Dichotomous Hamiltonians and Riccati equations for systems with unbounded control and observation operators

Christian Wyss

arxiv: 1907.05806 · v1 · pith:3PIQYXKRnew · submitted 2019-07-12 · 🧮 math.FA

Dichotomous Hamiltonians and Riccati equations for systems with unbounded control and observation operators

Christian Wyss This is my paper

Pith reviewed 2026-05-24 22:03 UTC · model grok-4.3

classification 🧮 math.FA

keywords Riccati equationHamiltonian operator matrixdichotomyinvariant subspacesunbounded operatorsexponential stabilitycompact resolventalgebraic Riccati equation

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The pith

A dichotomy property of the Hamiltonian operator matrix yields nonnegative and nonpositive solutions to the Riccati equation through invariant graph subspaces.

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

The paper develops a technique for solving the algebraic Riccati equation in infinite-dimensional systems that allow unbounded control and observation operators. It begins from the assumption that the associated Hamiltonian operator matrix has a dichotomy, meaning its spectrum splits into left and right half-planes without imaginary axis components. This property is used to construct two invariant graph subspaces. One subspace produces a nonnegative solution to the Riccati equation, while the other produces a nonpositive solution. When the original system generator has compact resolvent, the nonnegative solution is shown to be bounded and the feedback system it defines is exponentially stable.

Core claim

Using a dichotomy property of the associated Hamiltonian operator matrix, two invariant graph subspaces are constructed which yield a nonnegative and a nonpositive solution of the Riccati equation. The boundedness of the nonnegative solution and the exponential stability of the associated feedback system is proved for the case that the generator of the system has a compact resolvent.

What carries the argument

The dichotomy property of the Hamiltonian operator matrix, which splits the spectrum into stable and unstable parts with no spectrum on the imaginary axis, allowing the construction of invariant graph subspaces for the Riccati solutions.

If this is right

The construction gives a nonnegative solution to the control algebraic Riccati equation.
The construction gives a nonpositive solution to the control algebraic Riccati equation.
When the system generator has compact resolvent, the nonnegative solution is bounded.
The associated feedback system is exponentially stable under the compact resolvent assumption.

Where Pith is reading between the lines

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

The approach offers a spectral criterion for the existence of solutions without requiring direct solution of the Riccati equation.
This framework may apply to verifying stability in boundary-controlled PDE systems where operators are unbounded.
Extensions could explore whether similar subspace constructions work when the dichotomy is replaced by other spectral conditions.

Load-bearing premise

The Hamiltonian operator matrix possesses a dichotomy property, with its spectrum splitting into stable and unstable parts and no spectrum on the imaginary axis.

What would settle it

A system with unbounded control and observation operators and compact resolvent generator for which the Hamiltonian has a dichotomy but the resulting feedback system fails to be exponentially stable.

read the original abstract

The control algebraic Riccati equation is studied for a class of systems with unbounded control and observation operators. Using a dichotomy property of the associated Hamiltonian operator matrix, two invariant graph subspaces are constructed which yield a nonnegative and a nonpositive solution of the Riccati equation. The boundedness of the nonnegative solution and the exponential stability of the associated feedback system is proved for the case that the generator of the system has a compact resolvent.

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.

Desk Editor's Note private letter to a colleague

The paper builds Riccati solutions from an assumed Hamiltonian dichotomy for unbounded operators and adds a compact-resolvent stability result.

