Using dynamic extensions for the backstepping control of hyperbolic systems

Abdurrahman Irscheid; Joachim Deutscher; Nicole Gehring

REVIEW 3 major objections 5 minor 25 references

Using dynamic extensions for the backstepping control of hyperbolic systems

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A dynamic extension brings heterodirectional hyperbolic systems into a form where a static backstepping feedback assigns arbitrary target dynamics and complete input-output decoupling.

desk verdict A solid, genuinely new backstepping design that uses dynamic extensions to homogenize transport velocities; the decoupling section is terser than it should be, but the apparent gap closes on inspection. read the letter →

arxiv 2411.18965 v1 pith:JTATDCJR submitted 2024-11-28 eess.SY cs.SY

Nicole Gehring , Joachim Deutscher , Abdurrahman Irscheid This is my paper · ORCID

classification eess.SYcs.SY MSC 93C2093B5293D15

keywords dynamicextensionbacksteppingheterodirectionalhyperbolicsystemsboundarycontrolinput-outputdecouplingtransportvelocityhomogenizationPDE-ODEstatefeedbackdesign

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

General heterodirectional hyperbolic systems—transport PDEs whose components travel in opposite directions with different speeds—are usually controlled by static state feedback, which forces the target closed-loop system to contain specific local terms and makes complete input-output decoupling impossible. This paper claims that by first adding a dynamic extension, i.e., a controller that is itself a set of transport equations, one can homogenize all transport velocities so that every component travels with the slowest speed. On the extended system, a static backstepping feedback of the extended state assigns a freely parametrized closed-loop dynamics with arbitrary in-domain couplings, and a decoupling feedback makes each output component track its own delayed reference. If the kernel equations used in the design are well posed, the result is a systematic method for dynamic state feedback design, including for coupled PDE-ODE systems, without losing minimum-time stabilization.

What carries the argument

The load-bearing object is the dynamic extension, a controller dynamics of transport equations with speeds chosen so that all components of the extended state share one left-going speed $\lambda^-_{n^-}$ and one right-going speed $\lambda^+_{n^+}$ on $[0,1]$. This homogenization makes the kernel equations for the backstepping transformation scalar-like: each block of the kernel $L(z,\zeta)$ has only one characteristic family, so boundary values at $\zeta=z$ and $\zeta=0$ determine the solution. The Volterra integral transformation (35), with matrix gain $M(z)$ and kernel $L(z,\zeta)$, then carries the extended system into the target system (34) with freely chosen matrices $B_0,B_1,B(z),\bar{B}(z)$. The kernel equations (38) with boundary conditions determined by $M$ and $\bar{A}_0$ are the equations whose solvability the whole construction rests on.

What would settle it

Take a heterodirectional system with $n^-=n^+=2$, spatially varying speeds, $\mathop{\mathrm{rank}} Q_0=2$, and non-vanishing $\bar{A}^-_0(z)$; solve the reduced kernel equations (38a), (38b), (38d) with $\bar{B}(z)=0$ and check for a piecewise continuous $L^{--}$ and $L^{-+}$. If no such solution exists, the proposed decoupling transformation (42) is not well defined and the complete decoupling claim fails for that system.

Watch

Extended reading notes

Core claim

The central claim is that dynamic extensions lift the well-known limitation of static backstepping for heterodirectional hyperbolic systems. A preliminary Volterra transformation removes in-domain coupling, and the dynamic extension is chosen so that all left-going components of the extended state propagate with the slowest left-going speed and all right-going components with the slowest right-going speed on the same unit interval. In the resulting dynamically extended system, a Volterra integral transformation with a matrix gain $M(z)$ and kernel $L(z,\zeta)$ maps the extended system into a target system whose in-domain coupling matrix $\bar{B}(z)$, boundary matrices $B_0,B_1$, and distributed boundary kernel $B(\zeta)$ can be chosen freely. This is what enables assignment of general closed-loop dynamics and, as a special case, complete input-output decoupling with $y(t)=x^-(0,t)$ tracking $\bar{v}(t-\varphi^-_{n^-}(1))$ while the internal dynamics remains input-to-state stable. The same modular construction is claimed to transfer to PDE-ODE systems by combining the extension with an existing preliminary transformation. The paper's own condition is that the relevant kernel equations admit a solution; for the general target-system assignment this is proven under $\mathop{\mathrm{rank}} Q_0=n^-$, while the decoupling special case is asserted conditionally.

