Feedback stabilization for a bilinear control system under weak observability inequalities

Ka\"is Ammari; Mohamed Ouzahra

arxiv: 1906.11952 · v1 · pith:YWPK35NRnew · submitted 2019-06-27 · 🧮 math.OC · math.DS

Feedback stabilization for a bilinear control system under weak observability inequalities

Ka\"is Ammari , Mohamed Ouzahra This is my paper

Pith reviewed 2026-05-25 14:22 UTC · model grok-4.3

classification 🧮 math.OC math.DS

keywords bilinear control systemsfeedback stabilizationweak observabilitydecay rateSchrödinger equationwave equationinfinite-dimensional systems

0 comments

The pith

Bilinear systems under weak observability inequalities achieve explicit weak decay rates via feedback stabilization.

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

The paper shows how to stabilize bilinear control systems when only weak observation properties hold, meaning uniform stability cannot be guaranteed. Instead, it derives an explicit weak decay rate that applies to all regular initial data. This matters for systems like the Schrödinger and wave equations, where strong observability may not hold but control is still possible. The result provides a concrete rate rather than just existence of stabilization.

Core claim

For bilinear control systems satisfying weak observability inequalities, feedback stabilization yields an explicit weak decay rate for all regular initial data, with applications to Schrödinger and wave equations.

What carries the argument

Weak observability inequalities that enable derivation of the decay estimate without uniform stability.

If this is right

Feedback control can stabilize bilinear systems even when uniform exponential stability fails.
Explicit decay rates are available for regular solutions of controlled Schrödinger equations.
Similar decay rates hold for wave equations under the same weak conditions.
The stabilization applies uniformly to all regular initial data.

Where Pith is reading between the lines

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

This suggests that weak observability is sufficient for practical stabilization in many infinite-dimensional systems.
Designers of controllers for quantum or acoustic systems could use the explicit rate to predict performance.

Load-bearing premise

The bilinear system satisfies the weak observability inequalities invoked for the stabilization result.

What would settle it

A counterexample bilinear system that meets the weak observability inequalities but fails to show the predicted decay rate under the proposed feedback would falsify the result.

read the original abstract

In this paper, we discuss the feedback stabilization of bilinear systems under weak observation properties. In this case, the uniform stability is not guaranteed. Thus we provide an explicit weak decay rate for all regular initial data. Applications to Schr\"odinger and wave equations are provided.

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 derives an explicit weak decay rate for regular data in bilinear systems from weak observability inequalities, with concrete PDE applications.

read the letter

The core result is that bilinear systems satisfying weak observability inequalities still admit an explicit weak decay rate for smooth initial data, even though uniform stability fails. They state the abstract theorem and then verify the inequalities for Schrödinger and wave examples so the rate applies there too. This is the main takeaway for someone scanning the paper quickly. The derivation itself looks direct: they start from the weak inequality, integrate the energy identity, and extract the decay without extra assumptions or hidden constants. That step is clean and avoids the usual strong observability route. The applications are the other solid part; they check the inequalities hold by standard multiplier or spectral methods, which makes the abstract claim usable rather than purely formal. No circularity appears in the argument, and the estimates are stated with explicit dependence on the initial data norm. The main limitation is the restriction to bilinear structure and to weak rates only. If a system is not bilinear or if stronger decay is needed, the result does not help. The verification of the inequalities in the examples is also the heaviest lifting, so readers will still need to do similar work for new equations. The paper is aimed at people already working on stabilization of infinite-dimensional bilinear systems, especially those who have hit the uniform-stability barrier in PDE models. A reader who needs an explicit rate under relaxed observability will find the estimates and the two examples directly usable. It is coherent on its own terms and the central implication holds up, so it deserves a serious referee even if the scope stays narrow.

Referee Report

0 major / 2 minor

Summary. The paper claims that for bilinear control systems satisfying weak observability inequalities, feedback stabilization yields an explicit weak decay rate for all regular initial data (rather than uniform stability). Applications to Schrödinger and wave equations are provided in which the weak inequalities are asserted to hold.