read the letter

The main point is a construction that takes a dichotomy on the Hamiltonian operator matrix and produces two invariant graph subspaces, one giving a nonnegative Riccati solution and one a nonpositive one. For the case where the system generator has compact resolvent, it further shows the nonnegative solution is bounded and the closed-loop system is exponentially stable. This is the concrete advance over earlier bounded-operator results in the same line of work. The approach is direct: the subspaces are built from the stable and unstable parts of the spectrum, and the Riccati equation is recovered from the invariance. The compact-resolvent theorem is a clean extra statement that should be usable in applications where the generator is known to have that property. The obvious limitation is that the dichotomy itself is taken as a hypothesis rather than derived from A, B, and C. No general verifiable conditions on the original data are supplied to guarantee the spectral split with nothing on the imaginary axis, so the result applies only when that property can be checked separately. Domain questions for the unbounded case are also present by nature, though the abstract indicates the construction is set up to stay inside the right domains. The paper is written for people already working on infinite-dimensional Riccati equations and Hamiltonian methods. A reader in that narrow area will see a usable technical step; outsiders will find it too specialized. The claims are stated clearly enough and rest on a standard construction once the dichotomy is granted, so the work is worth sending to referees who can check the operator-domain details and the compact-resolvent argument.

Referee Report

2 major / 2 minor

Summary. The paper studies the control algebraic Riccati equation for infinite-dimensional systems with unbounded control and observation operators. Assuming a dichotomy property (spectral splitting with no imaginary-axis spectrum) of the associated Hamiltonian operator matrix, it constructs two invariant graph subspaces yielding a nonnegative and a nonpositive solution of the Riccati equation. For the special case in which the system generator has compact resolvent, boundedness of the nonnegative solution and exponential stability of the closed-loop feedback system are proved.

Significance. If the central construction is valid, the work supplies a useful abstract framework for obtaining stabilizing Riccati solutions in the presence of unbounded operators, a setting that arises in boundary-controlled PDEs. The compact-resolvent stability result supplies a concrete, verifiable conclusion under an additional structural hypothesis. The approach is parameter-free once the dichotomy is granted and therefore has potential for further application once verifiable conditions guaranteeing the dichotomy are identified.

major comments (2)

[Definition of the Hamiltonian matrix and §3 (construction of the graph subspaces)] The domain of the Hamiltonian operator matrix for unbounded B and C is not specified with sufficient precision. It is therefore unclear whether the constructed graph subspaces lie inside this domain and whether the resulting operators satisfy the Riccati equation in the strong (or mild) sense required when the control and observation operators are unbounded. This verification is load-bearing for the central claim that the graph subspaces yield solutions of the Riccati equation.
[Introduction and the statement of the main theorems] All subsequent results (nonnegativity, boundedness, and exponential stability) rest on the dichotomy assumption being given a priori. No general, checkable conditions on the triple (A,B,C) that guarantee the required spectral splitting are supplied; the manuscript therefore leaves open the question of when the hypothesis can be verified for concrete systems.

minor comments (2)

[§3] Notation for the graph subspaces and the associated projections should be introduced once and used consistently; several passages repeat the same construction with slightly varying symbols.
[Theorem on exponential stability] The compact-resolvent assumption is used only for the stability conclusion; its necessity or sufficiency for the dichotomy itself should be clarified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of the paper's significance. We respond point by point to the major comments below.

read point-by-point responses

Referee: [Definition of the Hamiltonian matrix and §3 (construction of the graph subspaces)] The domain of the Hamiltonian operator matrix for unbounded B and C is not specified with sufficient precision. It is therefore unclear whether the constructed graph subspaces lie inside this domain and whether the resulting operators satisfy the Riccati equation in the strong (or mild) sense required when the control and observation operators are unbounded. This verification is load-bearing for the central claim that the graph subspaces yield solutions of the Riccati equation.

Authors: We agree that greater precision is needed. In the revised manuscript we will explicitly define the domain of the Hamiltonian operator matrix (incorporating the graph norms induced by the unbounded B and C) and prove that both constructed graph subspaces are contained in this domain. We will further verify invariance and show that the resulting operators satisfy the Riccati equation in the strong sense appropriate for unbounded control and observation operators. This directly strengthens the central construction. revision: yes
Referee: [Introduction and the statement of the main theorems] All subsequent results (nonnegativity, boundedness, and exponential stability) rest on the dichotomy assumption being given a priori. No general, checkable conditions on the triple (A,B,C) that guarantee the required spectral splitting are supplied; the manuscript therefore leaves open the question of when the hypothesis can be verified for concrete systems.

