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This paper focuses on a class of functional diffusion systems with singularly perturbed regime switching, where the modulating Markov chain has a large state space and undergoes weak and strong interactions. By using the martingale method and weak convergence, this paper shows that the underlying system will weakly converge to a limit system, which is simpler than the original system. For a class of integro-differential diffusion system with singularly perturbed regime switching, as a class of special functional diffusion system, this paper demonstrates that if the limit system is moment exponentially stable, the original system with singular perturbation is also moment exponentially stable under suitable conditions. This result is interesting since the limit system is always simpler. Finally, an example is given to illustrate this result.

This paper considers a class of hybrid stochastic differential equations (SDEs) with different structures in different modes. In some modes, the coefficients of the SDEs satisfy the linear growth condition, while in the other modes, the coefficients are highly nonlinear. These systems are often unstable. This paper aims to design a discrete-time feedback control which is only put in a part of modes where the coefficients are highly nonlinear, such that these systems become stable. The stabilities concerned include H∞-stability and exponential stability in the moment, as well as almost sure stability. Finally, an example is given to illustrate these results.

This paper considers a given neutral functional system under regime switching whose solution grows at most polynomially. Can the system tolerate Brownian noise and colour noise to preserve the polynomial growth if the system is subject to another environment noise? The answer is positive. This paper shows that environmental noise and regime switching work together to make the original neutral differential system whose solution grows at most polynomially become a new neutral functional differential system whose solution will still preserve polynomial growth. This indicates that this neutral functional differential system can tolerate small noise without losing the property of polynomial growth, which implies robustness. This paper also proves that the criterion is suitable to neutral delay system which is as a special case of functional system. Finally, two examples are given to illustrate the main theory.