Partial Integral Equation (PIE) Framework

• Partial Integral Equation (PIE) Framework is a mathematical paradigm that reformulates PDEs into L2-based evolution equations using bounded integral operators.
• It leverages an algebra of PI operators to implicitly enforce boundary and continuity conditions, streamlining stability and performance analyses.
• The framework enables direct application of semidefinite programming techniques for control synthesis, observer design, and uncertainty quantification in complex systems.

The Partial Integral Equation (PIE) framework is a mathematical and computational paradigm for representing, analyzing, and controlling partial differential equations (PDEs) by reformulating their dynamics as first-order-in-time evolution equations in $L_2$ (square-integrable) spatial function spaces. Central to the PIE framework is the use of algebras of bounded Partial Integral (PI) operators that encompass all spatial actions of differentiation, integration, multiplication, and, crucially, the implicit enforcement of boundary and continuity constraints. The PIE representation enables direct application of convex optimization tools—primarily semidefinite programming (SDP) via Linear PI Inequalities (LPIs)—for stability analysis, observer and controller synthesis, and uncertainty quantification for both linear and certain classes of nonlinear PDEs.

1. Fundamentals of the PIE Framework

Let $u=u(t,s)$ denote a scalar or vector-valued state variable governed by a PDE on a spatial domain $s\in[a,b]$, possibly with time $t \geq 0$. Standard PDEs involve derivatives $\partial_s^k u$ and require intricate handling of boundary (Dirichlet, Neumann, Robin, periodic) and continuity conditions. In the PIE framework, the evolution equation is recast for a fundamental state

$v(t,s) := \partial_s^N u(t,s)$

where $N$ is the highest order of spatial differentiation present in the model. The mapping from $v$ (which lives in $L_2[a,b]$) back to the “physical” PDE state $u$ is implemented by a unitary PI operator $\mathcal{T}$, constructed uniquely from the boundary data. The resulting PIE

$\mathcal{T}\, \dot v = \mathcal{A}\,v$

is an operator equation in $L_2$, with all boundary and continuity information encoded in $\mathcal{T}, \mathcal{A}$ rather than imposed explicitly. For coupled systems, ODE-PDE interconnections, delay equations, or higher spatial dimensions, the PIE formalism incorporates extended PI operator classes and analogous lifting constructions.

2. Algebra of PI Operators and Implicit Boundary Handling

PI operators on $L_2[a,b]$ are defined by

$$(\mathcal{P}_{\{R_0,R_1,R_2\}}u)(s) = R_0(s)u(s) + \int_a^s R_1(s,\theta)u(\theta)d\theta + \int_s^b R_2(s,\theta)u(\theta)d\theta,$$

with $R_0 \in L_\infty[a,b]$, $R_1, R_2 \in L_2([a,b]^2)$. This class is closed under sum, scalar multiplication, operator composition, and $L_2$-adjoint, thus forming a *-algebra. The PIE framework extends these definitions to 2D and higher-dimensional spaces, block-structured (e.g., for ODE–PDE or multi-component systems), and tensor-product (for representing nonlinear terms or polynomials of fundamental states).

All necessary regularity and well-posedness (injectivity and surjectivity of the spatial derivative operator given boundary conditions) are transferred from the PDE to the domain and properties of the PI operators. The original boundary and continuity constraints on $u$ are absorbed by the construction of $\mathcal{T}$; in PIE form, all relevant states are unconstrained elements of $L_2$. For periodic boundary conditions, the PIE framework introduces decompositions aligning with the nullspace/image structure of the spatial operator, so that exponential stabilization is defined in terms of decay of the “zero-mean” component alone (Jagt et al., 28 Mar 2025).

3. PIE Representation of Linear and Nonlinear PDEs

Any scalar or coupled linear PDE of order $N$ with variable coefficients, on general boundary domains, admits a PIE lifting: $$u_t = \sum_{i=0}^N \alpha_i(s)\,\partial_s^i u \quad \to \quad \partial_t (\mathcal{T} v) = \mathcal{A} v,$$ where $v = \partial_s^N u$, $\mathcal{T},\mathcal{A}$ constructed as elements of $\Pi_2$ or $\Pi_3$ (PI-operator algebras). For quadratic (or more general polynomial semilinear) PDEs,

$$u_t = \sum_{i} \alpha_i \partial_s^i u + \sum_{i,j} \beta_{ij}(\partial_s^i u)(\partial_s^j u),$$

the quadratic terms can be represented in PIE form by tensor products in the PI operator algebra: if $v \in L_2[a,b]$, then $v \otimes v \in L_2([a,b]^2)$, and quadratic operator actions are written as new tensor-product PI operators. This forms a distributed analogue of the monomial basis in finite-dimensional polynomial systems and enables direct representation of polynomial nonlinearities (Jagt et al., 2023).