Authors: The manuscript deliberately takes the Hamiltonian dichotomy as a hypothesis; its contribution is the derivation of the Riccati solutions, nonnegativity, and (under compact resolvent) boundedness/stability from that assumption. General, verifiable conditions on (A,B,C) guaranteeing the spectral splitting are not supplied because they lie outside the paper's scope and typically depend on the concrete structure of the system. We will add a brief remark in the introduction acknowledging this as an important open direction, but the theorems themselves require no alteration. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation from explicit dichotomy hypothesis

full rationale

The paper takes the dichotomy property of the Hamiltonian operator matrix (spectral splitting into stable/unstable parts with no imaginary-axis spectrum) as an explicit input hypothesis and constructs the invariant graph subspaces yielding the nonnegative and nonpositive Riccati solutions directly from it. Boundedness of the nonnegative solution and exponential stability of the feedback system are then derived as consequences when the generator has compact resolvent. No equations or definitions reduce the claimed results to fitted quantities, self-referential constructions, or load-bearing self-citations; the argument is a standard implication from the stated spectral assumption and is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the dichotomy property of the Hamiltonian and on standard functional-analytic background for unbounded operators; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption The Hamiltonian operator matrix possesses a dichotomy property.
Invoked to guarantee the existence of the two invariant graph subspaces that solve the Riccati equation.

pith-pipeline@v0.9.0 · 5588 in / 1212 out tokens · 34889 ms · 2026-05-24T22:03:05.509591+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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H. Langer, A. C. M. Ran, B. A. van de Rotten. Invariant subspaces of inﬁ- nite dimensional Hamiltonians and solutions of the corresp onding Riccati equations. In Linear Operators and Matrices , volume 130 of Oper. Theory Adv. Appl., pages 235–254. Birkh¨ auser, Basel, 2002

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M. R. Opmeer, R. F. Curtain. New Riccati equations for well-posed linear systems. Systems Control Lett., 52(5) (2004), 339–347

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A. J. Pritchard, D. Salamon. The linear quadratic control problem for inﬁnite-dimensional systems with unbounded input and outp ut operators. SIAM J. Control Optim., 25(1) (1987), 121–144

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C. Tretter, C. Wyss. Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations . J. Evol. Equ., 14(1) (2014), 121–153

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Tucsnak, G

M. Tucsnak, G. Weiss. Observation and control for operator semigroups . Birkh¨ auser Advanced Texts. Birkh¨ auser Verlag, Basel, 2009

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Winklmeier, C

M. Winklmeier, C. Wyss. On the Spectral Decomposition of Dichotomous and Bisectorial Operators . Integral Equations Operator Theory, 82(1) (2015), 119–150

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C. Wyss. Hamiltonians with Riesz bases of generalised eigenvectors and Riccati equations. Indiana Univ. Math. J., 60 (2011), 1723–1766

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C. Wyss, B. Jacob, H. J. Zwart. Hamiltonians and Riccati equations for linear systems with unbounded control and observation oper ators. SIAM J. Control Optim., 50 (2012), 1518–1547. 40

work page 2012

[1] [1]

H. Bart, I. Gohberg, M. A. Kaashoek. Minimal factorization of matrix and operator functions , volume 1 of Operator Theory: Advances and Ap- plications. Birkh¨ auser Verlag, Basel, 1979

work page 1979

[2] [2]

J. M. Berezanski ˘ ı. Expansions in eigenfunctions of selfadjoint opera- tors. Translated from the Russian by R. Bolstein, J. M. Danskin, J . Rovnyak and L. Shulman. Translations of Mathematical Monog raphs, Vol

work page

[3] [3]

American Mathematical Society, Providence, R.I., 1968

work page 1968

[4] [4]

Bittanti, A

S. Bittanti, A. J. Laub, J. C. Willems, editors. The Riccati equation . Communications and Control Engineering Series. Springer- Verlag, Berlin, 1991

work page 1991

[5] [5]

Bub´ ak, C

P. Bub´ ak, C. V. M. van der Mee, A. C. M. Ran. Approximation of solutions of Riccati equations . SIAM J. Control Optim., 44(4) (2005), 1419–1435

work page 2005

[6] [6]

