NDLI: Stable Set-Valued Integration of Nonlinear Dynamic Systems using Affine Set-Parameterizations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Houska, Boris Villanueva, Mario E. Chachuat, BenoÎt
Copyright Year	2015
Abstract	Many set-valued integration algorithms for parametric ordinary differential equations (ODEs) implement a combination of Taylor series expansion with either interval arithmetic or Taylor model arithmetic. Due to the wrapping effect, the diameter of the solution-set enclosures computed with these algorithms typically diverges to infinity on finite integration horizons, even though the ODE trajectories themselves may be asymptotically stable. This paper starts by describing a new discretized set-valued integration algorithm that uses a predictor-validation approach to propagate generic affine set-parameterizations, whose images are guaranteed to enclose the ODE solution set. Sufficient conditions are then derived for this algorithm to be locally asymptotically stable, in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit (or equilibrium point). The key requirement here is quadratic Hausdorff convergence of function extensions in the chosen affine set-parameterization, which is proved to be the case, for instance, for Taylor models with ellipsoidal remainders. These stability properties are illustrated with the case study of a cubic oscillator system.
Starting Page	2307
Ending Page	2328
Page Count	22
File Format	PDF
ISSN	00361429
DOI	10.1137/140976807
e-ISSN	10957170
Issue Number	5
Volume Number	53
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-10-20
Access Restriction	Subscribed
Subject Keyword	reachability analysis set-valued integration Computational methods Asymptotic properties stability analysis affine set-parameterization ordinary differential equations Asymptotic stability convergence analysis Error bounds Nonconvex programming, global optimization Initial value problems Stability of manifolds of solutions General methods in interval analysis
Content Type	Text
Resource Type	Article
Subject	Computational Mathematics Applied Mathematics Numerical Analysis

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Libarray of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration

Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates

Convergence Analysis of Meshfree Approximation Schemes

On Taylor Model Based Integration of ODEs

Stable and Convergent Unsymmetric Meshless Collocation Methods

On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations

Internal Error Propagation in Explicit Runge--Kutta Methods

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results

Stable Set-Valued Integration of Nonlinear Dynamic Systems using Affine Set-Parameterizations

Similar Documents

Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration

Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates

Convergence Analysis of Meshfree Approximation Schemes

On Taylor Model Based Integration of ODEs

Stable and Convergent Unsymmetric Meshless Collocation Methods

On the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations

Internal Error Propagation in Explicit Runge--Kutta Methods

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results

Stable Set-Valued Integration of Nonlinear Dynamic Systems using Affine Set-Parameterizations