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 proceedingConference contribution

2 Citations (Scopus)

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 languageEnglish (US)
Title of host publicationProceedings of the IEEE Conference on Decision and Control
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1884-1890
Number of pages7
Volume2016-February
ISBN (Print)9781479978861
DOIs
StatePublished - Feb 8 2016
Event54th IEEE Conference on Decision and Control, CDC 2015 - Osaka, Japan
Duration: Dec 15 2015Dec 18 2015

Other

Other54th IEEE Conference on Decision and Control, CDC 2015
CountryJapan
CityOsaka
Period12/15/1512/18/15

Fingerprint

Lyapunov methods
Lyapunov Methods
Linear partial differential equation
Parabolic Partial Differential Equations
Partial differential equations
Stability Analysis
Polynomials
Polynomial
Partial differential equation
Lyapunov Functionals
Parameterise
Optimization
Population Growth
Sum of squares
Express
Decay
Derivatives
Norm
Derivative
Dependent

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Meyer, E., & Peet, M. (2016). Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. In Proceedings of the IEEE Conference on Decision and Control (Vol. 2016-February, pp. 1884-1890). [7402485] 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.

Proceedings of the IEEE Conference on Decision and Control. Vol. 2016-February Institute of Electrical and Electronics Engineers Inc., 2016. p. 1884-1890 7402485.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Meyer, E & Peet, M 2016, Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. in Proceedings of the IEEE Conference on Decision and Control. vol. 2016-February, 7402485, 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
Meyer E, Peet M. Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. In Proceedings of the IEEE Conference on Decision and Control. Vol. 2016-February. Institute of Electrical and Electronics Engineers Inc. 2016. p. 1884-1890. 7402485 https://doi.org/10.1109/CDC.2015.7402485
Meyer, Evgeny ; Peet, Matthew. / Stability analysis of parabolic linear PDEs with two spatial dimensions using Lyapunov method and SOS. Proceedings of the IEEE Conference on Decision and Control. Vol. 2016-February Institute of Electrical and Electronics Engineers Inc., 2016. pp. 1884-1890
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