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REVIEW 2 major objections 6 minor 47 references

LQ optimal control for infinite-dimensional passive systems

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Pith's one-line read For impedance energy-preserving infinite-dimensional passive systems, the unique stabilizing LQ-optimal control is $u(t) = -y(t)$ and the optimal cost operator is the identity; for scattering energy-preserving systems the optimal control…

desk verdict Solid generalization of LQ optimal control to unbounded passive systems, with a clean identity solution for energy-preserving nodes, but the finite-cost proof leans on a self-cited preprint and the beam example overreaches by one quantifier. read the letter →

arxiv 2506.03882 v1 pith:4NB64MOX submitted 2025-06-04 math.OC math.AP

Anthony Hastir , Birgit Jacob This is my paper
classification math.OCmath.AP MSC 49N1093C2593C2093B5247D06
keywords LQoptimalcontrolinfinite-dimensionalsystemspassivesystemnodesoperatorRiccatiequationboundaryport-HamiltonianEuler-Bernoullibeam
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper solves the linear-quadratic optimal control problem for a broad class of infinite-dimensional passive systems in which input and output operators may be unbounded. The central claim is that the finite cost condition—existence of an input with finite cost for every initial state—holds automatically for every impedance passive system node, and that the optimal cost operator is a contraction ($\Pi \le I$). For systems that preserve energy exactly, the paper proves more: the optimal cost operator is the identity, the unique stabilizing optimal control is negative output feedback $u(t) = -y(t)$ for impedance energy-preserving systems, and zero input for scattering energy-preserving systems. This yields an explicit solution of the operator Riccati equation and a direct factorization of the Popov function, without solving any spectral factorization problem. The results are recast for boundary control systems and first-order port-Hamiltonian systems, and are illustrated on an Euler-Bernoulli beam with shear force control.

What carries the argument

The central object is a system node, a closed operator $S = \begin{bmatrix} A\&B \\ C\&D \end{bmatrix}$ on $X \times U$ to $X \times Y$ that encodes the dynamics $\dot x = A\&B(x,u)$, $y = C\&D(x,u)$ while allowing unbounded input and output operators through the domain condition $D(S) = \{(x,u) : Ax + Bu \in X\}$. Passivity is expressed as algebraic inequalities on $S$: impedance passivity requires $2\operatorname{Re}\langle A\&B(x,u), x\rangle_X \le 2\operatorname{Re}\langle C\&D(x,u), u\rangle_U$, and scattering passivity requires $2\operatorname{Re}\langle A\&B(x,u), x\rangle_X \le \|u\|_U^2 - \|C\&D(x,u)\|_Y^2$. The argument combines three ingredients: the operator-node Riccati equation (13), introduced by the references as the Lur'e form of the Riccati equation for system nodes; the feedback interconnection $S_K$ obtained by closing the loop with $u = -y$, which is scattering passive and hence well-posed; and the energy-preservation identities that rewrite the cost as a sum of squares plus initial energy minus terminal energy. These identities let the paper minimize the cost by forcing the square terms to zero, yielding $u = -y$ or $u = 0$ directly.

What would settle it

Exhibit an impedance passive system node for which the closed loop under $u = -y$ is not well-posed, meaning its closed-loop transfer function is not bounded on the right half-plane; then the finite cost condition can fail, contradicting Theorem 1. Alternatively, for an impedance energy-preserving node with strongly stable $T_K$, find any stabilizing input different from $-y$ whose cost is strictly smaller than $\|x_0\|_X^2$.

