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Stochastic input-to-state stability and applications to global feedback stabilization. (English) Zbl 0953.93073

The author considers the stochastic composite system described by the following stochastic differential Ito equations

${{\Sigma }}_{1}:dx={f}_{1}\left(x,u\right)dt+{g}_{1}\left(x,u\right)dw\left(t\right)$
${{\Sigma }}_{2}:dy={f}_{2}\left(x,y,u\right)dt+{g}_{2}\left(x,y,u\right)dw\left(t\right)$

where $\left(x,y\right)\in {ℝ}^{n}×{ℝ}^{k}$ are the vectors of state, $u\in {ℝ}^{m}$ is the control (input) vector, ${f}_{i},{g}_{i},i=1,2$ are nonlinear vector functions of appropriate dimensions and $w\in ℝ$ is a standard Wiener process.

The author introduces new definitions of stochastic $\gamma$-input to state stability and derives sufficient conditions for global stabilization in two cases, i.e., by means of output (linear and bounded) static feedback and by a dynamic feedback controller.

MSC:
 93E15 Stochastic stability 93D15 Stabilization of systems by feedback 93D25 Input-output approaches to stability of control systems 93A15 Large scale systems