Load-bearing premise

The decoupling controller exists only if the reduced kernel equations that remove the local coupling in the left-going part have a solution; the paper assumes this without presenting a well-posedness proof for that special case.

Editorial extensions

If this is right

Static state feedback of the original state cannot assign target systems with arbitrary in-domain couplings; with the dynamic extension, these couplings become free design parameters.
Complete input-output decoupling of $y(t)=x^-(0,t)$ with respect to a new input $\bar{v}(t)$ is achieved, with closed-loop behavior $y(t)=\bar{v}(t-\varphi^-_{n^-}(1))$.
Minimum-time convergence is preserved: the homogenizing extension delays control action but does not increase the lower bound $T_{\min}=\varphi^-_{n^-}(1)+\varphi^+_{n^+}(1)$.
The same design applies to hyperbolic PDE-ODE systems, so the ODE at the unactuated boundary can be stabilized while the PDE part enjoys the same decoupling or target-system freedom.

Reading between the lines

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

This suggests that dynamic extensions could be used to shape not only transport velocities but also the characteristic boundary maps, for instance to enforce passivity or prescribed relative-degree properties.
A natural testable extension is to compute the decoupling kernel explicitly for general $n^-=n^+=2$ systems with spatially varying speeds; the paper demonstrates the construction on one concrete example rather than proving solvability for all such systems.
The decoupling construction resembles a PDE analogue of the Byrnes-Isidori normal form; making that analogy precise could yield a systematic zero-dynamics assignment procedure for hyperbolic systems.
One could investigate whether the homogenization idea extends to systems with non-strictly ordered or coincident velocities, where the characteristic decomposition changes qualitatively.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look. The genuinely new piece is the dynamic extension that homogenizes all transport velocities of a heterodirectional hyperbolic system, and then uses a static feedback of the extended state to assign arbitrary in-domain couplings and to do input-output decoupling. That is more than a repackaging of Redaud et al.: it turns their constant-velocity result into a systematic construction for spatially varying velocities, and it makes the decoupling design look natural. The derivation in Section 3 is careful and the example is genuine, not a cartoon.

The stability theorem rests on standard backstepping plus the kernel well-posedness cited from Hu et al., so that part is solid. The decoupling section is the soft spot. The paper states that if L^-- and L^-+ solve (38a), (38b), (38d) with \bar B=0, then (42) gives the decoupled form (43). It does not prove that this reduced kernel system is well-posed, and a first pass makes you worry about an extra \bar A^+_0 contribution coming through the unchanged \chi^+ dynamics. That worry does not survive: with \bar B=0, the homogeneous PDE for L^-+ together with the diagonal boundary condition L^-+(z,z)=0 forces L^-+=0 on the whole triangle. So the extra term vanishes and the reduced kernel equations reduce to a simple well-posed problem for L^-- with boundary data at \zeta=0. The authors should have said this in one sentence; the omission is minor but it is the only place where the reader has to fill in a real gap.

Remark 7 on minimum-time preservation is asserted without proof. The claim is plausible and in fact true by the bounded invertibility of the Volterra transformation plus the zero target system, but again a sentence or two would settle it. Section 4.3 on PDE-ODE systems is more of a sketch than a theorem; fine as an outlook, but do not treat it as a proved transfer.

Who gets value: people working in backstepping boundary control of hyperbolic PDEs, especially those interested in target-system flexibility and decoupling. The paper deserves a serious referee. It is a conditional accept, not a reject: fix the two missing justifications, tighten the PDE-ODE sketch, and it will be a useful contribution to Automatica or equivalent.

Referee Report

3 major / 5 minor

Summary. The paper proposes a dynamic-extension approach for backstepping boundary control of general heterodirectional hyperbolic PDE systems. A preliminary Volterra transformation removes the in-domain coupling, and a transport-type dynamic extension then homogenizes the transport velocities so that all negative-direction components share one velocity and all positive-direction components share another. The resulting extended system (28) is used to design a static feedback of the extended state, which assigns a target system with arbitrary in-domain couplings (Section 4.1) and, in Section 4.2, to achieve complete input-output decoupling. A sketch for hyperbolic PDE-ODE systems is given in Section 4.3, and a numerical example illustrates the decoupling design.

