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

TL;DR

PIETOOLS is a MATLAB toolbox that simplifies the construction, manipulation, and optimization of Partial Integral operators, facilitating analysis and control of infinite-dimensional systems like PDEs and TDS.

Contribution

The paper introduces PIETOOLS, a novel MATLAB toolbox with a new class of objects for handling PI operators, enabling algebraic operations and convex optimization for infinite-dimensional system analysis.

Findings

Successfully applied to stability analysis of PDEs

Enabled bounding of operator norms

Verified integral inequalities efficiently

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. +, *, ' e tc.) 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 RxL2. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential equations (PDE) and Time-delay 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 (LOIs) 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.