# Lyapunov Functions for Linear Stochastic Differential Equations: BMI Formulation of the Conditions

### Sigurdur Hafstein

#### Abstract

We present a bilinear matrix inequality (BMI) formulation of the conditions for a Lyapunov functions for autonomous, linear stochastic differential equations (SDEs). We review and collect useful results from the theory of stochastic stability of the null solution of an SDE. Further, we discuss the Itô- and Stratonovich interpretation and linearizations and Lyapunov functions for linear SDEs. Then we discuss the construction of Lyapunov functions for the damped pendulum, wihere the spring constant is modelled as a stochastic process. We implement in Matlab the characterization of its canonical Lyapunov function as BMI constraints and consider some practical implementation strategies. Further, we demonstrate that the general strategy is applicable to general autonomous and linear SDEs. Finally, we verify our findings by comparing with results from the literature.

Download#### Paper Citation

#### in Harvard Style

Hafstein S. (2019). **Lyapunov Functions for Linear Stochastic Differential Equations: BMI Formulation of the Conditions**.In *Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-758-380-3, pages 147-155. DOI: 10.5220/0008192201470155

#### in Bibtex Style

@conference{icinco19,

author={Sigurdur Hafstein},

title={Lyapunov Functions for Linear Stochastic Differential Equations: BMI Formulation of the Conditions},

booktitle={Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2019},

pages={147-155},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0008192201470155},

isbn={978-989-758-380-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Lyapunov Functions for Linear Stochastic Differential Equations: BMI Formulation of the Conditions

SN - 978-989-758-380-3

AU - Hafstein S.

PY - 2019

SP - 147

EP - 155

DO - 10.5220/0008192201470155