PIECEWISE QUADRATIC LYAPUNOV FUNCTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS BY LINEAR PROGRAMMING

Peter Giesl; Sigurdur Hafstein; Sareena Pokkakkillath

doi:10.3934/dcdsb.2024148

PIECEWISE QUADRATIC LYAPUNOV FUNCTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS BY LINEAR PROGRAMMING

Peter Giesl, Sigurdur Hafstein, Sareena Pokkakkillath

Faculty of Physical Sciences

University of Iceland

Research output: Contribution to journal › Article › peer-review

Abstract

We develop an algorithm to parameterize continuous and piecewise quadratic (CPQ) Lyapunov functions for stochastic differential equations (SDEs) using linear programming (LP). The algorithm is a non-trivial extension of algorithms to parameterize continuous and piecewise linear (CPA) Lyapunov functions for ordinary differential equations (ODEs), but since the conditions for a Lyapunov function for a stochastic system involve second order derivatives, CPA Lyapunov functions cannot exist for stochastic systems and hence CPQ Lyapunov functions are needed. We demonstrate our algorithm on two examples from the literature.

Original language	English
Pages (from-to)	2027-2050
Number of pages	24
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	30
Issue number	6
DOIs	https://doi.org/10.3934/dcdsb.2024148
Publication status	Published - Jun 2025

Bibliographical note

Other keywords

Lyapunov function
Stochastic differential equation
continuous piecewise quadratic function
linear programming
γ-basin of attraction

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10.3934/dcdsb.2024148

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title = "PIECEWISE QUADRATIC LYAPUNOV FUNCTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS BY LINEAR PROGRAMMING",

abstract = "We develop an algorithm to parameterize continuous and piecewise quadratic (CPQ) Lyapunov functions for stochastic differential equations (SDEs) using linear programming (LP). The algorithm is a non-trivial extension of algorithms to parameterize continuous and piecewise linear (CPA) Lyapunov functions for ordinary differential equations (ODEs), but since the conditions for a Lyapunov function for a stochastic system involve second order derivatives, CPA Lyapunov functions cannot exist for stochastic systems and hence CPQ Lyapunov functions are needed. We demonstrate our algorithm on two examples from the literature.",

keywords = "Lyapunov function, Stochastic differential equation, continuous piecewise quadratic function, linear programming, γ-basin of attraction",

author = "Peter Giesl and Sigurdur Hafstein and Sareena Pokkakkillath",

year = "2025",

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AU - Giesl, Peter

AU - Hafstein, Sigurdur

AU - Pokkakkillath, Sareena

PY - 2025/6

Y1 - 2025/6

N2 - We develop an algorithm to parameterize continuous and piecewise quadratic (CPQ) Lyapunov functions for stochastic differential equations (SDEs) using linear programming (LP). The algorithm is a non-trivial extension of algorithms to parameterize continuous and piecewise linear (CPA) Lyapunov functions for ordinary differential equations (ODEs), but since the conditions for a Lyapunov function for a stochastic system involve second order derivatives, CPA Lyapunov functions cannot exist for stochastic systems and hence CPQ Lyapunov functions are needed. We demonstrate our algorithm on two examples from the literature.

AB - We develop an algorithm to parameterize continuous and piecewise quadratic (CPQ) Lyapunov functions for stochastic differential equations (SDEs) using linear programming (LP). The algorithm is a non-trivial extension of algorithms to parameterize continuous and piecewise linear (CPA) Lyapunov functions for ordinary differential equations (ODEs), but since the conditions for a Lyapunov function for a stochastic system involve second order derivatives, CPA Lyapunov functions cannot exist for stochastic systems and hence CPQ Lyapunov functions are needed. We demonstrate our algorithm on two examples from the literature.

KW - Lyapunov function

KW - Stochastic differential equation

KW - continuous piecewise quadratic function

KW - linear programming

KW - γ-basin of attraction

UR - https://www.scopus.com/pages/publications/86000659280

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DO - 10.3934/dcdsb.2024148

M3 - Article

SN - 1531-3492

VL - 30

SP - 2027

EP - 2050

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 6

ER -

PIECEWISE QUADRATIC LYAPUNOV FUNCTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS BY LINEAR PROGRAMMING

Abstract

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