Significance. If correct, the paper makes a useful methodological contribution: it transfers the finite-dimensional concept of dynamic extensions to the backstepping framework for hyperbolic PDEs and shows that this enlarges the achievable closed-loop dynamics beyond static state feedback. The construction of the dynamic extension is explicit and modular, and the stability argument in Section 4.1 follows standard backstepping reasoning with kernel well-posedness cited from [11]. The paper also gives a concrete numerical demonstration of the decoupling idea. However, the central decoupling result in Section 4.2 rests on an unproved well-posedness claim for a reduced kernel system and on an omitted cancellation that is needed for the claimed decoupled form (43); these issues must be fixed before the main contribution can be fully accepted. The PDE-ODE transfer in Section 4.3 is only sketched rather than proved.

major comments (3)

[Section 4.2, Eqs. (42)-(43)] The decoupling construction is incomplete as written. Substituting the transformation (42) into the positive-component equation of (28a) introduces the term \int_0^z L^{-+}(z,\zeta)\bar A^+_0(\zeta)\,d\zeta\,\bar\chi^-(0,t) into the equation for \bar\chi^-, because \chi^+ obeys the PDE with source \bar A^+_0(z)\chi^-(0,t). This term is absent from the claimed target (43a) unless L^{-+}\equiv0. The paper does not prove that the reduced kernel equations (38a),(38b),(38d) with \bar B(z)=0 force L^{-+}=0, nor that the remaining reduced problem for L^{--} is well-posed. Lemma 4 addresses the full system (38) with the artificial boundary condition (40), not the reduced subset used here. The claim that (42) maps (28) into (43) therefore needs a dedicated lemma establishing L^{-+}=0 and the well-posedness of the reduced kernel equations; alternatively, if L^{-+} is not zero, the target dynamics must be revised.
[Section 4.3] The abstract states that the modularity of the design allows a straightforward transfer of all results to hyperbolic PDE-ODE systems, but Section 4.3 provides no theorem, no kernel equations for the combined plant-ODE problem, and no proof of stability or decoupling for the PDE-ODE case. The text only cites [7] and states that the extension is 'mostly straightforward.' If the PDE-ODE transfer is intended as a contribution, precise statements and proofs should be given; otherwise the section should be explicitly labeled as a sketch or outlook.
[Section 4.1, Lemma 4] The proof of Lemma 4 is only a citation to [11] together with a short sketch. Since the homogenized system has repeated velocities inside each block of \bar\Lambda, the characteristic structure degenerates relative to the distinct-velocity case in [11], so the well-posedness of (38) should be verified explicitly for this repeated-velocity setting, including the role of the artificial boundary condition (40). The sketch in Figure 2 is plausible, but the paper should state the precise regularity and uniqueness result it claims.

minor comments (5)

[Section 4.1, Lemma 4 and footnote 1] Footnote 1 says that the homogenized kernel equations are well-posed without additional artificial boundary conditions, while Lemma 4 explicitly introduces the artificial boundary condition (40). This apparent contradiction should be resolved by clarifying exactly which boundary conditions are needed.
[Remark 7] The claim that the dynamic feedback does not increase the minimum control time T_min is stated without proof. While the target system (34) with zero design matrices is clearly finite-time stable in time T_min, the statement that the original state x(z,t) vanishes for t>T_min requires tracking the effects of the preliminary transformation (5) and the dynamic extension; please add a proof or a precise reference.
[Theorem 6] The notion 'asymptotically stable pointwise in space' is used in the statement of Theorem 6 but is not defined, and no constructive conditions on B0, B1, B(z), and \bar B(z) are given beyond the trivial zero choice. Please define the stability notion and give at least one nontrivial class of target systems that satisfy the assumption.
[Section 3.3, Eq. (16)] In the definition of \bar a_0^-(z) in (16), the expression a_0^-(\lambda_1^-/\lambda_2^-\,z,t) contains a time argument t even though the left-hand side depends only on z. This appears to be a typo and should be corrected.
[Section 4.2] The phrase 'input-to-state stable' is used for the internal dynamics (45) without a definition adapted to the PDE setting. Please provide a definition or a reference to the relevant notion for infinite-dimensional systems.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is a direct backstepping design with external kernel-solvability support; the identified decoupling existence gap is a correctness risk, not a circular argument.

