Representation of linear PDEs with spatial integral terms as Partial Integral Equations

Sachin Shivakumar; Amritam Das; Matthew M. Peet

doi:10.23919/ACC55779.2023.10156465

Representation of linear PDEs with spatial integral terms as Partial Integral Equations

Sachin Shivakumar, Amritam Das, Matthew M. Peet

Control Systems

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

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Abstract

In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.

Original language	English
Title of host publication	2023 American Control Conference, ACC 2023
Publisher	Institute of Electrical and Electronics Engineers
Pages	1788-1793
Number of pages	6
ISBN (Electronic)	9798350328066
DOIs	https://doi.org/10.23919/ACC55779.2023.10156465
Publication status	Published - 3 Jul 2023
Event	2023 American Control Conference, ACC 2023 - San Diego, United States Duration: 31 May 2023 → 2 Jun 2023

Conference

Conference	2023 American Control Conference, ACC 2023
Abbreviated title	ACC 2023
Country/Territory	United States
City	San Diego
Period	31/05/23 → 2/06/23

Funding

ACKNOWLEDGEMENT This work was supported by the National Science Foundation under grants No. 1739990 and 1935453

Funders	Funder number
National Science Foundation	1739990, 1935453

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10.23919/ACC55779.2023.10156465

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title = "Representation of linear PDEs with spatial integral terms as Partial Integral Equations",

abstract = "In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.",

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Shivakumar, S, Das, A & Peet, MM 2023, Representation of linear PDEs with spatial integral terms as Partial Integral Equations. in 2023 American Control Conference, ACC 2023., 10156465, Institute of Electrical and Electronics Engineers, pp. 1788-1793, 2023 American Control Conference, ACC 2023, San Diego, California, United States, 31/05/23. https://doi.org/10.23919/ACC55779.2023.10156465

Representation of linear PDEs with spatial integral terms as Partial Integral Equations. / Shivakumar, Sachin; Das, Amritam; Peet, Matthew M.
2023 American Control Conference, ACC 2023. Institute of Electrical and Electronics Engineers, 2023. p. 1788-1793 10156465.

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

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T1 - Representation of linear PDEs with spatial integral terms as Partial Integral Equations

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N2 - In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.

AB - In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.

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BT - 2023 American Control Conference, ACC 2023

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T2 - 2023 American Control Conference, ACC 2023

Y2 - 31 May 2023 through 2 June 2023

ER -

Representation of linear PDEs with spatial integral terms as Partial Integral Equations

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