Continuous feedback stabilization for a class of affine stochastic nonlinear systems

Abstract

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On the simultaneous stabilization of stochastic nonlinear systems

“…Definition [15] A $$$ {C}^2 $$$ positive definite and proper function $$$ {V}_i,i\in I $$$, defined on $$$ {\mathbb{R}}^n $$$ is a SCLF for system $$$ {S}_i,i\in I $$$, if for any $$$ x\in {\mathbb{R}}^n\backslash \left\{0\right\} $$$ $$$$ \underset{u\in \mathbb{R}}{\operatorname{inf}}{\mathcal{L}}_u{V}_i(x)=\underset{u\in \mathbb{R}}{\operatorname{inf}}\left({a}_i(x){u}^2+{b}_i(x)u+{c}_i(x)\right)<0. $$$$ A SCLF $$$ {V}_i $$$ of system $$$ {S}_i,i\in I $$$, is said to satisfy the SCP, if for each $$$ \varepsilon >0 $$$, there is $$$ \delta >0 $$$ such that if …”

“…Theorem [15]. If $$$ {V}_i $$$ is a SCLF for system $$$ {S}_i,i\in I $$$, which satisfies the SCP and Assumption 1 holds, then the feedback law $$$$ {p}_i(x)=\left\{\begin{array}{ll}{v}_i(x),& \kern0.20em \mathrm{if}\kern0.50em {\Delta}_i(x)>0,\\ {}0,& \kern0.20em \mathrm{if}\kern0.50em {\Delta}_i(x)\le 0,\end{array}\right.$$…”

“…Theorem 1 [15]. If V i is a SCLF for system S i , i ∈ I, which satisfies the SCP and Assumption 1 holds, then the feedback law…”

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

“…Many real systems are nonlinear and are also prone to stochastic phenomena. These features have naturally attracted attention in the control literature with studies examining stabilization problems for stochastic nonlinear systems, and some interesting results obtained during the past few decades, see [1][2][3][4][5][6][7] and references therein. These results were obtained using classical stochastic theory as found, for instance, in [8] and [9].…”

“…In order to relax this restriction, Li and Liu [13] generalized the classical theorems on global stability to cover stochastic nonlinear systems without a local Lipschitz condition. Using the results in [13], several authors have obtained sufficient stabilizability conditions, for example, [5,14], allowing closed-loop systems to be merely continuous without requiring them to satisfy a local Lipschitz condition. It should be pointed out that all results mentioned above are about stability and stabilization in probability problems.…”