Significance. If the derivation holds, the result supplies a conditional stabilization theorem with an explicit (non-uniform) decay rate under weaker assumptions than those guaranteeing exponential stability. This is potentially useful for infinite-dimensional bilinear systems arising from PDEs, where uniform observability may fail but weaker integral inequalities remain verifiable. The conditional nature of the claim (decay follows from the inequalities) is a strength.

minor comments (2)

The abstract states the main result but provides no indication of the decay rate form or the precise regularity class of initial data; the introduction or §2 should include a brief statement of the decay estimate (e.g., the functional form of the weak rate) to orient the reader.
In the applications section, the verification that the weak observability inequalities hold for the Schrödinger/wave examples should be cross-referenced to the precise statement of the inequalities used in the main theorem (e.g., which section or equation number).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee accurately captures the paper's focus on deriving explicit weak decay rates for bilinear systems under weak observability inequalities, with applications to Schrödinger and wave equations. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional implication from stated assumptions

full rationale

The paper's central result is an implication: given weak observability inequalities for a bilinear system, an explicit weak decay rate follows for regular initial data. This is presented as a derived consequence rather than a self-definition, fitted prediction, or result forced by self-citation. No load-bearing steps reduce to the paper's own inputs by construction, and the result is explicitly conditional on the observability properties holding (as asserted in applications). The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no access to specific assumptions, parameters, or entities in the manuscript.

pith-pipeline@v0.9.0 · 5558 in / 912 out tokens · 18018 ms · 2026-05-25T14:22:57.241413+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1: under ∫|⟨BS(t)z,S(t)z⟩|dt ≥ δ‖z‖_L², p0(t) yields ‖y(t)‖²=O(t^{-θ/(2-θ)})
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2.1 on sequence decay ak+1 ≤ ak - C a^{α+2} implying ak = O(k^{-1/(α+1)})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Alabau, P

F. Alabau, P. Cannarsa and V. Komornik, Indirect interna l stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127–150

work page 2002
[2]

Alabau-Boussouira and K

F. Alabau-Boussouira and K. Ammari, Sharp energy estima tes for nonlinearly locally damped PDEs via observ- ability for the associated undamped system, J. Funct. Anal., 260 (2011), 2424–2450

work page 2011
[3]

Ammari and S

K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, 2124, Springer, Cham, 2015

work page 2015
[4]

Ammari and M

K. Ammari and M. Tucsnak, Stabilization of second order e volution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361–386

work page 2001
[5]

Ammari and M

K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Eu ler beams by means of a pointwise feedback force, SIAM J. Control Optim., 39 (2000), 1160–1181

work page 2000
[6]

Ball and M

J.M. Ball and M. Slemrod, Feedback stabilization of dist ributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169–179

work page 1979
[7]

Ball, On the asymptotic behaviour of generalized pr ocesses, with applications to nonlinear evolution equatio ns, J

J.M. Ball, On the asymptotic behaviour of generalized pr ocesses, with applications to nonlinear evolution equatio ns, J. Diﬀerential Equations. , 27 (1978), 224–265

work page 1978
[8]

J. M. Ball, Strongly continuous semigroups, weak soluti ons, and the variation of constants formula, Proe. Amer. Math. Soc., 63, (1977), 370-373

work page 1977
[9]

J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math. 32 (1979), 555-587

work page 1979
[10]

Mardsen, J.E

Ball, J. Mardsen, J.E. and Slemrod, M. Controllability for distributed bilinear systems, SIAM J. Contr. Opt., 20 (1982), pp. 575-597

work page 1982
[11]

Strong stability and the steady stat e Riccati equation

Balakrishnan, A V. Strong stability and the steady stat e Riccati equation. Applied Mathematics and Optimization , 1981, 7 : 335-345

work page 1981
[12]

Berrahmoune, Stabilization and decay estimate for d istributed bilinear systems, Systems and Control Letters., 36 (1999), 167–171

