Semidefinite approximations of invariant measures for polynomial systems
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F19%3A00334317" target="_blank" >RIV/68407700:21230/19:00334317 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.3934/dcdsb.2019165" target="_blank" >https://doi.org/10.3934/dcdsb.2019165</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.3934/dcdsb.2019165" target="_blank" >10.3934/dcdsb.2019165</a>
Alternative languages
Result language
angličtina
Original language name
Semidefinite approximations of invariant measures for polynomial systems
Original language description
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a conic optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies.Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure.The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure.We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Discrete and Continuous Dynamical Systems - B
ISSN
1531-3492
e-ISSN
1553-524X
Volume of the periodical
24
Issue of the periodical within the volume
12
Country of publishing house
US - UNITED STATES
Number of pages
26
Pages from-to
6745-6770
UT code for WoS article
000484545100021
EID of the result in the Scopus database
2-s2.0-85072559809