Advances in stabilization of highly nonlinear hybrid delay systems

Abstract

Given an unstable highly nonlinear hybrid stochastic differential delay equation (SDDE, also known as an SDDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDDE become exponentially stable? Recent work by Li and Mao in 2020 gave a positive answer when the delay in the given SDDE is a positive constant. It is also noted that in their paper the time lag in the feedback control is another constant. However, time delay in a real-world system is often a variable of time while it is difficult to implement the feedback control in practice if the time lag involved is a strict constant. Mathematically speaking, the stabilization problem becomes much harder if these delays are time-varying, in particular, if they are not differentiable. The aim of this paper is to tackle the stabilization problem under non-differentiable time delays. One more new feature in this paper is that the feedback control function used is bounded.