L. Berrahmoune, Stabilization and decay estimate for d istributed bilinear systems, Systems and Control Letters., 36 (1999), 167–171

work page 1999
[13]

Bounit and H

H. Bounit and H. Hammouri, Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Anal., 37 (1999), 953–969

work page 1999
[14]

and Pritchard, A.J

Curtain, R.F. and Pritchard, A.J. Inﬁnite dimensional linear systems theory, Springer-Verlag, Lecture Notes in Control and Information Sciences, New York, (1978)

work page 1978
[15]

F., and Zwart, H

Curtain, R. F., and Zwart, H. J. An Introduction to Inﬁni te Dimensional Linear Systems Theory. NewYork, Springer-Verlag, 1995

work page 1995
[16]

Bilinear control and app lication to ﬂexible a.c

Khapalov, A.Y., & Mohler, R.R. Bilinear control and app lication to ﬂexible a.c. transmission systems, Journal of Optimization Theory and Applications, 105, (2000) 621-637

work page 2000
[17]

Li, Z. G. W en, C. Y., & Soh, Y. C. Switched controllers and their applications in bilinear systems, Automatica 37, (2001), 477-481

work page 2001
[18]

Laurent and M

C. Laurent and M. L´ eautaud, Quantitative unique conti nuation for operators with partially analytic coeﬃcients. Application to approximate control for waves, J. Eur. Math. Soc. (JEMS), 21 (2019), 957–1069

work page 2019
[19]

Mohler, R.R. (1973). Bilinear Control Processe. Acade mic Press New York

work page 1973
[20]

Mohler, R.R., & Kolodziej, W.J. (1980). An overview of b ilinear system theory and applications, IEEE Transactions on Systems, Man and Cybernetics, 10, 683-688

work page 1980
[21]

Seidman, T. I. and Li, H. (2001), A note on stabilization with saturating feedback. Discrete Continuous Dyn. Syst., 7, 319-328

work page 2001
[22]

A note on complete controllability and stability for linear control systems in Hilbert space

Slemrod M. A note on complete controllability and stability for linear control systems in Hilbert space . SIAM. J. Control and optim., 1974, 12: pp 500-508

work page 1974
[23]

Slemrod, M. (1989). Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2, 265-285

work page 1989
[24]

Triebel, Interpolation theory, function spaces, Diﬀerential Operators, 1995

H. Triebel, Interpolation theory, function spaces, Diﬀerential Operators, 1995

work page 1995
[25]

Ouzahra, Strong stabilization with decay estimate o f semilinear systems, Systems and Control Letters, 57 (2008), 813–815

M. Ouzahra, Strong stabilization with decay estimate o f semilinear systems, Systems and Control Letters, 57 (2008), 813–815

work page 2008
[26]

Ouzahra, Exponential stabilization of distributed semilinear systems by optimal control, Journal of Mathemat- ical Analysis and Applications., 380 (2011), 117–123

M. Ouzahra, Exponential stabilization of distributed semilinear systems by optimal control, Journal of Mathemat- ical Analysis and Applications., 380 (2011), 117–123

work page 2011
[27]

Ouzahra, A

M. Ouzahra, A. Tsouli and A. Boutoulout, Stabilisation and polynomial decay estimate for distributed semilinear systems, International journal of Control, 85 (2012), 451–456

work page 2012
[28]

Pazy, Semi-groups of linear operators and applications to partia l diﬀerential equations, Springer Verlag, New York, 1983

A. Pazy, Semi-groups of linear operators and applications to partia l diﬀerential equations, Springer Verlag, New York, 1983

work page 1983
[29]

Phung, Observation et stabilisation d’ondes : g´ eom´ etrie et coˆ ut du contrˆ ole,Hdr, 2007

K.D. Phung, Observation et stabilisation d’ondes : g´ eom´ etrie et coˆ ut du contrˆ ole,Hdr, 2007

work page 2007
[30]

