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

This paper extends impulse-to-peak (I2P) optimal state-feedback control from ODEs to linear PDEs by leveraging the Partial Integral Equation (PIE) framework and Lyapunov theory to formulate the problem as a convex optimization with provable bounds and a constructive control design method.

Original authors: Tristan Thomas, Sachin Shivakumar, Javad Mohammadpour Velni

Published 2026-04-07

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Tristan Thomas, Sachin Shivakumar, Javad Mohammadpour Velni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the engineer in charge of a massive, complex system, like a power grid, a chemical plant, or even a battery in an electric car. Your job is to keep everything running smoothly.

Most engineers use tools that check if the system is stable over the long run. They ask: "If I push this button, will the system eventually settle down?" This is like checking if a car will eventually stop after you hit the brakes.

But what about the instant? What happens in the split second after a sudden jolt?

A lightning strike hits the grid.
A sudden spike in temperature occurs in a battery.
A gust of wind hits a bridge.

These are "impulses." If the system reacts too violently in that first second—spiking to a dangerous level—it could break, even if it eventually calms down. This is the Impulse-to-Peak (I2P) problem: How high does the system jump when you hit it hard, and can we design a controller to keep that jump as small as possible?

The Problem: The "Black Box" of Infinite Dimensions

For simple systems (like a single car engine), we have a perfect toolbox to solve this. But for complex systems described by Partial Differential Equations (PDEs)—which model things that change over both time and space (like heat spreading across a metal plate)—the toolbox was empty.

Why? Because PDEs are "infinite-dimensional." They have infinite variables (every point on the plate). Trying to solve them directly is like trying to count every single grain of sand on a beach to build a sandcastle. It's too messy, and the math gets stuck.

The Solution: The "PIE" Translation

The authors of this paper found a clever trick. They realized that while PDEs are messy, they can be translated into a different language called Partial Integral Equations (PIEs).

Think of it like this:

The PDE is a chaotic, screaming crowd of people (infinite variables).
The PIE is a calm, organized committee where everyone speaks in a structured, mathematical language.

The authors showed that you can translate any linear PDE into this "PIE language." Once translated, the infinite mess becomes manageable, almost like a giant spreadsheet (matrix) that computers can actually solve.

The Strategy: The "Mirror" Trick (Duality)

Here is the cleverest part of their method. They used a concept called Duality.

Imagine you want to push a heavy boulder up a hill (the control problem). It's hard to push from the bottom. But what if you could look at the problem in a mirror?

In the mirror, the hill looks like a valley.
Pushing the boulder up the hill in the real world is mathematically identical to pulling it down the valley in the mirror.

The authors proved that the "peak jump" of the original system is exactly the same as the "peak jump" of its mirror image (the dual system). By solving the problem in the "mirror" (the dual system), they could easily find the perfect controller for the real system. It's like solving a maze by looking at it from the exit; suddenly, the path becomes obvious.

The Result: A Safety Net for the Instant

Using this translation (PIE) and the mirror trick (Duality), they created a new set of rules (called Linear PI Inequalities or LPIs).

Analysis: They can now calculate the exact "safety limit" for how high a system will jump when hit.
Control: They can design a "smart controller" that automatically adjusts to keep that jump as small as possible.

Real-World Examples

The paper tested this on two classic problems:

The Unstable Reaction: Imagine a chemical reaction that wants to explode (run away). They designed a controller that acts like a super-fast shock absorber, keeping the explosion from ever getting too big.
The Transport Wave: Imagine a wave moving down a pipe. Even if it's stable, it might bounce too high. They designed a controller to "dampen" the wave, making the peak much lower than it would be on its own.

Why This Matters

Before this paper, engineers had to guess or use approximations that weren't guaranteed to be safe. If they were wrong, a battery could catch fire, or a bridge could shake apart.

This new method provides a mathematical guarantee. It says: "We have calculated the worst-case jump, and we have built a controller that proves the system will never exceed this safe limit."

In short: The authors built a universal translator that turns impossible-to-solve infinite problems into solvable puzzles, and then used a mirror trick to design the perfect safety net for systems that need to survive sudden shocks.

1. Problem Statement

The paper addresses the challenge of designing optimal state-feedback controllers for Linear Partial Differential Equations (PDEs) that minimize the Impulse-to-Peak (I2P) norm.

