control based on a trajectory of the state at discrete times in order to stabilise a given unstable hybrid stochastic differential equation (SDE). Since their event-triggered mechanism is based on a trajectory of the state, the corresponding event-triggered control can be implemented much more easily than the event-triggered controls in most of the existing papers where an event-triggered mechanism for SDEs is designed based on a criterion in terms of the mathematical expectation of the system state. As pointed out by Liu et al. (2025), there are few papers on the trajectory-based event-triggered control. We therefore see a need to develop it furthermore. We also note that the key condition imposed in Liu et al. (2025) is that both drift and diffusion coefficients of the given hybrid SDE need to satisfy the linear growth condition (namely bounded by linear functions). This is rather restrictive as many hybrid SDE models in the real world do not fulfil this condition (namely, they are super-linear or highly nonlinear). The aim of this paper is to tackle the stabilisation problem by trajectory-based event-triggered controls for a class of super-linear hybrid SDEs.