Positively Invariant Sets for ODEs and Numerical Integration

Peter Giesl; Sigurdur Hafstein; Iman Mehrabinezhad

doi:10.5220/0012189700003543

Positively Invariant Sets for ODEs and Numerical Integration

Peter Giesl, Sigurdur Hafstein, Iman Mehrabinezhad

University of Iceland

Research output: Contribution to journal › Conference article › peer-review

Abstract

We show that for an ordinary differential equation (ODE) with an exponentially stable equilibrium and any compact subset of its basin of attraction, we can find a larger compact set that is positively invariant for both the dynamics of the system and a numerical method to approximate its solution trajectories. We establish this for both one-step numerical integrators and multi-step integrators using sufficiently small time-steps. Further, we show how to localize such sets using continuously differentiable Lyapunov-like functions and numerically computed continuous, piecewise affine (CPA) Lyapunov-like functions.

Original language	English
Pages (from-to)	44-53
Number of pages	10
Journal	Proceedings of the International Conference on Informatics in Control, Automation and Robotics
Volume	1
DOIs	https://doi.org/10.5220/0012189700003543
Publication status	Published - 2023
Event	20th International Conference on Informatics in Control, Automation and Robotics, ICINCO 2023 - Rome, Italy Duration: 13 Nov 2023 → 15 Nov 2023

Bibliographical note

Other keywords

Numerical Integration
Ordinary Differential Equations
Positively Invariant Sets

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10.5220/0012189700003543

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title = "Positively Invariant Sets for ODEs and Numerical Integration",

abstract = "We show that for an ordinary differential equation (ODE) with an exponentially stable equilibrium and any compact subset of its basin of attraction, we can find a larger compact set that is positively invariant for both the dynamics of the system and a numerical method to approximate its solution trajectories. We establish this for both one-step numerical integrators and multi-step integrators using sufficiently small time-steps. Further, we show how to localize such sets using continuously differentiable Lyapunov-like functions and numerically computed continuous, piecewise affine (CPA) Lyapunov-like functions.",

keywords = "Numerical Integration, Ordinary Differential Equations, Positively Invariant Sets",

author = "Peter Giesl and Sigurdur Hafstein and Iman Mehrabinezhad",

note = "Publisher Copyright: {\textcopyright} 2023 by SCITEPRESS – Science and Technology Publications, Lda.; 20th International Conference on Informatics in Control, Automation and Robotics, ICINCO 2023 ; Conference date: 13-11-2023 Through 15-11-2023",

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language = "English",

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TY - JOUR

T1 - Positively Invariant Sets for ODEs and Numerical Integration

AU - Giesl, Peter

AU - Hafstein, Sigurdur

AU - Mehrabinezhad, Iman

PY - 2023

Y1 - 2023

N2 - We show that for an ordinary differential equation (ODE) with an exponentially stable equilibrium and any compact subset of its basin of attraction, we can find a larger compact set that is positively invariant for both the dynamics of the system and a numerical method to approximate its solution trajectories. We establish this for both one-step numerical integrators and multi-step integrators using sufficiently small time-steps. Further, we show how to localize such sets using continuously differentiable Lyapunov-like functions and numerically computed continuous, piecewise affine (CPA) Lyapunov-like functions.

AB - We show that for an ordinary differential equation (ODE) with an exponentially stable equilibrium and any compact subset of its basin of attraction, we can find a larger compact set that is positively invariant for both the dynamics of the system and a numerical method to approximate its solution trajectories. We establish this for both one-step numerical integrators and multi-step integrators using sufficiently small time-steps. Further, we show how to localize such sets using continuously differentiable Lyapunov-like functions and numerically computed continuous, piecewise affine (CPA) Lyapunov-like functions.

KW - Numerical Integration

KW - Ordinary Differential Equations

KW - Positively Invariant Sets

UR - https://www.scopus.com/pages/publications/85181554713

U2 - 10.5220/0012189700003543

DO - 10.5220/0012189700003543

M3 - Conference article

SN - 2184-2809

VL - 1

SP - 44

EP - 53

JO - Proceedings of the International Conference on Informatics in Control, Automation and Robotics

JF - Proceedings of the International Conference on Informatics in Control, Automation and Robotics

T2 - 20th International Conference on Informatics in Control, Automation and Robotics, ICINCO 2023

Y2 - 13 November 2023 through 15 November 2023

ER -

Positively Invariant Sets for ODEs and Numerical Integration

Abstract

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