F. M. Callier, L. Dumortier, J. Winkin. On the nonnegative self-adjoint solutions of the operator Riccati equation for inﬁnite-dim ensional systems . Integral Equations Operator Theory, 22(2) (1995), 162–195

work page 1995

[7] [7]

R. F. Curtain, H. J. Zwart. An Introduction to Inﬁnite Dimensional Linear Systems Theory. Springer, New York, 1995

work page 1995

[8] [8]

Engel, R

K.-J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000

work page 2000

[9] [9]

M. Haase. The functional calculus for sectorial operators , volume 169 of Operator Theory: Advances and Applications. Birkh¨ auser V erlag, Basel, 2006

work page 2006

[10] [10]

Jonas, H

P. Jonas, H. Langer. Some questions in the perturbation theory of J- nonnegative operators in Kre ˘ ın spaces. Math. Nachr., 114 (1983), 205– 226

work page 1983

[11] [11]

Jonas, C

P. Jonas, C. Trunk. On a class of analytic operator functions and their linearizations. Math. Nachr., 243 (2002), 92–133. 39

work page 2002

[12] [12]

S. G. Kre ˘ ın.Linear Diﬀerential Equations in Banach Space . Amer. Math. Soc., Providence, 1971

work page 1971

[13] [13]

C. R. Kuiper, H. J. Zwart. Connections between the algebraic Riccati equation and the Hamiltonian for Riesz-spectral systems . J. Math. Systems Estim. Control, 6(4) (1996), 1–48

work page 1996

[14] [14]

Lancaster, L

P. Lancaster, L. Rodman. Algebraic Riccati Equations. Oxford University Press, Oxford, 1995

work page 1995

[15] [15]

Langer, A

H. Langer, A. C. M. Ran, B. A. van de Rotten. Invariant subspaces of inﬁ- nite dimensional Hamiltonians and solutions of the corresp onding Riccati equations. In Linear Operators and Matrices , volume 130 of Oper. Theory Adv. Appl., pages 235–254. Birkh¨ auser, Basel, 2002

work page 2002

[16] [16]

M. R. Opmeer, R. F. Curtain. New Riccati equations for well-posed linear systems. Systems Control Lett., 52(5) (2004), 339–347

work page 2004

[17] [17]

A. J. Pritchard, D. Salamon. The linear quadratic control problem for inﬁnite-dimensional systems with unbounded input and outp ut operators. SIAM J. Control Optim., 25(1) (1987), 121–144

work page 1987

[18] [18]

O. Staﬀans. Well-posed linear systems , volume 103 of Encyclopedia of Mathematics and its Applications. Cambridge University Pr ess, Cam- bridge, 2005

work page 2005

[19] [19]

Tretter, C

C. Tretter, C. Wyss. Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations . J. Evol. Equ., 14(1) (2014), 121–153

work page 2014

[20] [20]

Tucsnak, G

M. Tucsnak, G. Weiss. Observation and control for operator semigroups . Birkh¨ auser Advanced Texts. Birkh¨ auser Verlag, Basel, 2009

work page 2009

[21] [21]

Weiss, G

M. Weiss, G. Weiss. Optimal control of stable weakly regular linear systems . Math. Control Signals Systems, 10(4) (1997), 287–330

work page 1997

[22] [22]

Winklmeier, C

M. Winklmeier, C. Wyss. On the Spectral Decomposition of Dichotomous and Bisectorial Operators . Integral Equations Operator Theory, 82(1) (2015), 119–150

work page 2015

[23] [23]

C. Wyss. Hamiltonians with Riesz bases of generalised eigenvectors and Riccati equations. Indiana Univ. Math. J., 60 (2011), 1723–1766

work page 2011

[24] [24]

C. Wyss, B. Jacob, H. J. Zwart. Hamiltonians and Riccati equations for linear systems with unbounded control and observation oper ators. SIAM J. Control Optim., 50 (2012), 1518–1547. 40

work page 2012