full rationale

The paper's derivation chain is constructive rather than circular. The dynamic extension is built from explicit spatial rescaling and transport-equation computations (Sections 3.2–3.4), and the feedback laws are obtained by solving backstepping kernel equations. There are no fitted parameters being relabeled as predictions, no output quantity is inserted into the model by definition, and the central closed-loop and decoupling claims are not assumed in the inputs of the derivation. The solvability of the full kernel equations (38) is attributed to the external result [11], not to the authors' own work, and the transformation (35) is explicitly 'inspired by [10, Rem. 6]', again an independent source. The paper does cite the authors' own [7] in the PDE-ODE section, but only to import an already established preliminary transformation; that citation is not load-bearing for the main dynamic-extension idea. The only substantive concern is that Section 4.2 conditions the decoupling result on existence of a solution to the reduced kernel equations (38a), (38b), (38d) with \bar B(z)=0 without proving that reduced existence claim; Lemma 4 covers the full system, not this special subset. That is a missing-support / well-posedness gap in an otherwise derived claim, not a circular step, because the decoupled dynamics is not presupposed by the kernel equations. Accordingly, no circularity is exhibited with the specificity required by the analysis rules.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard backstepping kernel well-posedness results (cited from [11]) and on the assumption that the homogenized kernel equations are solvable. The design matrices and artificial boundary conditions are free design degrees of freedom, not fitted parameters. No new physical entities are posited; the dynamic extension is a controller state.

free parameters (2)

target system matrices B0, B1, B(z), \bar B(z) in (34) = freely assignable design choices
These matrices define the desired closed-loop dynamics in the target system (34). They are chosen by the designer, not fitted to data, and the central claim is that arbitrary choices are achievable.
artificial kernel boundary condition R(z) in (40) = arbitrary piecewise continuous function
Used in Lemma 4 to ensure uniqueness of the kernel solution for (38). The choice does not affect the achievable target system but is a free parameter of the controller construction.

assumptions (4)

domain assumption Preliminary backstepping transformation (5) with kernel K solving (9) exists uniquely in piecewise C(T) for the heterodirectional system (2).
Invoked in Section 3.1 to map (2) into (7). The existence is cited from [11] and is standard in backstepping for hyperbolic PDEs.
domain assumption Kernel equations (38) for the homogenized system admit a unique piecewise continuous solution when rank Q0 = n- and an artificial boundary condition (40) is imposed.
Lemma 4 states this, citing [11]. The proof sketch uses the method of characteristics and successive approximations, but for the full equations only. The reduced decoupling kernel case is not covered.
standard math The initial value problem (37) for M(z) has a unique solution with det M(z) nonzero on [0,1].
Follows from standard ODE theory for linear systems (Kailath [15]); stated in Section 4.1.
domain assumption The target system (34) is asymptotically stable pointwise in space for the chosen design matrices.
Assumed in Theorem 6. The designer is free to select stable target dynamics; the theorem states stability of the closed loop under this assumption.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Using dynamic extensions for the backstepping control of hyperbolic systems." pith.science (2026). https://pith.science/paper/JTATDCJR

@misc{pith2026241118965,
  author       = {Pith},
  title        = {Pith review of: Using dynamic extensions for the backstepping control of hyperbolic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JTATDCJR}},
  note         = {Machine review of arXiv:2411.18965}
}

read the original abstract

This paper systematically introduces dynamic extensions for the boundary control of general heterodirectional hyperbolic PDE systems. These extensions, which are well known in the finite-dimensional setting, constitute the dynamics of state feedback controllers. They make it possible to achieve design goals beyond what can be accomplished by a static state feedback. The design of dynamic state feedback controllers is divided into first introducing an appropriate dynamic extension and then determining a static feedback of the extended state, which includes the system and controller state, to meet some design objective. In the paper, the dynamic extensions are chosen such that all transport velocities are homogenized on the unit spatial interval. Based on the dynamically extended system, a backstepping transformation allows to easily find a static state feedback that assigns a general dynamics to the closed-loop system, with arbitrary in-domain couplings. This new design flexibility is also used to determine a feedback that achieves complete input-output decoupling in the closed loop with ensured internal stability. It is shown that the modularity of this dynamic feedback design allows for a straightforward transfer of all results to hyperbolic PDE-ODE systems. An example demonstrates the new input-output decoupling approach by dynamic extension.