4. Lyapunov and Performance Analysis via Convex Optimization

Because all system dynamics (linear and certain nonlinear/polynomial) are captured in operator-algebraic form, Lyapunov analysis, stability, and performance objectives can be formulated as Linear PI Inequalities (LPIs)—direct generalizations of Linear Matrix Inequalities (LMIs) over the positive cone of PI operators. For example, the energy functional

$$V(u) = \langle u, P u \rangle_{L_2} = \langle \mathcal{T} v, P \mathcal{T} v \rangle_{L_2}$$

with $P=P^* \succeq \epsilon I \in \Pi_3$ leads to the sufficient stability condition

$$\mathcal{A}^* P \mathcal{T} + \mathcal{T}^* P \mathcal{A} \preceq -\delta \mathcal{T}^* \mathcal{T}.$$

For quadratic PDEs, an additional constraint ensures positivity-reduction of the cubic nonlinearity kernel in the Lyapunov functional’s derivative. Feasibility of these constraints (solved via SDP) yields exponential decay rates and, when sharp, recovers classical analytical results (e.g., Poincaré bounds for Burgers’ equation stability) (Jagt et al., 2023).

5. Extension to Higher Dimensions, Delays, and Coupled Systems

The PIE framework extends to multivariate PDEs—by constructing the inverse of the multivariate spatial differential operator inductively from its univariate inverses, subject to a commutative admissibility condition of 1D boundary blocks (Jagt et al., 20 Aug 2025). In two dimensions, the algebra of 2D-PI operators and associated state-space lifting yield an LPI-based theory that is directly applicable to coupled 2D PDEs (Jagt et al., 2021, Jagt et al., 7 Feb 2024). For systems with delays (DDEs, PDEs with state/boundary delays), PIE representations are constructed by casting the delay into distributed transport variables and applying the fundamental state lifting in both spatial and temporal “history” coordinates (Wu et al., 2020, Jagt et al., 2022). ODE–PDE cascades and interconnections are encoded via block PI operators (Shivakumar et al., 2020, Peet et al., 2020, Shivakumar et al., 2022).

6. Implementation, Computational Considerations, and Applications

The algebraic structure of PI operators admits a positive-cone parameterization by finite-dimensional polynomial bases (monomials, Chebyshev, Legendre), transforming operator inequalities into tractable SDP via block-matrix representations of PI kernels. The PIETOOLS MATLAB toolbox automates symbolic construction, parameterization, and solution of these problems, scaling efficiently to systems with dozens of coupled PDEs or complex boundary conditions. The approach produces sharp analytical results for a wide spectrum of PDE types (diffusion, reaction-diffusion, wave/beam, systems with spatial integrals and delays, boundary control/observer problems) without spatial discretization or auxiliary constraint imposition. All boundary/continuity constraints are handled natively in the operator algebra and lifting mappings (Peet, 2018, Peet et al., 2020, Jagt et al., 20 Aug 2025).

A concrete example: the PIE-based stability test for Burgers’ equation

$$u_t = u_{ss} + r\,u - u\,u_s, \quad u(t,0)=u(t,1)=0$$

proves global $L_2$-stability via SDP for $r\le \pi^2$, exactly matching the known sharp analytical bound (Jagt et al., 2023).

7. Significance, Limitations, and Future Directions

The PIE framework provides a canonical, boundary-free, algebraic state-space for a broad class of infinite-dimensional systems, enabling direct use of convex optimization formerly limited to ODEs and finite-dimensional LTI systems. It has unified various PDE types (1D/2D, with/without delays, with spatial integrals, general boundary conditions), and is extensible to polynomial nonlinearities. The key structural insight is the absorption of all constraints into the integral operator algebra, allowing manipulation entirely at the level of $L_2$-states and operator kernels.

Limitations include additional algebraic complexity in constructing the operator maps for high-order, multivariate, or nonstandard boundary conditions, and polynomial expansion degrees required for nontrivial geometries or high accuracy. For nonlinear terms beyond polynomials, or strongly nonlocal/measure-based effects, further generalization or new basis constructions may be required. Ongoing research addresses extensions to stochastic PDEs, data-driven models, and scalable parallel implementations.

The PIE framework and its operator-algebraic representation fundamentally shift the paradigm for PDE system analysis, observer/controller synthesis, and computational certification, offering a scalable, canonical, and convex approach for a wide spectrum of infinite-dimensional systems (Jagt et al., 2023, Peet, 2018, Jagt et al., 20 Aug 2025).