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

The authors would like to thank the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the Royal Society of Edinburgh (61294), the EPSRC (EP/K503174/1), the Natural Science Foundation of China (61773220, 61473334, 61876192, 61374085), the Ministry of Education (MOE) of China (MS2014DHDX020) for their financial support

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  • Given an unstable hybrid stochastic differential equation (SDE, also known as an SDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDE become asymptotically stable? The paper [14] by Mao et al. was the first to study the stabilisation by delay feedback controls for hybrid SDEs, though the stabilization by non-delay feedback controls had been well studied. A critical condition imposed in [14] is that both drift and diffusion coefficients of the given hybrid SDE need to satisfy the linear growth condition. However, many hybrid SDE models in the real world do not fulfill this condition (namely, they are highly nonlinear) and hence there is a need to develop a new theory for these highly nonlinear SDE models. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid SDEs whose coefficients satisfy the polynomial growth condition.

    Mathematics Subject Classification: Primary: 60H10, 60J10; Secondary: 93D15.


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  • Figure 4.1.  The computer simulation of the sample paths of the Markov chain and the SDDE (2.4) with control (4.2) and $ \tau = 0.06 $ using the Euler–Maruyama method with step size $ 10^{-4} $.

    Figure 4.2.  The computer simulation of the sample paths of the Markov chain and the SDDE (2.4) with control (4.3) and $ \tau = 0.01 $ using the Euler–Maruyama method with step size $ 10^{-4} $

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