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Almost sure stability of linear stochastic systems with Poisson process coefficients. (English) Zbl 0613.60053

Given the linear stochastic differential equation \[ (1)\quad dx(t)=Ax(t)dt+\sum^{m}_{i=1}B_ ix(t)dN_ i(t),\quad x(0)=x_ 0\in {\mathbb{R}}^ n\setminus \{0\}, \] with A and \(B_ i\) constant \(n\times n\) matrices and \(N_ i(t)\) independent Poisson processes with intensity \(\lambda_ i>0\), \(i=1,...,m.\)
The question of a.s. stability of x(t) is investigated in the framework of products of random matrices and Lyapunov exponents. Specifically, it is shown that the fundamental matrix of (1) has an a.s. exponential growth rate r, the sign of which decides about stability. A Khas’minskij type formula for r in terms of invariant measures on \(S^{n-1}\) is given. Several examples are treated.
A large deviations result is proved: If \(r<0\) then there exists a \(\gamma <0\) such that P(\(\sup_{s\geq t}| x(s)| \geq R)\leq M \exp (\gamma t)\), \(t\geq 0\). Finally, the problem of stabilizing an unstable system by feedback control is investigated.
Reviewer: L.Arnold

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
34F05 Ordinary differential equations and systems with randomness
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