Control of a reaction-diffusion PDE cascaded with a heat equation

Jun Min Wang; Ling Ling Su; Han Xiong Li

doi:10.1109/acc.2013.6579969

Control of a reaction-diffusion PDE cascaded with a heat equation

Jun Min Wang, Ling Ling Su, Han Xiong Li

School of Mathematics and Statistics

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

2 Citations (Scopus)

Abstract

We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the other boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H^-1(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H¹(0, 1) rather than the usual L²(0, 1).

Original language	English
Title of host publication	2013 American Control Conference, ACC 2013
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1012-1017
Number of pages	6
ISBN (Print)	9781479901777
DOIs	https://doi.org/10.1109/acc.2013.6579969
Publication status	Published - 2013
Event	2013 1st American Control Conference, ACC 2013 - Washington, DC, United States Duration: 17 Jun 2013 → 19 Jun 2013

Publication series

Name	Proceedings of the American Control Conference
ISSN (Print)	0743-1619

Conference

Conference	2013 1st American Control Conference, ACC 2013
Country/Territory	United States
City	Washington, DC
Period	17/06/13 → 19/06/13

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10.1109/acc.2013.6579969

Cite this

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title = "Control of a reaction-diffusion PDE cascaded with a heat equation",

abstract = "We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the other boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H-1(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H1(0, 1) rather than the usual L2(0, 1).",

author = "Wang, {Jun Min} and Su, {Ling Ling} and Li, {Han Xiong}",

year = "2013",

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language = "English",

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series = "Proceedings of the American Control Conference",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

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note = "2013 1st American Control Conference, ACC 2013 ; Conference date: 17-06-2013 Through 19-06-2013",

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Wang, JM, Su, LL & Li, HX 2013, Control of a reaction-diffusion PDE cascaded with a heat equation. in 2013 American Control Conference, ACC 2013., 6579969, Proceedings of the American Control Conference, Institute of Electrical and Electronics Engineers Inc., pp. 1012-1017, 2013 1st American Control Conference, ACC 2013, Washington, DC, United States, 17/06/13. https://doi.org/10.1109/acc.2013.6579969

Control of a reaction-diffusion PDE cascaded with a heat equation. / Wang, Jun Min; Su, Ling Ling; Li, Han Xiong.
2013 American Control Conference, ACC 2013. Institute of Electrical and Electronics Engineers Inc., 2013. p. 1012-1017 6579969 (Proceedings of the American Control Conference).

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

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AU - Su, Ling Ling

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PY - 2013

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N2 - We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the other boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H-1(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H1(0, 1) rather than the usual L2(0, 1).

AB - We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the other boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H-1(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H1(0, 1) rather than the usual L2(0, 1).

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BT - 2013 American Control Conference, ACC 2013

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2013 1st American Control Conference, ACC 2013

Y2 - 17 June 2013 through 19 June 2013

ER -

Control of a reaction-diffusion PDE cascaded with a heat equation

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