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kozyakin committed Dec 14, 2024
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Expand Up @@ -4712,17 +4712,17 @@ @ARTICLE{Dai:JFI14
zblnumber = "1372.93209",
doi = "10.1016/j.jfranklin.2014.01.010",
url = "https://www.sciencedirect.com/science/article/pii/S001600321400012X",
annote = "In this paper, there are shown the following two statements.
\begin{enumerate} \item[(1)] \emph{A discrete-time
annote = "In this paper, there are shown the following two statements.\par
(1) \emph{A discrete-time
Markovian jump linear system is uniformly exponentially
stable if and only if it is robustly periodically stable,
by using a Gel'fand--Berger--Wang formula proved here.}
\item[(2)] \emph{A random linear ODE driven by a semiflow
by using a Gel'fand--Berger--Wang formula proved here.}\par
(2) \emph{A random linear ODE driven by a semiflow
with closing by periodic orbits property is uniformly
exponentially stable if and only if it is robustly
periodically stable, by using Shantao Liao's perturbation
technique and the semi-uniform ergodic theorems.}
\end{enumerate} The proofs involve ergodic theory in both
technique and the semi-uniform ergodic theorems.}\par
The proofs involve ergodic theory in both
of the above two cases. In addition, counterexamples are
constructed to the robustness condition and to spectral
finiteness of linear cocycle.",
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specifically, let $\rho(A)$ stand for the spectral radius
of a matrix $A\in\mathbb{R}^{d\times d}$, then the outline
of results obtained in this paper
are:\begin{enumerate}\item[(1)] \emph{For the case $d=2$,
are:\par (1) \emph{For the case $d=2$,
$\boldsymbol{S}$ is absolutely stable (i.e.,
$\|S_{\sigma_{\!n}}\cdots S_{\sigma_{\!1}}\|\to0$ driven by
all switching signals $\sigma$) if and only if
$\rho(S_1),\rho(S_2)$ and $\rho(S_1S_2)$ all are less than
$1$;}\item[(2)] \emph{For the case $d=3$, $\boldsymbol{S}$
$1$;}\par (2) \emph{For the case $d=3$, $\boldsymbol{S}$
is absolutely stable if and only if $\rho(A)<1\;\forall
A\in\{S_1,S_2\}^\ell$ for $\ell=1,2,3,4,5,6$, and
$8$.}\end{enumerate} This further implies that for any
$8$.}\par This further implies that for any
$\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times d}$
with the generalized spectral radius
$\rho(\boldsymbol{S})=1$ where $d=2$ or $3$, if
Expand Down Expand Up @@ -7134,11 +7134,11 @@ @ARTICLE{GSM:IEEETAC07
the, convex hull of a set of Metzler matrices is a
necessary and sufficient condition for the asymptotic
stability of the associated switched linear system under
arbitrary switching. In this note, we show that (1) this
arbitrary switching. In this note, we show that\par(1) this
conjecture is true for systems constructed from a pair of
second-order Metzler matrices; (2) the conjecture is true
second-order Metzler matrices;\par(2) the conjecture is true
for systems constructed from an arbitrary finite number of
second-order Metzler matrices; and (3) the conjecture is in
second-order Metzler matrices; and\par(3) the conjecture is in
general false for higher order systems. The implications of
our results, both for the design of switched positive
linear systems, and for research directions that arise as a
Expand Down Expand Up @@ -16138,24 +16138,24 @@ @MISC{JSRToolbox
needed, for instance by setting a maximal computation time
or by fine-tuning a particular step in an algorithm. The
main steps of the default algorithm are the following:
\begin{itemize} \item Try to transform the problem into a
set of smaller independent problems. This is possible when
the matrices in the set $\Sigma$ are simultaneously
blocktriangularizable. \item If the matrices are
\par$\bullet$ Try to transform the problem into a set of
smaller independent problems. This is possible when the
matrices in the set $\Sigma$ are simultaneously
blocktriangularizable.\par$\bullet$ If the matrices are
nonnegative, start with the pruning algorithm in order to
get some bounds $[\beta^-, \beta^+]$ on the joint spectral
radius, then compute the joint conic radius, using the
positive orthant as cone, and the ellipsoidal norm
approximation using $[\beta^-, \beta^+]$ as initial bounds.
\item If some matrices have negative entries, start with a
variant of Gripenberg's algorithm in order to get some
initial bounds and a candidate product. This variant may
rescale the matrices during the computation in order to
avoid overflows. After this first step, compute the
ellipsoidal norm approximation and, if needed, try to
certify optimality or to find a better candidate product
using a balanced complex polytope method or a conitope
method, which is a lifted polytope method. \end{itemize}",
approximation using $[\beta^-, \beta^+]$ as initial
bounds.\par$\bullet$ If some matrices have negative
entries, start with a variant of Gripenberg's algorithm in
order to get some initial bounds and a candidate product.
This variant may rescale the matrices during the
computation in order to avoid overflows. After this first
step, compute the ellipsoidal norm approximation and, if
needed, try to certify optimality or to find a better
candidate product using a balanced complex polytope method
or a conitope method, which is a lifted polytope method.",
}

@INPROCEEDINGS{VHJ:ACM14,
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