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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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