Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs

Abstract: Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal ${H}_\infty$ control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.

• The paper introduces a novel PIE framework that reformulates linear PDE control problems into convex optimization tasks for impulse-to-peak norm minimization.
• It establishes strong duality and utilizes Lyapunov operator inequalities to derive finite-dimensional SDP relaxations that certify safety-critical performance.
• Numerical studies on transport, heat, and reaction-diffusion PDEs validate the approach by demonstrating reduced I2P norms and improved system stabilization.

Impulse-to-Peak Norm Optimal State-Feedback Control of Linear PDEs

Introduction and Motivation

The paper "Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs" (2604.03399) addresses the challenge of characterizing and synthesizing optimal state-feedback controllers that minimize the impulse-to-peak (I2P) output norm for systems governed by linear partial differential equations (PDEs). The I2P norm is crucial in safety-critical applications, as it certifies the system's worst-case instantaneous response to impulsive disturbances—an aspect not captured by traditional $H_\infty$ or LQR metrics.

For PDEs, existing control design paradigms either rely on finite-dimensional approximations (early lumping) or solve intractable infinite-dimensional Riccati/HJB equations (late lumping), often resulting in either significant performance degradation or loss of provable guarantees. The absence of a universal state-space representation for PDEs further complicates the direct extension of optimal I2P control techniques from ODEs.

This paper circumvents these issues by leveraging the Partial Integral Equation (PIE) representation, which provides a universal, bounded-operator-based state-space form for linear PDEs. The PIE framework, including the use of polynomial parameterized Partial Integral (PI) operators that are closed under algebraic operations, enables the deployment of convex optimization and Lyapunov-based analysis tools akin to those available for finite-dimensional systems via LMIs.

PIE Framework and Duality for PDE Control

The PIE representation reformulates linear PDEs using bounded PI operators, allowing PDE dynamics to be manipulated as operator equations in Hilbert spaces. These PI operators can be parameterized via polynomials and represented numerically as finite matrices, enabling direct optimization over operator inequalities.

A central theoretical result established in the paper is the strong duality of PIE systems: the impulse-to-peak norm, stability, and well-posedness properties are invariant under taking operator adjoints, i.e., for a PIE and its dual (obtained by adjoint operation on all parameters), the I2P norm is identical. This duality is essential because it enables the recasting of inherently non-convex controller synthesis problems into convex formulations by shifting the analysis to the dual system.

Optimization Characterization of I2P Norm for PDEs

The impulse-to-peak norm of a PIE system is formally defined via its response to an impulsive disturbance, transformed into an equivalent initial condition problem using the auxiliary PIE. The main technical contributions of the paper lie in deriving sufficient (convex) conditions for upper-bounding the I2P norm of PIEs, based on Lyapunov operator inequalities involving PI operator variables. Two main types of Lyapunov functionals are presented:

• Coercive Lyapunov Operator Approach: If there exists a $\gamma$ and positive PI operator $\mathcal{Q}$ satisfying a set of linear PI inequalities (analogous to LMI conditions), the I2P norm is upper-bounded by $\gamma$.
• Non-Coercive Lyapunov Operator Approach: Employing a broader class of Lyapunov functionals, the paper derives less conservative sufficient conditions in the form of PI operator block inequalities, still amenable to convex optimization.

Both approaches yield finite-dimensional SDP relaxations once PI operators are suitably parameterized, allowing computation of the smallest achievable $\gamma$ (I2P norm bound) via established optimization routines.

Convex Synthesis of I2P-Optimal State-Feedback for Linear PDEs

The proposed framework extends naturally to the synthesis of state-feedback controllers minimizing the I2P norm. By expressing the closed-loop system in PIE form, and adopting a variable substitution (using $Z = KQ$) following LMI-based S-procedure principles, the resulting controller synthesis conditions become convex in the optimization variables $(Q, Z)$.

The strong duality property ensures that these convex formulations are both feasible and non-conservative: the computed controller not only stabilizes the PIE/PDE system but also certifies the strict I2P performance bound.

Numerical Results and Key Claims

The paper validates the theoretical developments with numerical studies on canonical linear PDEs (transport and heat equations) and reaction-diffusion systems. Notable results include:

• Tight empirical upper bounds: For both transport and heat PDEs, computed I2P bounds from the proposed optimization match analytic bounds (e.g., for the heat equation, computed $\gamma=0.5$ equals the theoretical value $1/2$).
• Stabilization of unstable PDEs: In the reaction-diffusion example, the uncontrolled system is unstable with infinite I2P gain; the synthesized controller stabilizes the PDE and provides a concrete I2P bound ($\gamma=1.375$).
• Performance improvement in stable systems: For the transport equation, the designed controller reduces the I2P norm from $\gamma$0 (uncontrolled) to $\gamma$1 (controlled).

These empirical results demonstrate the viability of the PIE-based I2P control framework in both certifying and improving impulse-induced transient performance for PDE systems.

Implications and Future Directions

Practically, this framework enables certifiable safety guarantees for distributed parameter systems subject to impulsive disturbances—a key requirement in energy systems, thermo-fluid networks, and aerospace applications. Theoretically, it provides a systematic convex approach to performance-limited optimal control design for linear PDEs, avoiding the pitfalls of early/late lumping and the conservatism of classical multiplier techniques.

Future developments may include:

• Extension to nonlinear PDEs: Investigating how PIE representations can be adapted or generalized.
• Integration with robust/adaptive control: Merging PI operator-based methods with uncertainty quantification or adaptive update schemes.
• Scalable software tools: The PIETOOLS Matlab toolbox [shivakumar_pietools_2025] provides an initial implementation; further development could facilitate large-scale, high-dimensional PDE control.

Conclusion

The paper establishes a rigorous and tractable methodology for computing and optimizing the impulse-to-peak output norm for linear PDEs via the PIE framework. By exploiting operator duality and Lyapunov-based convex optimization, it provides both provable and constructive solutions to a class of problems previously inaccessible to systematic design. This advances the analysis and synthesis of feedback control for infinite-dimensional systems with critical transient performance requirements.