Figures

Figures reproduced from arXiv: 2411.18965 by the authors.

**Figure 2.** Typical characteristics visualize the kernel equat [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Norms ||χ(t)|| and ||w¯(t)|| of the dynamically extended system state and the controller state in closed loop [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Evolution of y(t), ¯v(t) and u(t). system and controller along the characteristic curves and applying the Euler method for numerical integration. For the new input, ¯v1(t) = (|2 sin(2πt)| − 1)h(t − 1.5) with the Heaviside function h(t) and ¯v2(t) = 0 are chosen in ord…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[11]

L. Hu, R. Vazquez, F. Di Meglio, and M. Krstic. Boundary exponential stabilization of 1-D inhomogeneous quasiline ar hyperbolic systems. SIAM J. Contr. Optim. , 57:963–998, 2019

work page 2019
[7]

Deutscher, N

J. Deutscher, N. Gehring, and R. Kern. Output feedback control of general linear heterodirectional hyperbolic PD E- ODE systems with spatially-varying coeﬃcients. Int. J. Contr., 92(10):2274–2290, 2019. 10

work page 2019
[1]

Auriol and F

J. Auriol and F. Di Meglio. Minimum time control of heterodirectional linear coupled hyperbolic PDEs. Automatica, 71:300–307, 2016

work page 2016
[2]

Charlet, J

B. Charlet, J. L´ evine, and R. Marino. Suﬃcient conditio ns for dynamic state feedback linearization. SIAM J. Contr. Optim., 29:38–57, 1991

work page 1991
[3]

Coron, L

J.-M. Coron, L. Hu, and G. Olive. Finite-time boundary stabilization of general linear hyperbolic balance laws vi a Fredholm backstepping transformation. Automatica, 84:95– 100, 2017

work page 2017
[4]

Coron, R

J.-M. Coron, R. Vazquez, M. Krstic, and G. Bastin. Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J. Contr. Optim. , 51(3):2005–2035, 2013

work page 2005
[5]

Curtain and H

R. Curtain and H. Zwart. Introduction to Inﬁnite- Dimensional Systems Theory: A State-Space Approach . Springer New York, NY, 2020

work page 2020
[6]

R.F. Curtain. Decoupling in inﬁnite dimensions. Syst. Contr. Lett., 5:249–254, 1985

work page 1985

Show all 25 references

[8]

Di Meglio, F

F. Di Meglio, F. Bribiesca Argomedo, L. Hu, and M. Krstic. Stabilization of coupled linear heterodirectional hyperb olic PDE-ODE systems. Automatica, 87:281–289, 2018

2018
[9]

Gehring, A

N. Gehring, A. Irscheid, J. Deutscher, F. W oittennek, an d J. Rudolph. Control of distributed-parameter systems usin g normal forms: an introduction. at - Automatisierungstechnik, 71(3):624–645, 2023

2023
[10]

L. Hu, F. Di Meglio, and R. Vazquez. Control of homodirectional and general heterodirectional linear cou pled hyperbolic PDEs. IEEE Trans. Automat. Contr. , 61:3301– 3313, 2016

2016
[12]

Ilchmann, T

A. Ilchmann, T. Selig, and C. Trunk. The Byrnes-Isidori form for inﬁnite-dimensional systems. SIAM J. Contr. Optim. , 54:1504–1534, 2016

2016
[13]

A. Isidori. Nonlinear Control Systems . Springer London, 1995

1995
[14]

Jacob, K.A

B. Jacob, K.A. Morris, and H. Zwart. Zero dynamics for networks of waves. Automatica, 103:310–321, 2019

2019
[15]

T. Kailath. Linear Systems. Prentice-Hall Englewood Cliﬀs, NJ, 1980

1980
[16]

Kn¨ uppel, F

T. Kn¨ uppel, F. W oittennek, I. Boussaada, H. Mounier, and S.-I. Niculescu. Flatness-based control for a non-line ar spatially distributed model of a drilling system. In A. Seur et, H. ¨Ozbay, C. Bonnet, and H. Mounier, editors, Low- Complexity Controllers for Time-Delay Syst...

