Evolution of measures in nonsmooth dynamical systems: Formalisms and computation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F25%3A00388292" target="_blank" >RIV/68407700:21230/25:00388292 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.automatica.2025.112402" target="_blank" >https://doi.org/10.1016/j.automatica.2025.112402</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.automatica.2025.112402" target="_blank" >10.1016/j.automatica.2025.112402</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Evolution of measures in nonsmooth dynamical systems: Formalisms and computation

Popis výsledku v původním jazyce

This article develops mathematical formalisms and provides numerical methods for studying the evolution of measures in nonsmooth dynamical systems using the continuity equation. The nonsmooth dynamical system is described by an evolution variational inequality and we derive the continuity equation associated with this system class using three different formalisms. The first formalism consists of using the superposition principle to describe the continuity equation for a measure that disintegrates into a probability measure supported on the set of vector fields and another measure representing the distribution of system trajectories at each time instant. The second formalism is based on the regularization of the nonsmooth vector field and describing the measure as the limit of a sequence of measures associated with the regularization parameter. In doing so, we obtain quantitative bounds on the Wasserstein metric between measure solutions of the regularized vector field and the limiting measure associated with the nonsmooth vector field. The third formalism uses a time-stepping algorithm to model a time-discretized evolution of the measures and show that the absolutely continuous trajectories associated with the continuity equation are recovered in the limit as the sampling time goes to zero. We also validate each formalism with numerical examples. For the first formalism, we use polynomial optimization techniques and the moment-SOS hierarchy to obtain approximate moments of the measures. For the second formalism, we illustrate the bounds on the Wasserstein metric for an academic example for which the closed-form expression of the Wasserstein metric can be calculated. For the third formalism, we illustrate the time-stepping based algorithm for measure evolution on an example that shows the effect of the concentration of measures. (c) 2025 Published by Elsevier Ltd.

Název v anglickém jazyce

Evolution of measures in nonsmooth dynamical systems: Formalisms and computation

Popis výsledku anglicky

This article develops mathematical formalisms and provides numerical methods for studying the evolution of measures in nonsmooth dynamical systems using the continuity equation. The nonsmooth dynamical system is described by an evolution variational inequality and we derive the continuity equation associated with this system class using three different formalisms. The first formalism consists of using the superposition principle to describe the continuity equation for a measure that disintegrates into a probability measure supported on the set of vector fields and another measure representing the distribution of system trajectories at each time instant. The second formalism is based on the regularization of the nonsmooth vector field and describing the measure as the limit of a sequence of measures associated with the regularization parameter. In doing so, we obtain quantitative bounds on the Wasserstein metric between measure solutions of the regularized vector field and the limiting measure associated with the nonsmooth vector field. The third formalism uses a time-stepping algorithm to model a time-discretized evolution of the measures and show that the absolutely continuous trajectories associated with the continuity equation are recovered in the limit as the sampling time goes to zero. We also validate each formalism with numerical examples. For the first formalism, we use polynomial optimization techniques and the moment-SOS hierarchy to obtain approximate moments of the measures. For the second formalism, we illustrate the bounds on the Wasserstein metric for an academic example for which the closed-form expression of the Wasserstein metric can be calculated. For the third formalism, we illustrate the time-stepping based algorithm for measure evolution on an example that shows the effect of the concentration of measures. (c) 2025 Published by Elsevier Ltd.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

20205 - Automation and control systems

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Automatica

ISSN

0005-1098

e-ISSN

1873-2836

Svazek periodika

179

Číslo periodika v rámci svazku

September

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

15

Strana od-do

—

Kód UT WoS článku

001511906500004

EID výsledku v databázi Scopus

2-s2.0-105007343897