Context: While optimal control for Ordinary Differential Equations (ODEs) using I2P metrics is well-established, extending these techniques to PDEs has been difficult.
Limitations of Existing Methods:
- $H_\infty$ /LQR Metrics: These capture energy-to-energy or energy-to-peak behaviors but fail to certify safety-critical transient spikes (e.g., instantaneous current or temperature spikes in batteries) which are critical for safety.
- Approximation Methods (Early-lumping): Discretizing PDEs into ODEs introduces truncation errors, leading to spillover effects and a lack of guaranteed closed-loop stability or performance bounds.
- Direct PDE Methods (Late-lumping): Approaches using Hamilton-Jacobi-Bellman (HJB) equations or Riccati PDEs often require solving nonlinear PDEs with unbounded operators, which is computationally intractable for most systems.
- Existing Lyapunov Approaches: Previous work provided Lyapunov characterizations but lacked a constructive method to synthesize controllers.
Goal: Develop a computationally efficient, general algorithm to design I2P-optimal state-feedback controllers for a broad class of linear PDEs with provable performance bounds and no conservatism.

2. Methodology

The authors leverage the Partial Integral Equation (PIE) framework, a universal state-space-like representation for linear PDEs, to transform the infinite-dimensional control problem into a solvable convex optimization problem.

A. The PIE Framework

Representation: Instead of dealing with unbounded differential operators directly, the PDE is represented as a PIE using Partial Integral (PI) operators. These operators are bounded, linear, and form a $*$ -subalgebra (closed under addition, composition, and adjoint), similar to finite-dimensional matrices.
Formulation: A linear PDE is converted into a system of the form:
$\partial_t (T x)(t) = Ax(t) + Bw(t)$
$z(t) = Cx(t) + Dw(t)$
where $T, A, B, C, D$ are PI operators.

B. Duality and Norm Characterization

Dual System: The paper defines a dual PIE system. Crucially, it proves that the I2P norm of the primal system is equal to the I2P norm of its dual system.
Impulse-to-Peak Definition: The I2P norm is defined as the maximum peak output ( $L_\infty$ ) resulting from an impulsive input ( $\delta(t)$ ). The authors show this is equivalent to the peak output resulting from a specific initial condition.
Convex Optimization (LPIs): Using Lyapunov stability theory, the I2P norm bounding problem is formulated as a Linear PI Inequality (LPI).
- The problem is transformed from an infinite-dimensional search over operators to a finite-dimensional search over matrix coefficients.
- This is achieved by parameterizing positive PI operators using positive matrices and polynomial kernels, converting the problem into a Linear Matrix Inequality (LMI) solvable via interior-point methods.

C. Controller Synthesis

State-Feedback Design: To find the optimal controller $u(t) = Kx(t)$, the authors use a change of variables ($Z = KQ$) to convert the bilinear constraints (which arise from the product of the Lyapunov function and the controller gain) into linear convex constraints.
Strong Duality: The paper establishes strong duality between the primal and dual optimization formulations, ensuring that the solution found is optimal and not just a conservative bound.

3. Key Contributions

I2P Norm Characterization: The first characterization of the I2P norm for linear PDEs using the PIE framework.
Duality Proof: A rigorous proof that the I2P norm of a PIE is identical to that of its dual system, enabling the use of dual formulations for controller synthesis.
Constructive Synthesis: Derivation of solvable convex optimization (LPI) formulations for both I2P norm upper-bounding and optimal state-feedback control synthesis.
Implementation: The methodology is implemented using the open-source PIETOOLS toolbox in MATLAB, making it accessible for practical application.

4. Results

The authors validated their theoretical results on four numerical examples:

Example 1 (Transport PDE): Computed the I2P norm upper bound. The primal and dual LPI solutions yielded $\gamma \approx 0.1684$ and $0.1667$, closely matching the theoretical bound of $1/6$ .
Example 2 (Heat Equation): Computed the I2P norm for a heat equation. The result $\gamma = 0.5000$ exactly matched the theoretical bound of $1/2$ .
Example 3 (Unstable Reaction-Diffusion): Designed a stabilizing state-feedback controller for an inherently unstable PDE. The method successfully found a controller achieving a finite I2P norm of $\gamma = 1.375$ , stabilizing the system and bounding the peak output.
Example 4 (Transport Equation with Control): Designed a controller to suppress vibrations in a stable transport equation. The controlled system reduced the I2P norm from $0.1678$ (uncontrolled) to $0.1247$, demonstrating the ability to improve transient performance.

5. Significance

Safety-Critical Control: This work provides a rigorous mathematical tool to guarantee that transient spikes (peaks) in PDE systems remain within safe limits, addressing a gap left by standard $H_\infty$ or LQR control.
Generality and Exactness: Unlike approximation methods, this approach works directly on the infinite-dimensional PDE model, avoiding spillover effects and providing provable, non-conservative bounds.
Computational Feasibility: By utilizing the PIE framework and LPIs, the paper bridges the gap between theoretical PDE control and practical, computationally tractable design, allowing for the synthesis of optimal controllers for complex distributed parameter systems.
Foundation for Future Work: The establishment of strong duality and constructive synthesis methods opens the door for applying similar techniques to other non-standard performance metrics in PDE control.