Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent

Magne Erlandsen; Tomas J. Meijer; W.P.M.H. Heemels; Sebastiaan J.A.M. van den Eijnden

doi:10.1109/TAC.2026.3655632

Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent

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

Abstract

Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions. Several conditions for guaranteeing continuity of PWQ functions over state-space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.

Original language	English
Article number	11359492
Number of pages	8
Journal	IEEE Transactions on Automatic Control
Volume	XX
DOIs	https://doi.org/10.1109/TAC.2026.3655632
Publication status	E-pub ahead of print - 20 Jan 2026

Funding

The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement PROACTHIS, no. 101055384.

Keywords

Linear matrix inequalities
Lyapunov methods
Piecewise linear systems
Piecewise quadratic functions
lyapunov methods
piecewise linear systems
piecewise quadratic functions

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10.1109/TAC.2026.3655632

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title = "Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent",

abstract = "Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions. Several conditions for guaranteeing continuity of PWQ functions over state-space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.",

keywords = "Linear matrix inequalities, Lyapunov methods, Piecewise linear systems, Piecewise quadratic functions, lyapunov methods, piecewise linear systems, piecewise quadratic functions",

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AU - Erlandsen, Magne

AU - Meijer, Tomas J.

AU - Heemels, W.P.M.H.

AU - van den Eijnden, Sebastiaan J.A.M.

PY - 2026/1/20

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N2 - Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions. Several conditions for guaranteeing continuity of PWQ functions over state-space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.

AB - Analysis of continuous-time piecewise linear systems based on piecewise quadratic (PWQ) Lyapunov functions typically requires continuity of these functions. Several conditions for guaranteeing continuity of PWQ functions over state-space partitions can be found in the literature. In this technical note, we show that these continuity conditions are equivalent over so-called simplicial conic partitions. As a consequence, the choice of which condition to impose can be based solely on practical considerations such as specific application or numerical aspects, without introducing additional conservatism in the analysis.

KW - Linear matrix inequalities

KW - Lyapunov methods

KW - Piecewise linear systems

KW - Piecewise quadratic functions

KW - lyapunov methods

KW - piecewise linear systems

KW - piecewise quadratic functions

UR - https://www.scopus.com/pages/publications/105028444778

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M3 - Article

SN - 0018-9286

VL - XX

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

M1 - 11359492

ER -

Continuity Conditions for Piecewise Quadratic Functions on Simplicial Conic Partitions are Equivalent

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Keywords

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