One of the main stability theories for stochastic systems is stochastic Lyapanuv stability theory, it is the same as Lyapanuv theory for deterministic systems.
the main idea is that for the stochastic system:
dx=f(x)dt+g(x)dwt
the differential operator LV(infinitesimal generator- the derivative of the Lyapanuv function) be negative definite.
there is another assumption for this theory:
f(0)=g(0)=0
and this implies that at equilibrium point (here x_e=0) the disturbance vanishes automatically.
what I want to know is that is it a reasonable assumption?
i.e in engineering context, is it reasonable to assumed that the disturbance will vanish at the equilibrium point?
From my practical experience if f(0)=0, then g(0) doesn't equal 0, because of the sensors noises.
If I am reading your question right, from a very practical point - no - at least with open computational systems. A system may transiently be at equilibrium and for temporal reasons we see no disturbance and little noise. There may, or may not, be periodicity around the disturbance and if there is periodicity, its my be outside the range of the control loop (e.g. a control loop may operate over a 30s range and the periodicity may be 24 hrs / weekly /monthly or annually).
This hypothesis is not necessary in any cases. In reality this is not a hypothesis but a property inherent in the mathematical system used to model the phenomenon. For example, consider a stochastic epidemiological model described by a stochastic differential equation ( this is the case e.g. in the stochastic perturbed S-I-S epidemic model ) :
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dI(t) = f(I(t))dt+g(I(t))dw_t, (1)
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where I(t) is the number the infectious individuals in the population at time t. The corresponding deterministic model is given by
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dI(t)=f(I(t))dt. (2)
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In general, only humains are the main host of the disease studied and in this case, we must have f(I_e) = 0. Then "I_e = 0" is said to be the " disease free-equilibrium " of the deterministic model (2). Then the unstability or stability of the disease free-equilibrium determines whether the disease will persist in a population or the disease simply dies out.
- If we have the stability of the disease free-equilibrium "I_e = 0", then I(t) vanishes at infinity.
- If the disease free-equilibrium "I_e = 0" is unstable, then "lim_inf I(t) > 0" when t tends to infinity.
The stochastic model (1) is obtained by disturbing one of the parameters from deterministic (2). The choice of the parameter to
disturbing determines if the disease free-equilibrium I_e of the
deterministic (1) model is also disease free-equilibrium of the stochastic model (2) that is
f(I_e) = g(I_e) = 0.
Then of course there may be cases where f(I_e) and g(I_e) are not the
same, that is the disease free-equilibrium "I_e" of deterministic model (2) is not a disease free-equilibrium of the stochastic model (1). In this case, the Lyapunov technique with the differential operator LV is not usable but we can use another analysis based on the stochastic process and differential equation theory to determines the asymptotic behavior of the stochastic model (2) around "I_e."
Thank you all
Thanks Boubacar Sidiki Kouyate for your comprehensive answer.
So it seems that if a system is stochastic, most of the time the process noise will not necessarily vanishes at the equilibrium point and it is more a mathematical assumption that helps us to study the stability of the trajectories of the stochastic system using Lyapunov theory, not a practical assumption.
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