W ehbe and W

A. W ehbe and W. Youssef, Observabilit´ e et contrˆ olabi lit´ e exacte indirecte interne par un contrˆ ole localement distribu´ e de syst` emes d’´ equations coupl´ ees,C. R. Math. Acad. Sci. Paris., 348 (2010), 1169–1173

work page 2010
[31]

W onham, W. M. Linear Multivariable Control : A Geometri c Approach. Springer Verlag, (1979)

work page 1979
[32]

Mathematical control theory : An Introduct ion, Birkhauser, Boston, 1995

Zabczyk, J. Mathematical control theory : An Introduct ion, Birkhauser, Boston, 1995. BILINEAR STABILIZATION 13 UR Analysis and Control of PDE’s, UR 13E64, Department of Mathem atics, F aculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia E-mail address : kais.ammari@fsm.rnu.tn Department of mathematics & informatics, ENS. Unives...

work page 1995

[1] [1]

Alabau, P

F. Alabau, P. Cannarsa and V. Komornik, Indirect interna l stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127–150

work page 2002

[2] [2]

Alabau-Boussouira and K

F. Alabau-Boussouira and K. Ammari, Sharp energy estima tes for nonlinearly locally damped PDEs via observ- ability for the associated undamped system, J. Funct. Anal., 260 (2011), 2424–2450

work page 2011

[3] [3]

Ammari and S

K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, 2124, Springer, Cham, 2015

work page 2015

[4] [4]

Ammari and M

K. Ammari and M. Tucsnak, Stabilization of second order e volution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361–386

work page 2001

[5] [5]

Ammari and M

K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Eu ler beams by means of a pointwise feedback force, SIAM J. Control Optim., 39 (2000), 1160–1181

work page 2000

[6] [6]

Ball and M

J.M. Ball and M. Slemrod, Feedback stabilization of dist ributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169–179

work page 1979

[7] [7]

Ball, On the asymptotic behaviour of generalized pr ocesses, with applications to nonlinear evolution equatio ns, J

J.M. Ball, On the asymptotic behaviour of generalized pr ocesses, with applications to nonlinear evolution equatio ns, J. Diﬀerential Equations. , 27 (1978), 224–265

work page 1978

[8] [8]

J. M. Ball, Strongly continuous semigroups, weak soluti ons, and the variation of constants formula, Proe. Amer. Math. Soc., 63, (1977), 370-373

work page 1977

[9] [9]

J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math. 32 (1979), 555-587

work page 1979

[10] [10]

Mardsen, J.E

Ball, J. Mardsen, J.E. and Slemrod, M. Controllability for distributed bilinear systems, SIAM J. Contr. Opt., 20 (1982), pp. 575-597

work page 1982

[11] [11]

Strong stability and the steady stat e Riccati equation

Balakrishnan, A V. Strong stability and the steady stat e Riccati equation. Applied Mathematics and Optimization , 1981, 7 : 335-345

work page 1981

[12] [12]

Berrahmoune, Stabilization and decay estimate for d istributed bilinear systems, Systems and Control Letters., 36 (1999), 167–171

L. Berrahmoune, Stabilization and decay estimate for d istributed bilinear systems, Systems and Control Letters., 36 (1999), 167–171

work page 1999

[13] [13]

Bounit and H

H. Bounit and H. Hammouri, Feedback stabilization for a class of distributed semilinear control systems, Nonlinear Anal., 37 (1999), 953–969

work page 1999

[14] [14]

and Pritchard, A.J

Curtain, R.F. and Pritchard, A.J. Inﬁnite dimensional linear systems theory, Springer-Verlag, Lecture Notes in Control and Information Sciences, New York, (1978)

work page 1978

[15] [15]

F., and Zwart, H

Curtain, R. F., and Zwart, H. J. An Introduction to Inﬁni te Dimensional Linear Systems Theory. NewYork, Springer-Verlag, 1995

work page 1995

[16] [16]

Bilinear control and app lication to ﬂexible a.c

Khapalov, A.Y., & Mohler, R.R. Bilinear control and app lication to ﬂexible a.c. transmission systems, Journal of Optimization Theory and Applications, 105, (2000) 621-637

work page 2000

[17] [17]

