Evolution of measures in nonsmooth dynamical systems: Formalisms and computation

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Evolution of measures in nonsmooth dynamical systems: Formalisms and computation

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

20205 - Automation and control systems

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

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

Name of the periodical

Automatica

ISSN

0005-1098

e-ISSN

1873-2836

Volume of the periodical

179

Issue of the periodical within the volume

September

Country of publishing house

GB - UNITED KINGDOM

Number of pages

15

Pages from-to

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UT code for WoS article

001511906500004

EID of the result in the Scopus database

2-s2.0-105007343897