# Stability analysis of ODE disturbed by a random variable [closed]

I have a question concerning the stability analysis for a kind of differential equation taking the form $$\dot x=Ax+Bw,$$ where $A\in \mathbb{R}^{n \times n}$, $B\in \mathbb{R}^{n \times m}$ are constant matrices and $w \in \mathbb{R}^m$ is a normal random variable, i.e., $w\sim \mathcal{N}(0,W)$ with $W$ be a symmetric and positive definite matrix.

Since the zero is not an equilibrium of the system, the Lyapunov analysis does not make sense. When the input-to-state stability analysis is considered, the robust control theory does not apply due to the unboundness of $w$. By resorting to stochastic stability in the sense of mean square or almost surely, the Ito formula seems to be invalid.

HOW to carry out the stability analysis of this kind of systems? Any pointer will be helpful. Thanks!

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## closed as off-topic by Benoît Kloeckner, Ryan Budney, Chris Godsil, Carlo Beenakker, David WhiteNov 3 '13 at 14:44

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It seems to me that both Lyapunov analysis and control theory make perfect sense in this application. What you cannot hope to achieve is convergence to a fixed equilibrium point. Concepts such as input-to-state stability were formulated precisely to study systems of this kind. –  Pait Aug 11 '13 at 21:13
As mentioned in an answer, the randomness is mostly irrelevant since $w$ is a random variable, not a random process. You are reduced to studying the systems $\dot x = Ax +v$ with $v$ a fixed vector, this is standard and seems not suitable for MO. –  Benoît Kloeckner Nov 3 '13 at 9:55

## 2 Answers

This is a stochastic differential system of equations. I think this can help. You can also check the full syllabus of the course here with the proper references and other material to download.

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Thank you for your valuable references. Since $w$ is just a random variable rather than a random process, I am not sure wether or not the standard stochastic analyis could be used here. –  W. Nyway Jul 13 '13 at 2:47

Randomness is irrelevant here. For any value of $w$, the change of variables $x=y-A^{-1}Bw$ reduces your system to $\dot y=Ay$ that you know how to treat.

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Thanks to Bakhtin. The matrix $A$ is not necessarily invertable. –  W. Nyway Jul 13 '13 at 2:41
In the general situation, take the Jordan decomposition of $A$ and project the dynamics onto generalized eigenspaces corresponding to zero and nonzero eigenvalues. The randomness is still only mildly relevant and boils down to how a typical value of $Bw$ projects. –  Yuri Bakhtin Jul 13 '13 at 20:37
Let me try as your suggestion. Anyway, thank you again. –  W. Nyway Jul 15 '13 at 0:43
This change of variables puts in play the derivative of $w$, which may be problematic when $w$ is a random variable. –  Pait Aug 11 '13 at 21:09