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arxiv: 2604.03399 · v1 · submitted 2026-04-03 · 🧮 math.OC · cs.SY· eess.SY

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

Tristan Thomas , Sachin Shivakumar , Javad Mohammadpour Velni This is my paper

Pith reviewed 2026-05-13 18:07 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords impulse-to-peak normpartial integral equationsPDE controloptimal state feedbackconvex optimizationLyapunov methodsinfinite-dimensional systems
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The pith

Partial integral equations turn impulse-to-peak analysis and optimal state-feedback design for linear PDEs into solvable convex optimization problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates the impulse-to-peak response problem for linear partial differential equations as a convex optimization task that produces explicit upper bounds on the I2P norm. It then exploits strong duality between primal and dual forms of that optimization to construct state-feedback controllers that achieve the optimal I2P performance. The method rests on the exact representation of the PDE as a partial integral equation together with standard Lyapunov arguments. A reader would care because the same framework previously handled stability and H-infinity control, so it now supplies a uniform convex route from analysis to controller synthesis for distributed systems that previously lacked transfer-function tools.

Core claim

Representing linear PDEs exactly as partial integral equations allows the impulse-to-peak response analysis problem to be written as a convex optimization whose solution yields provable bounds on the I2P norm. Strong duality between the resulting primal and dual problems then supplies a constructive procedure that produces state-feedback controllers minimizing that norm, demonstrated on several example PDEs.

What carries the argument

Partial integral equation representation of the PDE, paired with Lyapunov-based convex optimization and strong duality between its primal and dual formulations.

If this is right

  • Explicit, computable upper bounds on the impulse-to-peak norm become available for any linear PDE that admits a partial integral equation representation.
  • Optimal state-feedback gains can be obtained by solving a single convex program whose dual yields the controller parameters.
  • The same PIE-Lyapunov pipeline now covers I2P performance in addition to stability and H-infinity norms.
  • Controller synthesis no longer requires a transfer-function description of the distributed system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical solution of the semidefinite programs on spatially discretized grids would immediately produce implementable controllers for specific PDEs such as the heat or wave equation.
  • The duality construction could be reused for other induced norms once their analysis LMI is available inside the PIE framework.
  • Comparison of the resulting closed-loop I2P performance against classical finite-dimensional approximations would quantify the benefit of working directly in the infinite-dimensional setting.

Load-bearing premise

The partial integral equation is an exact and complete state-space model of the linear PDE, so the convex optimization produces bounds and controllers without hidden approximation error.

What would settle it

A concrete linear PDE whose simulated or analytically computed impulse-to-peak norm exceeds the value returned by the optimization procedure.

Figures

Figures reproduced from arXiv: 2604.03399 by Javad Mohammadpour Velni, Sachin Shivakumar, Tristan Thomas.

Figure 4
Figure 4. Figure 4: The regulated output, R 1 0 x(t, s)ds, for the controlled (−−) and uncontrolled (−) transport equation. The control reduces the peak response under impulsive input in comparison to uncontrolled system. develop Lyapunov-based convex optimization formulation of the impulse-to-peak norm bounding problem for arbitrary lin￾ear PDEs. We showed that the bounds so obtained are prov￾able and have no conservatism. M… view at source ↗
Figure 2
Figure 2. Figure 2: The regulated output, R 1 0 x(t, s)ds, for the reaction-diffusion equation. The control stabilizes the system and satisfies peak output bounds. z(t) = Z 1 0 x(t, s) ds, x(t, 1) = 0. Although the transport equation is naturally stable, the con￾trol action can be utilized to suppress vibrations. Similar to unstable reaction-diffusion PDE, we search for an optimal controller using LPIs in Cor. 1 by bisecting … view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the PDE state for the transport equation [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to extend impulse-to-peak (I2P) analysis and optimal state-feedback control from ODEs to linear PDEs by representing the PDEs as partial integral equations (PIEs). It formulates the I2P problem as a convex optimization using Lyapunov techniques to obtain provable bounds on the I2P norm, establishes strong duality between primal and dual problems to construct an optimal controller, and demonstrates the approach on examples.

Significance. If the convexity, provable bounds, and strong duality hold, the work would provide a valuable general framework for I2P control of PDEs, addressing the historical lack of suitable state-space representations. The constructive controller via duality and the use of the established PIE framework for Lyapunov-based optimization represent a clear strength, potentially enabling non-conservative designs for a range of linear PDEs.

major comments (1)
  1. Abstract and the section establishing strong duality: the claim that strong duality holds for the primal/dual I2P pair in the infinite-dimensional PIE setting is load-bearing for the constructive controller, yet no explicit verification of a constraint qualification (such as a Slater-type interior-point condition on the operator variables) is indicated; without it the duality gap may be positive and the dual solution may yield only a lower bound rather than the exact optimal I2P norm or gain.
minor comments (2)
  1. Ensure the definition of the I2P norm and its relation to the impulse response operator is stated with explicit reference to the PIE state-space realization to allow direct comparison with the classical ODE case.
  2. In the examples section, clarify how the infinite-dimensional convex programs are discretized or solved numerically and report the resulting I2P bounds alongside any simulation validation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of constraint qualifications when claiming strong duality in the infinite-dimensional PIE setting. We address the major comment below and will revise the manuscript to include the requested verification.

read point-by-point responses

  1. Referee: Abstract and the section establishing strong duality: the claim that strong duality holds for the primal/dual I2P pair in the infinite-dimensional PIE setting is load-bearing for the constructive controller, yet no explicit verification of a constraint qualification (such as a Slater-type interior-point condition on the operator variables) is indicated; without it the duality gap may be positive and the dual solution may yield only a lower bound rather than the exact optimal I2P norm or gain.

    Authors: We agree that an explicit constraint qualification is required to guarantee strong duality for the infinite-dimensional semidefinite programs arising from the PIE formulation. In the revised manuscript we will add a new remark (placed immediately after the statement of strong duality) that verifies a Slater-type interior-point condition. Concretely, we exhibit a strictly feasible point for the primal problem by selecting Lyapunov operators that are strictly positive definite on the PIE state space; such operators exist because the open-loop system is assumed exponentially stable, allowing a uniform positive margin in the Lyapunov inequalities. This construction ensures the duality gap is zero, so the dual solution recovers the exact optimal I2P norm and the associated state-feedback gain is optimal. revision: yes

Circularity Check

0 steps flagged

No circularity: PIE-based I2P formulation and duality are independent extensions

full rationale

The derivation applies standard Lyapunov stability and convex optimization techniques to the existing PIE state-space representation of PDEs, then invokes strong duality on the resulting primal/dual pair to obtain the state-feedback controller. No equation reduces to a prior fitted quantity or self-defined input by construction; the I2P norm bounds and optimal gains are obtained from the optimization problem itself rather than being presupposed. The PIE framework is treated as an external input (recently proposed), and the paper's contribution is the specific I2P convex program and duality argument, which remain falsifiable against external benchmarks without self-referential collapse.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the validity of the PIE representation for the target PDE class and on standard results from Lyapunov stability and convex duality; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The partial integral equation representation is a valid and complete state-space model for the linear PDEs under study
    The abstract treats the recently proposed PIE framework as the enabling state-space representation without re-deriving its equivalence to the original PDE.

pith-pipeline@v0.9.0 · 5489 in / 1350 out tokens · 56271 ms · 2026-05-13T18:07:49.609967+00:00 · methodology

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