SCIENCE CHINA Information Sciences, Volume 61, Issue 7:
070213(2018)
https://doi.org/10.1007/s11432-017-9297-1

More info

- ReceivedAug 22, 2017
- AcceptedOct 27, 2017
- PublishedJun 13, 2018

Although the mean square stabilization of hybrid systems byfeedback control based on discrete-time observations of state and mode has been studied by severalauthors since 2013,the corresponding almost sure stabilization problemhas received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system$\dot~x(t)=f(x(t))$ by a linear discrete-time stochastic feedback control $Ax([t/\T]\T){\rm~d}B(t)$ (namely the stochastically controlled systemhas the form ${\rm~d}x(t)=f(x(t)){\rm~d}t~+~Ax([t/\T]\T){\rm~d}B(t)$), where $B(t)$ is a scalar Brownian, $\T~>0$, and $[t/\T]$ is the integer part of $t/\T$. In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system$\dot~x(t)=f(x(t),r(t))$ by nonlinear discrete-time stochastic feedback control $u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$ (so the stochastically controlled system is a hybrid stochastic system of the form ${\rm~d}x(t)=f(x(t),r(t)){\rm~d}t~+~u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$), where $B(t)$ is a multi-dimensional Brownian motion and $r(t)$ is a Markov chain.

The authors would like to thank Leverhulme Trust (Grant No. RF-2015-385), Royal Society (Grant No. WM160014, Royal Society Wolfson Research Merit Award), Royal Society and Newton Fund (Grant No. NA160317, Royal Society–Newton Advanced Fellowship), Engineering and Physics Sciences Research Council (Grant No. EP/K503174/1), National Natural Science Foundation of China (Grant Nos. 61503190, 61473334, 61403207), Natural Science Foundation of Jiangsu Province (Grant Nos. BK20150927, BK20131000), and Ministry of Education (MOE) of China (Grant No. MS2014DHDX020) for their financial support. The first author would like to thank Chinese Scholarship Council for awarding him the scholarship to visit the University of Strathclyde for one year.

**Appendix**

Once Assumption

$$ a_{ij}\le 0 \hbox{ for all } i\not =j. $$

Here, we cite a useful lemma on $M$-matrices.

(1) $A$ is a nonsingular $M$-matrix.

(2) $A$ is semi-positive; that is, there exists $x~>~0$ in $\RR^N$ such that $Ax~>~0$.

(3) $A^{-1}$ exists and its elements are all nonnegative.

(4) All the leading principal minors of $A$ are positive; that is

$$ \left| \begin{matrix} a_{11} & \cdots & a_{1k} \\ \vdots & &\vdots \\ a_{k1} & \cdots & a_{kk} \end{matrix} \right| > 0, \hbox{\rm for every } k=1, 2,\ldots, N. $$

We also need another classical result.

$$ \sum_{j=1}^N a_{ij} >0, \forall i=1,2,\ldots, N, $$

$\hbox{\rm~det}~A~>~0$.Now, let us propose a condition that

\begin{equation} \left| \begin{array}{cccc} -(\a_1 + 0.5\r_1^2 -\s_1^2) & -\g_{12} & \cdots & -\g_{1N} \\ -(\a_2 + 0.5\r_2^2 -\s_2^2) & -\g_{22} & \cdots & -\g_{2N} \\ \vdots & \vdots & \cdots &\vdots \\ -(\a_N + 0.5\r_N^2 -\s_N^2) & -\g_{N2} & \cdots & -\g_{NN} \end{array} \right| >0, \tag{50}\end{equation}

where $\a_i,~\r_i,~~\s_i$ are the constants specified in AssumptionIt was shown in

\begin{equation} \hbox{ for some $u\in \SM$, $\g_{iu} > 0$, for all $i\not= u$, } \tag{51}\end{equation}

condition (\begin{equation} \sum_{i=1}^N \pi_i (\a_i + 0.5\r_i^2 -\s_i^2) < 0, \tag{52}\end{equation}

\begin{equation} \hbox{for some $u\in \SM$, } \g_{iu} \ve (\s_i^2 - 0.5\r_i^2 -\a_i) > 0, \hbox{for all $i\not= u$}. \tag{53}\end{equation}

proof Without loss of generality, we may assume that the state $u=N$ in condition (

\begin{equation} \g_{iN} \ve (\s_i^2 - 0.5\r_i^2 -\a_i) > 0, \hbox{for all $1\le i \le N-1$}. \tag{54}\end{equation}

If not, by switching state $u$ with $N$, we need to reorder the states of the Markov chain $r(t)$ that is, rename state $u$ as $N$ and $N$ as $u$. Consequently, the determinant in the left-hand side of ( By

\begin{align} \hbox{\rm det} {\cal A}(p) > 0. \tag{55} \end{align}

On the other hand, for each $i=1,2,\ldots,~N-1$, either $\g_{iN}~>~0$ or $\g_{iN}=0$. In the case when $\g_{iN}~>~0$, we clearly have$$ \theta_i(p) > -\g_{iN}, \hbox{for all sufficiently small $p\in (0,1)$}; $$

$$ \theta_i(p) > 0 = -\g_{iN}, \hbox{for all sufficiently small $p\in (0,1)$.} $$

\begin{align} \theta_i(p) > -\g_{iN}, i=1,2,\ldots, N-1, \tag{56} \end{align}

$$ {\cal A}_k(p) := \left[ \begin{array}{cccc} \theta_1(p)-\g_{11} & -\g_{12} & \cdots & -\g_{1k} \\ -\g_{21} & \theta_1(p)-\g_{22} & \cdots & -\g_{2k} \\ \vdots & \vdots & \cdots &\vdots \\ -\g_{k1} & -\g_{k2} & \cdots & \theta_k(p)-\g_{kk} \end{array} \right] $$

$$ \theta_i(p) - \sum_{j=1}^k \g_{ij} = \theta_i(p) + \sum_{j=k+1}^N \g_{ij} \ge \theta_i(p) +\g_{iN} >0 $$

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