Semidefinite approximations of invariant measures for polynomial systems

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Semidefinite approximations of invariant measures for polynomial systems

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Semidefinite approximations of invariant measures for polynomial systems

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete and Continuous Dynamical Systems - B

ISSN

1531-3492

e-ISSN

1553-524X

Svazek periodika

24

Číslo periodika v rámci svazku

12

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

26

Strana od-do

6745-6770

Kód UT WoS článku

000484545100021

EID výsledku v databázi Scopus

2-s2.0-85072559809