Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS

Evgeny Meyer; Matthew Peet

doi:10.1109/CDC.2015.7402485

Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS

Evgeny Meyer, Matthew Peet

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

4 Scopus citations

Abstract

In this paper, we address stability of parabolic linear Partial Differential Equations (PDEs). We consider PDEs with two spatial variables and spatially dependent polynomial coefficients. We parameterize a class of Lyapunov functionals and their time derivatives by polynomials and express stability as optimization over polynomials. We use Sum-of-Squares and Positivstellensatz results to numerically search for a solution to the optimization over polynomials. We also show that our algorithm can be used to estimate the rate of decay of the solution to PDE in the L2 norm. Finally, we validate the technique by applying our conditions to the 2D biological KISS PDE model of population growth and an additional example.

Original language	English (US)
Title of host publication	54rd IEEE Conference on Decision and Control,CDC 2015
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1884-1890
Number of pages	7
ISBN (Electronic)	9781479978861
DOIs	https://doi.org/10.1109/CDC.2015.7402485
State	Published - Feb 8 2015
Event	54th IEEE Conference on Decision and Control, CDC 2015 - Osaka, Japan Duration: Dec 15 2015 → Dec 18 2015

Publication series

Name	Proceedings of the IEEE Conference on Decision and Control
Volume	54rd IEEE Conference on Decision and Control,CDC 2015
ISSN (Print)	0743-1546
ISSN (Electronic)	2576-2370

Other

Other	54th IEEE Conference on Decision and Control, CDC 2015
Country/Territory	Japan
City	Osaka
Period	12/15/15 → 12/18/15

ASJC Scopus subject areas

Control and Systems Engineering
Modeling and Simulation
Control and Optimization

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10.1109/CDC.2015.7402485

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Meyer, E., & Peet, M. (2015). Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. In 54rd IEEE Conference on Decision and Control,CDC 2015 (pp. 1884-1890). Article 7402485 (Proceedings of the IEEE Conference on Decision and Control; Vol. 54rd IEEE Conference on Decision and Control,CDC 2015). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2015.7402485

Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. / Meyer, Evgeny; Peet, Matthew.
54rd IEEE Conference on Decision and Control,CDC 2015. Institute of Electrical and Electronics Engineers Inc., 2015. p. 1884-1890 7402485 (Proceedings of the IEEE Conference on Decision and Control; Vol. 54rd IEEE Conference on Decision and Control,CDC 2015).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Meyer, E & Peet, M 2015, Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. in 54rd IEEE Conference on Decision and Control,CDC 2015., 7402485, Proceedings of the IEEE Conference on Decision and Control, vol. 54rd IEEE Conference on Decision and Control,CDC 2015, Institute of Electrical and Electronics Engineers Inc., pp. 1884-1890, 54th IEEE Conference on Decision and Control, CDC 2015, Osaka, Japan, 12/15/15. https://doi.org/10.1109/CDC.2015.7402485

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AB - In this paper, we address stability of parabolic linear Partial Differential Equations (PDEs). We consider PDEs with two spatial variables and spatially dependent polynomial coefficients. We parameterize a class of Lyapunov functionals and their time derivatives by polynomials and express stability as optimization over polynomials. We use Sum-of-Squares and Positivstellensatz results to numerically search for a solution to the optimization over polynomials. We also show that our algorithm can be used to estimate the rate of decay of the solution to PDE in the L2 norm. Finally, we validate the technique by applying our conditions to the 2D biological KISS PDE model of population growth and an additional example.

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Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS

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