Stabilisation of highly non‐linear continuous‐time hybrid stochastic differential delay equations by discrete‐time feedback control

Abstract

In this paper, we consider how to use discrete-time state feedback to stabilize hybrid stochastic differential delay equations. The coefficients of these stochastic differential delay equations do not satisfy the conventional linear growth conditions, but are highly nonlinear. Using the Lyapunov functional method, we show that a discrete feedback controller u(x([t/τ]τ), r(t), t) can be designed to make the solutions of such controlled hybrid stochastic differential delay equations asymptotically stable and exponentially stable. The upper bound of the discrete observation interval τ is also given in the article. Finally, a numerical examples are given to illustrate our theory.

Feedback delay control of highly nonlinear stochastic functional differential equations with discrete‐time state observations

“…Inspired by this, some scholars have extended this controller based on discrete observations to more general systems, and some have applied it to stochastic stabilization by intermittent control and have achieved many results. 24,25 Recently, based on discrete observation data ([ ∕ ] ), Fei and his collaborators 26,27 designed feedback controllers for highly nonlinear hybrid systems, and studied the asymptotic and exponential stability of the controlled systems. Furthermore, considering that there may be a time lag 0 in the signal transmission of feedback control, Qiu et al 28 designed a more realistic controller ( , ([ ∕ ] − 0 ), ( )) to stabilize the unstable hybrid SDEs.…”

Advances in Discrete-State-Feedback Stabilization of Highly Nonlinear Hybrid Systems by Razumikhin Technique

“…Step 2: The function V has been given by η i |x| 2 + |x| q1+1 in (27). Since q ≥ 2q 1 + 2q 2 − 2, it is easy to see that sup −τ ≤s≤t EV (x(s), s, r(s))…”

“…Meanwhile, the Khasminskii-type condition is given based on Assumption 3.1 since q ≥ 2q 1 + 2q 2 − 2. These two assumptions can also be found in [16] and [27]. But here we have a little stronger restriction on q that q ≥ 2q 1 + 2q 2 − 2, rather than q ≥ 2q 1 ∨ (q 1 + 2q 2 − 1).…”

Stabilization of highly nonlinear neutral stochastic systems with Markovian switching by periodically intermittent feedback control

“…Mao 18 proposed the stabilization problem of continuous‐time hybrid SDEs based on discrete‐time FC. Continuous‐time FC and discrete‐time FC were applied to stabilize highly nonlinear hybrid stochastic systems in References 19,20 and References 21,22, respectively. In addition, References 22 and 23 studied discrete‐time FC for highly nonlinear continuous‐time hybrid SDEs and continuous‐time FC for highly nonlinear neutral SDEs, respectively.…”

“…As we know, the linear growth condition and the global Lipschitz condition is not used to guarantee the unique global solution of highly nonlinear NHSSs. In References 6,21‐23,33,34, the polynomial growth condition and the local Lipschitz condition are established to obtain the unique global solution of highly nonlinear stochastic hybrid systems. Especially, the polynomial growth condition and the Khasminskii‐type condition will cause a lot of challenge in the proof.…”