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Extended reading notes

Core claim

The paper's main result is that for an impedance passive system node $S$ with cost $J(x_0,u) = \int_0^\infty (\|u(t)\|_U^2 + \|y(t)\|_Y^2)dt$, the finite cost condition is always satisfied and the optimal cost operator satisfies $\Pi \le I$. If $S$ is impedance energy preserving and the semigroup $T_K$ generated by the negative output feedback interconnection is strongly stable, then $\Pi = I$ and the unique stabilizing optimal control is $u(t) = -y(t)$; in that case the operator-node Riccati equation (13) holds with $E\&F\begin{bmatrix}x\\u\end{bmatrix} = C\&D\begin{bmatrix}x\\u\end{bmatrix} + u$. For a scattering passive system node the same conclusions hold with $\Pi \le I$, and if the node is scattering energy preserving with strongly stable open-loop semigroup $T$, then $\Pi = I$ and the unique stabilizing optimal control is $u(t) = 0$, with $E\&F\begin{bmatrix}x\\u\end{bmatrix} = \sqrt{2}u$. The paper further shows that the Popov function factorizes as $\chi(s) = P(s) + I$ in the impedance energy-preserving case and $\chi(s) = \sqrt{2}I$ in the scattering energy-preserving case, providing explicit spectral factors. It then translates these theorems into algebraic conditions for boundary control systems and for a class of first-order port-Hamiltonian systems, and derives an explicit optimal boundary control for an Euler-Bernoulli beam with shear force control.

Load-bearing premise

The proof that the finite cost condition always holds depends on an imported, unproved result from a preprint by one of the authors: that negative output feedback applied to any impedance passive system node always produces a well-posed closed-loop system; if that result fails, the first part of the main theorem collapses.

Editorial extensions

If this is right

  • Every impedance passive system node automatically satisfies the finite cost condition, so no separate optimizability check is needed before solving the LQ problem.
  • For impedance energy-preserving systems, the optimal regulator is the static output feedback $u(t) = -y(t)$ with optimal cost $\|x_0\|_X^2$; the Riccati equation is solved explicitly rather than requiring iteration.
  • For scattering energy-preserving systems, the best input is zero, so the uncontrolled trajectory is already LQ-optimal whenever its semigroup is strongly stable.
  • The Popov function factorizes directly as $\chi = P + I$ (impedance energy preserving) or $\chi = \sqrt{2}I$ (scattering energy preserving), giving spectral factors without a separate factorization algorithm.
  • For boundary control and port-Hamiltonian systems, the optimality conditions reduce to matrix inequalities on boundary operators, and for the Euler-Bernoulli beam an explicit boundary feedback law is obtained.
  • For the Euler-Bernoulli beam with shear force control, the optimal input is $u_{\rm opt}(t) = -\mu\bigl(\varepsilon \frac{\partial^3 w_{\rm opt}}{\partial \zeta^3}(1,t) - \frac{\partial w_{\rm opt}}{\partial t}(1,t)\bigr)$ with $\mu = (\sqrt{1+\varepsilon^2}-\varepsilon)^{-1}$, and the optimal cost operator is $\Pi = \mu^{-1} I$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The identity $\Pi = I$ for energy-preserving systems suggests a general principle: when the passivity inequality is an equality, the LQ problem is solved by the energy balance itself, and the Riccati equation reduces to a sum-of-squares identity. A testable extension is whether strictly passive systems with a passivity deficit proportional to $\|u\|^2$ always have $\Pi = cI$ for some scalar $c$, a
  • The spectral factor $\chi = P + I$ for impedance energy-preserving nodes bypasses the usual regular spectral factorization step; one could check whether a similar direct factorization survives for non-energy-preserving passive nodes where the passivity deficit rescales the gain, as the beam's $\mu$ does.
  • The finite cost conclusion rests on the external result that negative output feedback on any impedance passive system node yields a well-posed scattering passive node; if that result fails in a wider class (for instance with unbounded feedthrough), the theorem would restrict to nodes where the closed loop is known to be well-posed by other means.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies the LQ optimal control problem with cost J(x0,u)=∫0∞(∥u(t)∥²_U+∥y(t)∥²_Y)dt for infinite-dimensional passive systems formulated as system nodes, which allows unbounded input and output operators. For impedance passive system nodes, the paper claims that the finite cost condition always holds and that the optimal cost operator satisfies Π≤I; for impedance energy preserving nodes with a strongly stable closed loop under u=-y, it claims Π=I, that the unique stabilizing optimal control is u=-y, and that the operator-node Riccati equation has the explicit solution E&F[x;u]=C&D[x;u]+u. For scattering passive nodes, the analogous claims are finite cost, Π≤I, and, for scattering energy preserving nodes with strongly stable semigroup, u=0 is the unique stabilizing optimal control, Π=I, and E&F[x;u]=√2u. The paper also derives spectral factorizations of the Popov function in both energy-preserving cases, translates the main results to boundary control systems and to a class of first-order port-Hamiltonian systems, and applies them to an Euler-Bernoulli beam with shear force control, where it claims a unique optimal feedback u=-µy and optimal cost operator Π=µ^{-1}I.

