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Decentralized stabilization of a class of interconnected stochastic nonlinear systems. (English) Zbl 0983.93003

The authors consider an interconnected large-scale stochastic system consisting of $N$ subsystems given by the following nonlinear stochastic differential equations

$d{x}_{ij}={x}_{i,j+1}dt+{\varphi }_{ij}\left({\overline{x}}_{ij}\right)d{w}_{i}+{h}_{ij}\left({\overline{x}}_{ij},{X}_{i1}\right),\phantom{\rule{1.em}{0ex}}1\le j\le {n}_{i}-1$
$d{x}_{i{n}_{i}}={u}_{i}dt+{\varphi }_{i{n}_{i}}\left({\overline{x}}_{i{n}_{i}}\right)d{w}_{i}+{h}_{i{n}_{i}}\left({\overline{x}}_{i{n}_{i}},{X}_{i1}\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\overline{x}}_{ij}={\left[{x}_{i1},\cdots ,{x}_{ij}\right]}^{T}$, $j=1,\cdots ,{n}_{i}$, $i=1,\cdots ,N$ are the states of the $i$th subsystem ${X}_{i1}=\left[{x}_{11},{x}_{21},\cdots ,{x}_{i-1,1}$, ${x}_{i+1,1},\cdots ,{x}_{N1}{\right]}^{T}$, ${u}_{i}\in ℝ$ is the control of the $i$th subsystem, ${w}_{i}$, $i=1,\cdots ,N$ are ${r}_{i}$-dimensional standard Wiener processes, ${\varphi }_{ij}\left({\overline{x}}_{ij}\right)$, $j=1,\cdots ,{n}_{i}$ are ${r}_{i}$-dimensional smooth row vector functions with ${\varphi }_{ij}\left(0\right)=0$, and ${h}_{ij}\left({\overline{x}}_{ij},{X}_{i1}\right)$, $j=1,\cdots ,{n}_{i}$ are ${r}_{i}$-dimensional smooth interconnections between the $i$th subsystem and $j$th subsystem with ${h}_{ij}\left(0,0\right)=0$. The interconnections are bounded by strong nonlinear functions that contain first-order and higher-order polynomials as special cases.

The authors present a decentralized control such that the closed-loop interconnected system (1) is globally asymptotically stable in probability for all admissible interconnections. They show that the decentralized global stabilization via both state feedback and output feedback can be solved by a Lyapunov-based recursive design method.

##### MSC:
 93A14 Decentralized systems 93D15 Stabilization of systems by feedback 93A15 Large scale systems 93E15 Stochastic stability