Abstract
This paper is concerned with the almost sure exponential stability of the n-dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ1(xt,t),r(t),t)dt+g(ψ2(xt,t),r(t),t)dB(t), where xt={x(t+u):−τ≤u≤0} is a C([−τ,0];Rn)-valued process, B(t) is an m-dimensional Brownian motion while r(t) is a Markov chain. We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t)=f(y(t),r(t),t)dt+g(y(t),r(t),t)dB(t) is almost surely exponentially stable, then there exists a positive number τ⁎ such that the SFDE is also almost surely exponentially stable as long as τ<τ⁎. We also describe a method to determine τ⁎ which can be computed numerically in practice.
Highlights
This paper is concerned with the almost sure exponential stability of the n-dimensional nonlinear hybrid stochastic functional differential equation (SFDE) of the form dx(t) = f (ψ1(xt, t), r(t), t)dt + g(ψ2(xt, t), r(t), t)dB(t). (1.1)Here B(t) is an m-dimensional Brownian motion, r(t) is a Markov chain on the finite state space S = {1, 2, · · ·, N }, xt = {x(t + s) : −τ ≤ s ≤ 0}, τ is a positive number, ψ1, ψ2 : C([−τ, 0]; Rn) × R+ → Rn, f : Rn × S × R+ → Rn and g : Rn × S × R+ → Rn×m
We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t) = f (y(t), r(t), t)dt + g(y(t), r(t), t)dB(t) is almost surely exponentially stable, there exists a positive number τ ∗ such that the SFDE is almost surely exponentially stable as long as τ < τ ∗
In this paper we investigated the almost sure exponential stability of the n-dimensional nonlinear hybrid SFDE (2.1)
Summary
Where B(t) is a scalar Brownian motion and σ is positive number They showed that the SDDE (1.2) is almost surely exponentially stable provided the time delay τ is sufficiently small. Where σ is positive number and ψ is a Lipschitz continuous functional from C([−τ, 0]; R) to R such that inf |φ(s)| ≤ |ψ(φ)| ≤ sup |φ(s)|, ∀φ ∈ C([−τ, 0]; R) He showed that equation (1.3) is almost surely exponentially stable provided τ is sufficiently small. Is almost surely exponentially stable, so is the SDDE (1.4) provided the time delays are sufficiently small The reason why it has taken almost 20 years to make these progresses in this area is because SFDEs (including SDDEs) are infinite-dimensional systems which are significantly different from SDEs. For example, it is straightforward to show that the linear scalar SDE dx(t) = σx(t)dB(t) is almost surely exponentially stable by applying the Itô formula to log(x(t))
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