Significance. If the results hold, the structural conclusions are attractive and potentially useful: in the energy-preserving case the Riccati equation is solved by an explicit identity, the optimal cost operator is the identity, and the optimal stabilizing control has a simple form. The algebraic checks of the operator-node Riccati equation and the spectral factorizations are clean and explicit, and the translation to boundary control and port-Hamiltonian systems widens the applicability beyond the bounded-input-output setting. The main caveats are that the finite-cost argument for impedance passive nodes rests on an unproved, self-cited preprint result, and that several statements claim global optimality while the proofs only establish optimality among stabilizing controls. Both issues are localizable and fixable, but they are load-bearing for the abstract's central claims.

major comments (2)
  1. [Section 3.1, Theorem 1 and Remark 2] The proof of the finite cost condition in Theorem 1 depends on the unproved external result [30, Thm. 2.5]. The manuscript uses this result to assert that the negative output feedback u=-y turns an impedance passive system node S into a well-posed scattering passive system node S_K, but it neither states the exact theorem nor verifies its hypotheses against Definitions 4 and 7. In particular, it is not clear whether [30, Thm. 2.5] allows S to be non-well-posed, whether K=I satisfies the condition 'Re K ≥ I' in the intended sense, and whether additional assumptions on U or on the feedthrough operator are needed. The subsequent energy-balance estimate for J(x0,-y) requires the existence of the closed-loop trajectory T_K^t x0 and of the corresponding output y, so without this theorem the finite cost condition for impedance passive nodes is not established. Because Theorems 5 and 7 inherit the finite cost conclusion from Theorem 1, the gap propagates. The authors should either prove the needed version of [30, Thm. 2.5] in the paper, give a published reference whose hypotheses are checked here, or add well-posedness of the closed-loop system as an explicit assumption.
  2. [Sections 3.1-3.2, 4-6, Theorems 1, 3, 5-8 and Proposition 1] The identity Π=I (or Π=µ^{-1}I) is derived from optimality among stabilizing controls, while Π was defined as the infimum over all u∈L²(0,∞;U). In the proofs of Theorems 1 and 3 the completed-square expression is minimized after imposing lim_{t→∞}∥x(t)∥=0, and the proof of Proposition 1 says explicitly 'Because we look for a stabilizing optimal control'. No argument is supplied that a global minimizer must be stabilizing or that a non-stabilizing control cannot produce a smaller value of (11). Thus the global optimality claims in the abstract and in Proposition 1, and the unconditional identification Π=I in Theorems 1 and 3, are not supported by the given proofs. The same issue appears in Theorems 5-8, whose statements say 'the unique optimal control' without a stabilizing qualifier. The manuscript should either prove that minimizers of (11) are necessarily stabilizing, for example via a detectability or observability argument, or restrict the stated optimality to the class of stabilizing controls consistently throughout the paper.
minor comments (6)
  1. [Section 1, Introduction] The phrase 'passive systems with impedance and dispersion' should presumably be 'impedance and scattering'; as written, 'dispersion' does not match the notions studied in the paper.
  2. [Section 2.2, Remark 1] The operator K in the phrases '-K ∈ L(U) is an admissible output feedback operator' and 'if Re K ≥ I' is never defined; presumably it should be K=I, or the parameter K should be introduced explicitly before the remark.
  3. [Section 6, Proposition 1] The displayed formula 'J(w0, uopt)=µ^{-1}∥x0∥_X' mixes the notation w0 and x0 and omits the square; it should read J(x0,u_opt)=µ^{-1}∥x0∥²_X with the initial state x0=(w(·,0),∂_t w(·,0))^T.
  4. [Section 3.2, proof of Theorem 3] The map in the last displayed formula is written 'E&F : D(s)→U' with a lowercase s; this should be 'D(S)'.
  5. [Equations (14) and surrounding text] The definition of D(A_K) in (14) uses an existential quantifier '∃v∈U' and then writes v(x); the uniqueness of v for each x should be stated explicitly, since A_K must be single-valued.
  6. [Section 5, Theorems 7 and 8] The phrases 'the partial differential equations (18a) together with the boundary conditions (24) is strongly stable' and the analogous phrase with (25) should use the plural verb 'are' instead of 'is'.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity; the Π=I and u=-y results follow from energy balance, but the finite-cost proof depends on a self-cited preprint for closed-loop well-posedness.

