# Positively Invariant Sets for ODEs and Numerical Integration

### Peter Giesl, Sigurdur Hafstein, Iman Mehrabinezhad

#### 2023

#### 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.

Download#### Paper Citation

#### in Harvard Style

Giesl P., Hafstein S. and Mehrabinezhad I. (2023). **Positively Invariant Sets for ODEs and Numerical Integration**. In *Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO*; ISBN 978-989-758-670-5, SciTePress, pages 44-53. DOI: 10.5220/0012189700003543

#### in Bibtex Style

@conference{icinco23,

author={Peter Giesl and Sigurdur Hafstein and Iman Mehrabinezhad},

title={Positively Invariant Sets for ODEs and Numerical Integration},

booktitle={Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO},

year={2023},

pages={44-53},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0012189700003543},

isbn={978-989-758-670-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO

TI - Positively Invariant Sets for ODEs and Numerical Integration

SN - 978-989-758-670-5

AU - Giesl P.

AU - Hafstein S.

AU - Mehrabinezhad I.

PY - 2023

SP - 44

EP - 53

DO - 10.5220/0012189700003543

PB - SciTePress