PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

Sachin Shivakumar; Amritam Das; Matthew M. Peet

doi:10.23919/ACC45564.2020.9147712

PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

Sachin Shivakumar, Amritam Das, Matthew M. Peet

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

18 Citations (Scopus)

Abstract

In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g. +, ' etc.) is defined. PI operators are a generalization of bounded linear operators on infinite-dimensional spaces that form a-subalgebra with two binary operations (addition and composition) on the space × L₂. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential Equations (PDE) and Timedelay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOI) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.

Original language	English
Title of host publication	2020 American Control Conference, ACC 2020
Publisher	Institute of Electrical and Electronics Engineers
Pages	2667-2672
Number of pages	6
ISBN (Electronic)	9781538682661
DOIs	https://doi.org/10.23919/ACC45564.2020.9147712
Publication status	Published - Jul 2020
Event	2020 American Control Conference, ACC 2020 - Denver, United States Duration: 1 Jul 2020 → 3 Jul 2020 http://acc2020.a2c2.org/

Conference

Conference	2020 American Control Conference, ACC 2020
Abbreviated title	ACC 2020
Country/Territory	United States
City	Denver
Period	1/07/20 → 3/07/20
Internet address	http://acc2020.a2c2.org/

Funding

X. CONCLUSIONS In this paper, we have provided a guide to the new MAT-LAB toolbox PIETOOLS for manipulation and optimization of PI operators. We have provided details on declaration of PI operator objects, manipulation of PI operators, declaration of PI decision variables, addition of operator equality and inequality constraints, solution of PI optimization problems, and extraction of feasible operators. We have demonstrated the practical usage of PIETOOLS, including scripts for analysis and control of PDEs and systems with delay, as well as bounding operator norms and proving integral inequalities. These examples and descriptions illustrate both the syntax of available features and the necessary components of any PIETOOLS script. Finally, we note that PIETOOLS is still under active development. Ongoing efforts focus on identifying and balancing the degree structures in sos opineq. ACKNOWLEDGMENT This work was supported by Office of Naval Research Award N00014-17-1-2117 and National Science Foundation under Grants No. 1739990 and 1935453.

Funders	Funder number
National Science Foundation	1935453

Access to Document

10.23919/ACC45564.2020.9147712

Cite this

@inproceedings{af9d01a413e5453f80174e02dccc9830,

title = "PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators",

abstract = "In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g. +, ' etc.) is defined. PI operators are a generalization of bounded linear operators on infinite-dimensional spaces that form a-subalgebra with two binary operations (addition and composition) on the space × L2. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential Equations (PDE) and Timedelay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOI) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.",

author = "Sachin Shivakumar and Amritam Das and Peet, {Matthew M.}",

year = "2020",

month = jul,

doi = "10.23919/ACC45564.2020.9147712",

language = "English",

pages = "2667--2672",

booktitle = "2020 American Control Conference, ACC 2020",

publisher = "Institute of Electrical and Electronics Engineers",

address = "United States",

note = "2020 American Control Conference, ACC 2020, ACC 2020 ; Conference date: 01-07-2020 Through 03-07-2020",

url = "http://acc2020.a2c2.org/",

}

Shivakumar, S, Das, A & Peet, MM 2020, PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators. in 2020 American Control Conference, ACC 2020., 9147712, Institute of Electrical and Electronics Engineers, pp. 2667-2672, 2020 American Control Conference, ACC 2020, Denver, Colorado, United States, 1/07/20. https://doi.org/10.23919/ACC45564.2020.9147712

PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators. / Shivakumar, Sachin; Das, Amritam; Peet, Matthew M.
2020 American Control Conference, ACC 2020. Institute of Electrical and Electronics Engineers, 2020. p. 2667-2672 9147712.

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

TY - GEN

T1 - PIETOOLS

T2 - 2020 American Control Conference, ACC 2020

AU - Shivakumar, Sachin

AU - Das, Amritam

AU - Peet, Matthew M.

PY - 2020/7

Y1 - 2020/7

N2 - In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g. +, ' etc.) is defined. PI operators are a generalization of bounded linear operators on infinite-dimensional spaces that form a-subalgebra with two binary operations (addition and composition) on the space × L2. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential Equations (PDE) and Timedelay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOI) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.

AB - In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g. +, ' etc.) is defined. PI operators are a generalization of bounded linear operators on infinite-dimensional spaces that form a-subalgebra with two binary operations (addition and composition) on the space × L2. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential Equations (PDE) and Timedelay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOI) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.

UR - http://www.scopus.com/inward/record.url?scp=85089568848&partnerID=8YFLogxK

U2 - 10.23919/ACC45564.2020.9147712

DO - 10.23919/ACC45564.2020.9147712

M3 - Conference contribution

AN - SCOPUS:85089568848

SP - 2667

EP - 2672

BT - 2020 American Control Conference, ACC 2020

PB - Institute of Electrical and Electronics Engineers

Y2 - 1 July 2020 through 3 July 2020

ER -

PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

Abstract

Conference

Funding

Access to Document

Other files and links

Fingerprint

PIETOOLS 2020a

Cite this

PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

Abstract

Conference

Funding

Access to Document

Other files and links

Fingerprint

Datasets

PIETOOLS 2020a

Cite this