full rationale

The derivation is not circular in the sense of Eq. X = Eq. Y by construction. Theorem 1 proves the finite cost condition by choosing u=-y and using [30, Thm. 2.5] to assert that the closed-loop node S_K is a well-posed scattering passive system node. This is a load-bearing dependency on a preprint coauthored by A. Hastir, so the paper is not fully self-contained at that step. However, [30] is an external result about monotone output feedback, not a disguised version of the paper's LQ conclusion; it is parameter-free, has stated assumptions, and can be checked independently. Once well-posedness of S_K is granted, the identities Π=I and the optimal feedback u=-y (resp. u=0 for scattering) are obtained directly from the passivity energy balance and the strong-stability assumption, not from the definition of the optimal control or from any fitted quantity. The Riccati verifications are algebraic checks. Thus there is no significant circularity; score reflects only the self-citation reliance.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It builds on standard system node theory and several cited results; the most fragile input is a self-cited preprint [30] for the closed-loop well-posedness.

assumptions (5)
  • domain assumption The algebraic characterization of passivity in terms of system node inequalities is equivalent to the integral passivity inequalities (Staffans 2002, Thm 3.3, 3.4, 4.2, 4.6).
    Used in Definitions 7 and 8 and throughout the proofs of Theorems 1 and 3.
  • domain assumption The operator node Riccati equation characterizes the LQ optimal control problem: the optimal input solves E&F[x;u]=0 (Opmeer-Staffans 2014, Opmeer 2014).
    Bridges the Riccati equation to the optimal control in Theorems 1 and 3 and Proposition 1.
  • domain assumption Negative output feedback u=-y for an impedance passive system node yields a scattering passive (hence well-posed) system node S_K (Hastir-Paunonen 2025, Theorem 2.5, arXiv preprint).
    Essential for the finite cost condition proof in Theorem 1; the cited theorem is a self-cited preprint.
  • domain assumption For port-Hamiltonian systems, Assumption 1 guarantees that A generates a C0-semigroup and that (G,L,K) is a boundary node (Jacob-Zwart 2012, Thm 11.3.2; Jacob-Morris-Zwart 2015).
    Used in Section 5 to apply the boundary control results.
  • domain assumption The closed-loop beam operator Aopt generates an exponentially stable semigroup for any alpha>0 (Conrad-Morgul 1998, Lemma 3.1).
    Used in Proposition 1 to ensure the proposed feedback is stabilizing.

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Cite this review

Pith. "Pith review of LQ optimal control for infinite-dimensional passive systems." pith.science (2026). https://pith.science/paper/4NB64MOX

@misc{pith2026250603882,
  author       = {Pith},
  title        = {Pith review of: LQ optimal control for infinite-dimensional passive systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NB64MOX}},
  note         = {Machine review of arXiv:2506.03882}
}
read the original abstract

We study the Linear-Quadratic optimal control problem for a general class of infinite-dimensional passive systems, allowing for unbounded input and output operators. We show that under mild assumptions, the finite cost condition is always satisfied. Moreover, we show that the optimal cost operator is a contraction. In the case where the system is energy preserving, the optimal cost operator is shown to be the identity, which allows to deduce easily the unique stabilizing optimal control. In this case, we derive an explicit solution to an adapted operator Riccati equation. We apply our results to boundary control systems, first-order port-Hamiltonian systems and an Euler-Bernoulli beam with shear force control.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.