Linear Partial Integral Inequalities

• Linear partial integral inequalities are functional inequalities that generalize classical bounds by encoding quadratic and polynomial relations among function derivatives under boundary constraints.
• They integrate techniques from functional analysis, polynomial optimization, and semidefinite programming to yield tractable certificates for stability, robustness, and performance in PDE, PIE, and delayed systems.
• Applications include designing Lyapunov functionals, $L_2$-gain analysis, and robust control synthesis, demonstrated through SDP-based numerical examples and operator-theoretic frameworks.

Linear partial integral inequalities are a class of functional inequalities central to the analysis of infinite-dimensional systems, especially those governed by partial differential equations (PDEs), partial integral equations (PIEs), and interconnected PDE-ODE systems. These inequalities generalize classical integral inequalities (such as Jensen and Wirtinger bounds) by rigorously encoding quadratic (and more generally polynomial) relations among functions and their derivatives, usually subject to boundary and coupling constraints. Their modern theory synthesizes functional analysis, polynomial optimization, and semidefinite programming, enabling tractable certificates for positivity, stability, robustness, and performance bounds for distributed parameter and delay systems.

1. Mathematical Definitions and General Framework

A canonical setting is a bounded domain $\Omega \subset \mathbb{R}^d$ with $\mathbf{u}:\Omega\to\mathbb{R}^n$, possibly with prescribed smoothness and boundary conditions. A linear partial integral inequality (LPII) typically takes the form: $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$ where $v_{\theta}(\mathbf{u})(x)$ is the vector collecting all partial derivatives $D^\alpha \mathbf{u}(x)$ up to order $\theta$, and $F(x)\in\mathbb{R}^{n_v \times n_v}$ is a matrix-valued function, often polynomial in $x$ and the coefficients of $v_\theta(\mathbf{u})$.

In broader operator-theoretic terms (notably for PIE and PI operator settings), a linear partial integral operator $\mathcal{L}$ acts on a Hilbert space $\mathbf{u}:\Omega\to\mathbb{R}^n$0 via kernels: $\mathbf{u}:\Omega\to\mathbb{R}^n$1 and the corresponding PI inequality requires

$\mathbf{u}:\Omega\to\mathbb{R}^n$2

with inner product specified by $\mathbf{u}:\Omega\to\mathbb{R}^n$3 on the function component and Euclidean on $\mathbf{u}:\Omega\to\mathbb{R}^n$4 (Shivakumar et al., 2019).

LPIs generalize to block-operator settings—a key feature in robust and optimal control.

2. Functional Certificates and Polynomial Positivity

A foundational principle is the equivalence between the positivity of the quadratic (functional) form defined by the integral and pointwise or operator positivity of an associated matrix or PI-operator:

• Integration-by-parts Nullspace: For scalar or vector PDEs, one can construct multipliers $\mathbf{u}:\Omega\to\mathbb{R}^n$5 such that the boundary terms vanish and auxiliary (null) integrals are introduced. The sum $\mathbf{u}:\Omega\to\mathbb{R}^n$6, where $\mathbf{u}:\Omega\to\mathbb{R}^n$7 collects divergence/contribution from multipliers, yields a pointwise LMI: $\mathbf{u}:\Omega\to\mathbb{R}^n$8 for all $\mathbf{u}:\Omega\to\mathbb{R}^n$9 (Valmorbida et al., 2014).
• PI and LOI Frameworks: In the operator domain, positivity of an operator $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$0 (self-adjoint, defined by polynomial kernels) is equivalent to the existence of a sum-of-squares (SOS) Gram matrix representation in polynomial bases. This reduces infinite-dimensional positivity constraints to finite-dimensional semidefinite programs (Shivakumar et al., 2019).

For polynomial matrix $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$1 (as above), Putinar’s Positivstellensatz applies: $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$2 on a semialgebraic set can be certified by searching for $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$3 positive semidefinite matrices so that $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$4, with $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$5 characterizing the domain (such as $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$6 on $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$7) (Valmorbida et al., 2014).

3. Semidefinite Programming and Computational Tools

The practical verification and optimization over LPIs leverage sum-of-squares (SOS) relaxations and semidefinite programming (SDP):

• SOS/SDP Encoding: All polynomial coefficients (including for multipliers, nullspace functions, and PI operator kernels) are decision variables. The constraints that a PI operator be positive semidefinite are translated into Linear Matrix Inequalities (LMIs) for the (Gram matrix) coefficients of polynomial bases (Valmorbida et al., 2014, Shivakumar et al., 2019, Fantuzzi, 2022).
• PIETOOLS: A MATLAB toolbox facilitating efficient algebraic manipulation, operator positivity constraints, and optimization of PI operators. It packages routines for symbolic kernel definition, positivity constraints, and operator-valued objective and constraint management, with interfaces to numerical SDP solvers such as SeDuMi or MOSEK (Shivakumar et al., 2019).
• Moment-SOS Hierarchy: For functional inequalities involving general nonlinearities in $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$8, $$\int_{\Omega} v_{\theta}(\mathbf{u})(x)^\top F(x) v_{\theta}(\mathbf{u})(x) \, dx \geq 0$$9, etc., a measure-theoretic lifting allows encoding via a moment relaxation—moments of the occupation and boundary measures encode all relevant information, and the positivity is enforced via moment and localizing matrices (Fantuzzi, 2022).

This unification enables convex and scalable optimization for certification and synthesis tasks.

