Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control

Given an unstable hybrid stochastic differential equation (SDDE, also known as an SDDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDDE become asymptotically stable? If the feedback control is based on the current state, the stabilisation problem has been studied. However, there is little known when the feedback control is based on the past state. The problem becomes even harder when the coefficients of the underlying hybrid SDDE do not satisfy the linear growth condition (namely, the coefficients are highly nonlinear). The aim of this paper is to tackle the stabilisation problem for a given unstable highly nonlinear hybrid SDDE.