PIETOOLS: MATLAB Toolbox for PIE Analysis

Updated 2 July 2026

PIETOOLS is a MATLAB toolbox that represents infinite-dimensional systems using Partial Integral Equations (PIEs) and PI operators.
It automates conversion of diverse PDEs, DDEs, and coupled ODE–PDE systems into boundary-free operator formulations for convex optimization.
The toolbox integrates semidefinite programming and symbolic manipulation to achieve precise stability analysis, observer/controller synthesis, and robust μ-analysis.

PIETOOLS is a MATLAB-based software suite for the analysis, control, and robust optimization of infinite-dimensional systems represented as Partial Integral Equations (PIEs). Leveraging an algebra of Partial Integral (PI) operators, PIETOOLS generalizes Linear Matrix Inequality (LMI) methods to the operator level, enabling convex optimization over classes of partial differential equations (PDEs), delay-differential equations (DDEs), and distributed-parameter ODE–PDE/DDE–PDE systems. Its architecture integrates advanced operator-theoretic formulations, semidefinite programming, and symbolic representation, providing a tractable and numerically precise environment for stability analysis, input-output norm estimation, observer/controller synthesis, and robust/μ-analysis of complex dynamical systems (Shivakumar et al., 2019, Shivakumar et al., 29 Jan 2025, Shivakumar et al., 2020, Lenssen et al., 18 Nov 2025).

1. Foundational Concepts: Partial Integral Equations and PI Operators

The core mathematical structure in PIETOOLS is the Partial Integral Equation (PIE) formalism, which encodes the dynamics of infinite-dimensional systems using PI operators that act on spaces such as $\mathbb{R}^m \times L_2^n[a,b]$ (or multi-dimensional analogues). A 4-PI operator $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ is parameterized by a matrix $P$ , two function-valued matrices $Q_1$ , $Q_2$ , and a 3-kernel operator $R$ (Shivakumar et al., 2019):

$\mathcal{P} \begin{bmatrix} x \ y(\cdot) \end{bmatrix} = \begin{bmatrix} P\,x + \int_a^b Q_1(\theta)\,y(\theta)\, d\theta \ Q_2(s) x + R_0(s)\, y(s) + \int_a^s R_1(s,\theta) y(\theta) d\theta + \int_s^b R_2(s,\theta) y(\theta) d\theta \end{bmatrix}.$

These PI operators form a $*$ -subalgebra under addition, composition, and adjoint (Shivakumar et al., 2019). In finite dimensions, the Lyapunov and control synthesis problems are posed as LMIs on matrices; in the PIE setting, these generalize to linear PI inequalities (LPIs) where the variable is a PI operator (Shivakumar et al., 2020). This abstraction is essential for developing convex stability and performance certificates for infinite-dimensional, spatially-distributed, and delay systems (Peet, 2018, Jagt et al., 2022, Peet, 2019).

2. PIE Representation and System Conversion

PIETOOLS automates the conversion of a wide range of system classes into PIE representation. Any suitably well-posed linear ODE–PDE, DDE, or coupled system can be mapped to an algebraic PIE, where all boundary, continuity, and delay constraints are absorbed into the operator structure (Shivakumar et al., 29 Jan 2025, Shivakumar et al., 2020, Jagt et al., 2022, Shivakumar et al., 2022, Peet, 2018). The “fundamental state”—typically the highest-order spatial derivative—forms an unconstrained state in an appropriate $L_2$ space. The conversion process is critical for transforming implicit constraint-laden models into boundary-free operator equations amenable to convex optimization.

The user specifies the PDE, boundary conditions, distributed delays, or ODE couplings in a symbolic MATLAB format. PIETOOLS applies analytic formulas, often relying on generalized Cauchy integrals or Gohberg–Kaashoek–Lay inversion, to compute the mapping operators (e.g., T, A, B, C, D) (Shivakumar et al., 2022, Jagt et al., 20 Aug 2025, Shivakumar et al., 2021). For multi-dimensional domains, PIETOOLS constructs multivariate PI operators using tensor products of univariate kernels, imposing explicit consistency conditions for domain intersections (Jagt et al., 20 Aug 2025).

3. Convex Operator Inequalities: Stability and Performance Analysis

Central to PIETOOLS is the implementation of LPI-based certificates for system properties. To test exponential stability or input-output performance, PIETOOLS parameterizes self-adjoint positive PI operators (often via sum-of-squares or finite-dimensional basis expansions) and formulates operator inequalities that are direct analogues of LMIs (Shivakumar et al., 2020, Jagt et al., 20 Aug 2025, Peet, 2018, Jagt et al., 2022). For example, the Lyapunov stability for a PIE

$T \dot{x}(t) = A x(t)$

is certified by the feasibility of

$\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 0

where $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 1 is a PI operator (Peet, 2018, Shivakumar et al., 29 Jan 2025). For $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 2/ $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 3-gain and impulse-to-peak norm bounds, block-operator KYP conditions are used (Shivakumar et al., 2020, Thomas et al., 3 Apr 2026).

PIETOOLS translates these infinite-dimensional inequalities into finite semidefinite programs (SDPs) by expanding operator kernels over polynomial or monomial bases and encoding positivity via matrix constraints (Jagt et al., 2022, Thomas et al., 3 Apr 2026). Feasibility guarantees non-conservative, numerically certified stability and performance results for a wide variety of ODE–PDE, DDE, and multivariate PDE models.

