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Robust stability and stabilization of the family of jumping stochastic systems. (English) Zbl 0942.93043
The author investigates exponential stability in mean square of the trivial solution of a linear stochastic differential equation driven by white noise, whose coefficients are governed by a finite-state Markov process $$i_t$$ and which is controlled by a nonlinear feedback $$u(t)$$.
Sufficient conditions are derived which (for given and arbitrary intensities of the white noise) yield the stability property in question, whatever the jump intensity $$q_{ij}$$ of the Markovian background noise $$i_t$$ may be (robustness).
Method: The problem is reduced to the algebraic problem of finding a positive definite matrix $$H$$ which satisfies a system of nonlinear matrix equations labeled by the states $$i$$ of the Markovian background noise.

##### MSC:
 93E15 Stochastic stability in control theory 93D15 Stabilization of systems by feedback 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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