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 proceedingConference contributionAcademicpeer-review

15 Citations (Scopus)


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 languageEnglish
Title of host publication2020 American Control Conference, ACC 2020
PublisherInstitute of Electrical and Electronics Engineers
Number of pages6
ISBN (Electronic)9781538682661
Publication statusPublished - Jul 2020
Event2020 American Control Conference, ACC 2020 - Denver, United States
Duration: 1 Jul 20203 Jul 2020


Conference2020 American Control Conference, ACC 2020
Abbreviated titleACC 2020
Country/TerritoryUnited States
Internet address


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.

FundersFunder number
National Science Foundation1935453
Office of Naval ResearchN00014-17-1-2117
Directorate for Computer and Information Science and Engineering1739990


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