4. Synthesis: Observer, Controller, and Robust Optimization

PIETOOLS supports the synthesis of optimal state feedback controllers, Luenberger observers, and robust controllers under uncertainty. Synthesis problems are posed as convex LPIs over PI operator variables. For $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 4-optimal state-feedback and observer design, the duality theory for PIEs implies the convexification variable change (e.g., $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 5 for controllers, $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 6 for observers), mirroring finite-dimensional results (Shivakumar et al., 2020, Wu et al., 2020, Jagt et al., 2024, Shivakumar et al., 29 Jan 2025).

Robust and $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 7-optimal synthesis in the presence of structured uncertainty uses Integral Quadratic Constraint (IQC) multipliers, also modeled as PI operators (Lenssen et al., 18 Nov 2025). The design procedure involves joint optimization over Lyapunov PI operators and IQC multipliers, yielding upper bounds on $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 8 or worst-case gain with significantly reduced conservatism compared to unstructured analysis.

For nonlinear or polynomial PDE dynamics, PIETOOLS interfaces with distributed sum-of-squares (SOS) representations in tensor-product PI operator algebras, enabling local stability analysis via polynomial kernel expansions (Richardson et al., 1 Apr 2026).

5. Software Architecture, User Workflow, and Example Applications

PIETOOLS is structured as a MATLAB toolbox, introducing the opvar (operator variable) and pio classes for symbolic and numerical manipulation of PI operators (Shivakumar et al., 2019). Operator arithmetic—addition, composition, adjoint—is overloaded. The toolbox integrates with SOSTOOLS/MULTIPOLY for polynomial manipulations and interfaces with SDP solvers (e.g., SeDuMi, MOSEK, SDPT3) for LPI feasibility (Shivakumar et al., 2019, Shivakumar et al., 29 Jan 2025, Shivakumar et al., 2021).

A typical user workflow (Shivakumar et al., 29 Jan 2025, Shivakumar et al., 2021):

Define spatial, temporal, and auxiliary variables (e.g., pvar s, t, theta).
Specify system equations, boundary conditions, and inputs/outputs through symbolic PDE or DDE models.
Convert to PIE form using conversion routines (e.g., convert(PDE,'pie')).
Declare PI-operator decision variables and form LPI constraints via built-in functions (sosprogram, sos_posopvar, sos_opineq).
Solve the resulting SDP and extract operator-valued solutions (sossolve, sosgetsol_opvar).
Construct synthesized controllers or observers and simulate open- or closed-loop PIE systems using PIESIM.

PIETOOLS ships with end-to-end demo scripts for canonical PDEs (e.g., tip-damped wave equation, 2D heat and plate, reaction–diffusion with delay, boundary ODE–PDE couplings), and supports batch benchmarking for scalability studies (Peet, 2018, Thomas et al., 3 Apr 2026, Jagt et al., 2022).

6. Performance, Scalability, and Comparison

Numerical results indicate that PIETOOLS achieves high accuracy—matching analytic or high-fidelity finite-difference results to four significant digits in $\mathcal{P}: \mathbb{R}^m \times L_2^n[a,b] \to \mathbb{R}^p \times L_2^q[a,b]$ 9 and impulse-peak norms (Jagt et al., 2022, Thomas et al., 3 Apr 2026, Wu et al., 2020)—and can handle problems with up to tens of PDE states, operator degrees up to four, and moderate polynomial basis sizes in seconds to a few minutes on contemporary desktops (Peet, 2018, Jagt et al., 20 Aug 2025, Shivakumar et al., 29 Jan 2025). The direct operator-theoretic approach eliminates the need for spatial discretization or ad hoc state augmentation.

Compared to ad hoc SOS techniques, PIETOOLS significantly extends scalability, handles higher-order and multi-dimensional PDEs, supports robust IQC/μ-analysis, and provides closed-form controller/observer recovery via analytic inversion of PI operators (in particular, for structured PI operators in the separable or rank-one-integral class) (Jagt et al., 2024, Lenssen et al., 18 Nov 2025, Shivakumar et al., 2022). For delay-coupled PDEs or spatially distributed uncertainties, the reduction in conservatism relative to unstructured $P$ 0-gain methods can reach 30–40% (Lenssen et al., 18 Nov 2025).

7. Limitations, Extensions, and Future Directions

PIETOOLS, as of its latest documented versions, is restricted to linear, time-invariant systems with polynomial or $P$ 1 coefficients, and to positive operators parameterized via polynomial bases (Shivakumar et al., 29 Jan 2025, Jagt et al., 2022, Shivakumar et al., 2022). Extensions under development address nonlinear PDEs (via SOS-in-tensor-algebras), periodic boundary conditions (with quotient-space representations), time-varying and switching PIE systems, and multivariate domains beyond hyper-rectangles (Richardson et al., 1 Apr 2026, Jagt et al., 20 Aug 2025, Jagt et al., 2022).

Theoretical limitations include the need for prior model well-posedness checks (the software does not certify Lumer–Phillips conditions or boundary-input regularity), and computational scaling in the polynomial degree and state dimension. Nonlinear generalizations, trajectory-level performance analysis (e.g., input–state–output $P$ 2– $P$ 3 norms), and scalable policy-synthesis for semi-infinite constraints remain active research topics.