Skip to content

Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces

Research output: Contribution to journalArticle

Standard

Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces. / Szalai, Robert.

In: Journal of Nonlinear Science, Vol. 29, No. 3, 15.06.2019, p. 897-960.

Research output: Contribution to journalArticle

Harvard

APA

Vancouver

Author

Szalai, Robert. / Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces. In: Journal of Nonlinear Science. 2019 ; Vol. 29, No. 3. pp. 897-960.

Bibtex

@article{0a2972d48be04ed19dda461db9644b59,
title = "Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces",
abstract = "In this paper, a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence, the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper, we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced-order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.",
keywords = "Piecewise-smooth, Model reduction, Invariant manifolds, Non-dense domain",
author = "Robert Szalai",
year = "2019",
month = "6",
day = "15",
doi = "10.1007/s00332-018-9508-4",
language = "English",
volume = "29",
pages = "897--960",
journal = "Journal of Nonlinear Science",
issn = "0938-8974",
publisher = "Springer US",
number = "3",

}

RIS - suitable for import to EndNote

TY - JOUR

T1 - Model Reduction of Non-densely Defined Piecewise-Smooth Systems in Banach Spaces

AU - Szalai, Robert

PY - 2019/6/15

Y1 - 2019/6/15

N2 - In this paper, a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence, the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper, we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced-order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.

AB - In this paper, a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence, the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper, we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced-order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.

KW - Piecewise-smooth

KW - Model reduction

KW - Invariant manifolds

KW - Non-dense domain

UR - http://www.scopus.com/inward/record.url?scp=85055993498&partnerID=8YFLogxK

U2 - 10.1007/s00332-018-9508-4

DO - 10.1007/s00332-018-9508-4

M3 - Article

AN - SCOPUS:85055993498

VL - 29

SP - 897

EP - 960

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 3

ER -