2014
[17]

Krstic and A

M. Krstic and A. Smyshlyaev. Boundary Control of PDEs: A Course on Backstepping Designs . SIAM Philadelphia, 2008

2008
[18]

L´ evine

J. L´ evine. Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Springer Berlin, Heidelberg, 2009

2009
[19]

Morse and W.M

A.S. Morse and W.M. W onham. Decoupling and pole assignment by dynamic compensation. SIAM J. Contr. , 8:317–337, 1970

1970
[20]

Otsuka, H

N. Otsuka, H. Inaba, and T. Oide. Decoupling by state feedback in inﬁnite-dimensional systems. IMA J. Math. Contr. Inform. , 7:125–141, 1990

1990
[21]

Redaud, J

J. Redaud, J. Auriol, and Y. Le Gorrec. In-domain dampin g assignment of a Timoshenko-beam using state feedback boundary control. In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 5405–5410, 2022

2022
[22]

Redaud, J

J. Redaud, J. Auriol, and Y. Le Gorrec. In domain dissipa tion assignment of boundary controlled Port-Hamiltonian syste ms using backstepping. Syst. Contr. Lett. , 185:105722, 2024

2024
[23]

Reger and L

J. Reger and L. Triska. Dynamic extensions for exact backstepping control of systems in pure feedback form. In 2019 IEEE 58th Conference on Decision and Control (CDC) , pages 480–486, 2019

2019
[24]

Vazquez, M

R. Vazquez, M. Krstic, and J.-M. Coron. Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system. In 2011 IEEE 50th Conference on Decision and Control and European Control Conference (CDC-ECC), pages 4937–4942, 2011

2011
[25]

W oittennek

F. W oittennek. Flatness based feedback design for hyperbolic distributed parameter systems with spatially varying coeﬃcients. IF AC Proceedings Volumes, 46(26):37– 42, 2013. 1st IF AC W orkshop on Control of Systems Governed by Partial Diﬀerential Equations (CPDE). 11

2013

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [11]

L. Hu, R. Vazquez, F. Di Meglio, and M. Krstic. Boundary exponential stabilization of 1-D inhomogeneous quasiline ar hyperbolic systems. SIAM J. Contr. Optim. , 57:963–998, 2019

work page 2019

[2] [7]

Deutscher, N

J. Deutscher, N. Gehring, and R. Kern. Output feedback control of general linear heterodirectional hyperbolic PD E- ODE systems with spatially-varying coeﬃcients. Int. J. Contr., 92(10):2274–2290, 2019. 10

work page 2019

[3] [1]

Auriol and F

J. Auriol and F. Di Meglio. Minimum time control of heterodirectional linear coupled hyperbolic PDEs. Automatica, 71:300–307, 2016

work page 2016

[4] [2]

Charlet, J

B. Charlet, J. L´ evine, and R. Marino. Suﬃcient conditio ns for dynamic state feedback linearization. SIAM J. Contr. Optim., 29:38–57, 1991

work page 1991

[5] [3]

Coron, L

J.-M. Coron, L. Hu, and G. Olive. Finite-time boundary stabilization of general linear hyperbolic balance laws vi a Fredholm backstepping transformation. Automatica, 84:95– 100, 2017

work page 2017

[6] [4]

Coron, R

J.-M. Coron, R. Vazquez, M. Krstic, and G. Bastin. Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J. Contr. Optim. , 51(3):2005–2035, 2013

work page 2005

[7] [5]

Curtain and H

R. Curtain and H. Zwart. Introduction to Inﬁnite- Dimensional Systems Theory: A State-Space Approach . Springer New York, NY, 2020

work page 2020

[8] [6]

R.F. Curtain. Decoupling in inﬁnite dimensions. Syst. Contr. Lett., 5:249–254, 1985

work page 1985