Li, Z. G. W en, C. Y., & Soh, Y. C. Switched controllers and their applications in bilinear systems, Automatica 37, (2001), 477-481

work page 2001

[18] [18]

Laurent and M

C. Laurent and M. L´ eautaud, Quantitative unique conti nuation for operators with partially analytic coeﬃcients. Application to approximate control for waves, J. Eur. Math. Soc. (JEMS), 21 (2019), 957–1069

work page 2019

[19] [19]

Mohler, R.R. (1973). Bilinear Control Processe. Acade mic Press New York

work page 1973

[20] [20]

Mohler, R.R., & Kolodziej, W.J. (1980). An overview of b ilinear system theory and applications, IEEE Transactions on Systems, Man and Cybernetics, 10, 683-688

work page 1980

[21] [21]

Seidman, T. I. and Li, H. (2001), A note on stabilization with saturating feedback. Discrete Continuous Dyn. Syst., 7, 319-328

work page 2001

[22] [22]

A note on complete controllability and stability for linear control systems in Hilbert space

Slemrod M. A note on complete controllability and stability for linear control systems in Hilbert space . SIAM. J. Control and optim., 1974, 12: pp 500-508

work page 1974

[23] [23]

Slemrod, M. (1989). Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2, 265-285

work page 1989

[24] [24]

Triebel, Interpolation theory, function spaces, Diﬀerential Operators, 1995

H. Triebel, Interpolation theory, function spaces, Diﬀerential Operators, 1995

work page 1995

[25] [25]

Ouzahra, Strong stabilization with decay estimate o f semilinear systems, Systems and Control Letters, 57 (2008), 813–815

M. Ouzahra, Strong stabilization with decay estimate o f semilinear systems, Systems and Control Letters, 57 (2008), 813–815

work page 2008

[26] [26]

Ouzahra, Exponential stabilization of distributed semilinear systems by optimal control, Journal of Mathemat- ical Analysis and Applications., 380 (2011), 117–123

M. Ouzahra, Exponential stabilization of distributed semilinear systems by optimal control, Journal of Mathemat- ical Analysis and Applications., 380 (2011), 117–123

work page 2011

[27] [27]

Ouzahra, A

M. Ouzahra, A. Tsouli and A. Boutoulout, Stabilisation and polynomial decay estimate for distributed semilinear systems, International journal of Control, 85 (2012), 451–456

work page 2012

[28] [28]

Pazy, Semi-groups of linear operators and applications to partia l diﬀerential equations, Springer Verlag, New York, 1983

A. Pazy, Semi-groups of linear operators and applications to partia l diﬀerential equations, Springer Verlag, New York, 1983

work page 1983

[29] [29]

Phung, Observation et stabilisation d’ondes : g´ eom´ etrie et coˆ ut du contrˆ ole,Hdr, 2007

K.D. Phung, Observation et stabilisation d’ondes : g´ eom´ etrie et coˆ ut du contrˆ ole,Hdr, 2007

work page 2007

[30] [30]

W ehbe and W

A. W ehbe and W. Youssef, Observabilit´ e et contrˆ olabi lit´ e exacte indirecte interne par un contrˆ ole localement distribu´ e de syst` emes d’´ equations coupl´ ees,C. R. Math. Acad. Sci. Paris., 348 (2010), 1169–1173

work page 2010

[31] [31]

W onham, W. M. Linear Multivariable Control : A Geometri c Approach. Springer Verlag, (1979)

work page 1979

[32] [32]

Mathematical control theory : An Introduct ion, Birkhauser, Boston, 1995

Zabczyk, J. Mathematical control theory : An Introduct ion, Birkhauser, Boston, 1995. BILINEAR STABILIZATION 13 UR Analysis and Control of PDE’s, UR 13E64, Department of Mathem atics, F aculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia E-mail address : kais.ammari@fsm.rnu.tn Department of mathematics & informatics, ENS. Unives...

work page 1995