4. Master Inequalities, Least-Squares Principle, and Hierarchical Tightening

Contemporary theory provides a complete characterization of LPIs by introducing two "master" classes:

• Class I ("tight" without slack): Inequalities of the form

$v_{\theta}(\mathbf{u})(x)$0

where $v_{\theta}(\mathbf{u})(x)$1 is the Gram matrix for a basis $v_{\theta}(\mathbf{u})(x)$2, $v_{\theta}(\mathbf{u})(x)$3, and $v_{\theta}(\mathbf{u})(x)$4 denotes the symmetrization (Feng et al., 2023).

• Class II (free-matrix type): Introduces decision matrices that generalize slack variables and allows for tightest lower bounds equivalent to Class I via optimization.
• Least-Squares Principle: The tightest lower bound in the hierarchy is derived from the $v_{\theta}(\mathbf{u})(x)$5-norm projection of $v_{\theta}(\mathbf{u})(x)$6 onto the subspace spanned by the chosen basis functions $v_{\theta}(\mathbf{u})(x)$7. As the basis becomes dense (Schauder basis), the lower bound converges to the exact upper integral, demonstrating asymptotic tightness (Feng et al., 2023).

Classical inequalities such as Jensen's and Wirtinger's are recovered as specializations of this framework.

5. Applications: Stability, Performance, and Control of Infinite-Dimensional Systems

LPIs are foundational in:

• Lyapunov Functionals for PDEs: Formulation of quadratic Lyapunov functionals and their derivatives as LPIs yields tractable SDP-based criteria for exponential stability. For example, the exponential stability of a linear PDE can be proved by feasibility of a matrix positivity certificate derived from an associated LPI (Valmorbida et al., 2014, Feng et al., 2023).
• $v_{\theta}(\mathbf{u})(x)$8-Gain and $v_{\theta}(\mathbf{u})(x)$9-Optimal Estimation: The $D^\alpha \mathbf{u}(x)$0 operator gain and $D^\alpha \mathbf{u}(x)$1 estimator synthesis problems for PDE and PIE systems admit convex LPI formulations. A KYP-type LPI on PI operators characterizes the induced $D^\alpha \mathbf{u}(x)$2-gain, and $D^\alpha \mathbf{u}(x)$3-optimal observer design reduces to a block-operator LMI on the PIE state (Jagt et al., 2022, Braghini et al., 2024).
• Robustness and $D^\alpha \mathbf{u}(x)$4-Analysis: IQC theory and structured singular value ($D^\alpha \mathbf{u}(x)$5) analysis are lifted to the PIE setting by encoding IQCs as operator-valued multipliers within LPIs. This yields convex certificates for robust stability and controller synthesis, exploiting dynamic IQCs for conservatism reduction (Lenssen et al., 18 Nov 2025).
• Coupled PDE-ODE and Delay Systems: Hierarchical LPI methods address boundary, coupling, and distributed delays, with systematic recipes for deriving LMI-based stability and performance tests (Feng et al., 2023).

Numerical examples confirm that SDP-derived certificates closely match analytical optima (e.g., recovery of sharp constants in Poincaré and Wirtinger inequalities) (Valmorbida et al., 2014, Fantuzzi, 2022).

6. Unified Operator-Theoretic and Measure-Theoretic Approaches

Two principal mathematical liftings unify the treatment of LPIIs:

• Operator-Theoretic: PI and 4-PI operator algebras, forming adjoint-closed, composition-closed spaces, allow parametrization and certification via polynomial kernels as described above (Shivakumar et al., 2019, Braghini et al., 2024).
• Measure-Theoretic/Moment Approach: Occupation measures and boundary measures encode all functional constraints (domain, PDE, boundary) and divergence theorems. Linear functionals of dependent variables and their derivatives are linear in these measures, and a hierarchy of moment relaxations provides approximating SDPs whose convergence is monotonic as relaxation order increases (Fantuzzi, 2022).

This duality ensures that the convex-analytic and algebraic-geometric perspectives are harmonized within computational optimization.

7. Examples, Numerical Implementation, and Scalability

Several canonical examples illustrate the range and power of linear partial integral inequalities:

• Poincaré Inequality: Tight computation of the optimal constant on $D^\alpha \mathbf{u}(x)$6 using an SDP encoding returns $D^\alpha \mathbf{u}(x)$7 as polynomial degree increases.
• Energy Decay in Transport and Heat Equations: Suitable weighting, integrand, and multipliers enable certification of exponential decay for parameterized classes.
• Stability of Coupled Reaction-Diffusion and PDE-ODE Systems: Optimization over polynomial matrix weights and boundary-coupling inequalities produces computable and tight bounds close to direct simulation results (Valmorbida et al., 2014, Jagt et al., 2022, Braghini et al., 2024, Feng et al., 2023).
• Operator Norms and Integral Operators: Calculation of the $D^\alpha \mathbf{u}(x)$8 norm for Volterra and related integral operators as in PIETOOLS matches analytic results (Shivakumar et al., 2019).

Tooling (PIETOOLS, SOSTOOLS) automates operator decision variable handling, LMI assembling, and solution extraction, scaling polynomially in both operator and kernel degree, and enabling practical use for moderately high-dimensional problems (Shivakumar et al., 2019, Jagt et al., 2022).

---

The modern theory of linear partial integral inequalities provides a convex, systematic, and computationally tractable framework for the analysis and synthesis of infinite-dimensional dynamical systems. It subsumes classical integral inequalities, enables robust and optimal control design via convex programming, and tightly connects functional analysis, operator algebras, and real algebraic geometry (Valmorbida et al., 2014, Shivakumar et al., 2019, Jagt et al., 2022, Lenssen et al., 18 Nov 2025, Braghini et al., 2024, Fantuzzi, 2022, Feng et al., 2023).