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# pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations

• * Corresponding author: Yi Zhang
• In this paper, we discuss the $p$th moment exponential stabilization of continuous-time hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. The hybrid stochastic functional differential equations are also known as stochastic functional differential equations with the Markovian switching. We follow Mao's paper to consider the auxiliary system whose control is based on continuous-time state observation. The lemma is provided that if the $p$th moment of the solution $y(t)$ of the auxiliary system decays exponentially then the same with the $p$th moment of the functional $y_t$. With the help of this lemma, the criterion for $p$th moment exponential stability of the primary system is given, and the margin of the duration of the discrete-time state observation is presented. Then the special case like the linear system is considered and the discrete-time feedback control is designed.

Mathematics Subject Classification: 34H15, 34K50, 93E03, 93E15.

 Citation: • • Figure 1.  Computer simulation of the paths of $r(t)$, $x_1(t)$ and $x_2(t)$ for the discrete-time controlled system (17) using the Euler-Maruyama method with step size $10^{-5}$, delay $\tau = 0.01$ and initial values $r(0) = 1$, $\xi_1 \equiv -20$ and $\xi_2 \equiv 10$ on $[-\tau,0]$

Figure 2.  Computer simulation of the paths of $r(t)$, $x_1(t)$ and $x_2(t)$ for the discretetime controlled system (18) with $\alpha = 10^{-4}$ using the Euler-Maruyama method with step size $10^{-5}$, delay $\tau = 0.01$ and initial values $r(0) = 1$, $\xi_1 \equiv -20$ and $\xi_2 \equiv 10$ on $[-\tau,0]$

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