Input-to-state stability for parabolic boundary control: Linear and semi-linear systems

Felix Schwenninger

REVIEW 3 major objections 4 minor 50 references

Input-to-state stability for parabolic boundary control: Linear and semi-linear systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read For parabolic boundary control systems, input-to-state stability is governed by the fractional regularity of the boundary input operator; the paper proves the semilinear version under a dissipativity condition.

desk verdict A useful ISS survey whose central semilinear theorem is not supported by its own headline example: cubic f fails the linear-growth hypothesis. read the letter →

arxiv 1908.08317 v1 pith:SQB66URV submitted 2019-08-22 math.OC math.APmath.FA

Felix Schwenninger This is my paper · ORCID

classification math.OCmath.APmath.FA MSC 93C2035K5847D0693D2093C05

keywords input-to-statestabilityboundarycontrolparabolicequationssemilinearsystemsanalyticsemigroupsinterpolationspacesadmissibilityLyapunovestimates

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 establishes a general route to input-to-state stability (ISS) for parabolic equations controlled through the boundary: the state at time $t$ is bounded by an exponentially decaying term in the initial state plus a term in the $L^q$ norm of the input. For linear systems, the decisive quantity is how much the boundary input operator smooths data when measured in the interpolation spaces $X_\alpha$; if it maps into $X_{-1+\alpha}$, then $L^q$-ISS holds for every $q > 1/\alpha$. The paper's main new result extends this to semilinear equations: when the linear part is self-adjoint and bounded above, the boundary operator maps into $X_{-1/2}$, and the nonlinearity obeys a dissipativity condition, the system has a unique global mild solution (the integrated form of the PDE) and is $L^q$-ISS for every $q \ge 2$. This matters because it converts the question of whether a controlled parabolic PDE is stable under external inputs into a checkable trace-regularity computation, and it upgrades earlier one-dimensional results to higher-dimensional domains and general elliptic operators.

What carries the argument

The machinery is the abstraction of a boundary control system as an abstract Cauchy problem in an extrapolation space. Writing $A$ for the semigroup generator obtained by restricting the differential operator to the kernel of the boundary operator $B$, and $B_0$ for a right inverse of $B$, every classical solution is represented by the variation-of-constants formula $x(t)=T(t)x_0+\int_0^t T_{-1}(t-s)\mathcal{B}u(s)\,ds$ with $\mathcal{B}=AB_0-A_{-1}B_0\in\mathcal{L}(U,X_{-1})$; this is the Fattorini trick, a standard rewrite that turns boundary control into interior forcing in a larger space. The condition $\mathcal{B}\in\mathcal{L}(U,X_{-1+\alpha})$ controls the singularity of the analytic-semigroup kernel $T_{-1}(t-s)\mathcal{B}$, which behaves like $(t-s)^{-1+\alpha}$ and is integrable when $q>1/\alpha$. For the semilinear theorem the second ingredient is a Lyapunov energy estimate: along classical solutions, $\frac12\frac{d}{dt}\|x\|^2 \le ((1-m_1-\varepsilon)\omega_A+m_2)\|x\|^2 + C\|u\|^2$, and Gronwall's inequality turns this into $L^2$-ISS, with Hölder's inequality extending it to every $q\ge 2$. The space $X_{-1/2}$ is the interpolation space one half-step less regular than $X$; it is the regularity level reached by Neumann or Robin boundary traces.

What would settle it

A concrete check is the Dirichlet-controlled heat equation on a $C^2$ domain with $X=L^2(\Omega)$ and $U=L^2(\partial\Omega)$: the paper's trace calculation gives $\mathcal{B}\in\mathcal{L}(U,X_{-3/4})$, not $\mathcal{L}(U,X_{-1/2})$, because the Neumann trace is unbounded from $H^1(\Omega)$ to $L^2(\partial\Omega)$. If one could construct $x_0=0$ and inputs $u_n$ with $\|u_n\|_{L^2(0,t;U)}\le 1$ for which the semilinear Dirichlet problem with $f(x)=-x^3$ has unbounded $L^2(\Omega)$-norms at a fixed time, the $q\ge 2$ conclusion would be false exactly where the $X_{-1/2}$ assumption is absent; the classical non-$L^2$-admissibility example for the linear Dirichlet problem indicates such a construction exists.

Watch

Extended reading notes

Core claim

The central claim is that for parabolic boundary control systems, $L^q$-input-to-state stability is governed by the fractional regularity of the boundary input operator, and that this same regularity condition survives semilinear perturbations. In the linear case, if $A$ generates an exponentially stable analytic semigroup and the associated input operator $\mathcal{B}$ lies in $\mathcal{L}(U,X_{-1+\alpha})$, the mild solution satisfies an $L^q$-ISS estimate for every $q>1/\alpha$. The main semilinear theorem states that if $A$ is self-adjoint and $\langle Ax,x\rangle\le \omega_A\|x\|^2$, if $\mathcal{B}\in\mathcal{L}(U,X_{-1/2})$, and if $f$ is locally Hölder continuous in time, Lipschitz in the state, of at most linear growth in the $X_{1/2}$-norm, and satisfies $\langle f(t,x),x\rangle\le -m_1\langle Ax,x\rangle + m_2\|x\|^2$ with $(1-m_1)\omega_A+m_2<0$, then for every $x_0\in X_{1/2}$ and every $u\in W^{2,1}(\mathbb{R}_+;U)$ with $A_{-1}x_0+\mathcal{B}u(0)\in X$ there is a unique global mild solution, classical on $(0,\infty)$, and the estimate $\|x(t)\|_X\le C_1 e^{-\omega t}\|x_0\|_X + C_2\|u\|_{L^q(0,t;U)}$ holds for all $q\ge 2$.

Load-bearing premise

The semilinear theorem's conclusion for every $q\ge 2$ rests on the regularity condition $\mathcal{B}\in\mathcal{L}(U,X_{-1/2})$: the boundary input operator must map into the interpolation space half a derivative less regular than the state space $X$. For Dirichlet control on $L^2(\Omega)$ this condition fails, and then the energy estimate (3.9) no longer yields the ISS bound.

Editorial extensions

If this is right

For Neumann or Robin boundary control of the heat equation on a $C^2$ domain, trace theorems place the input operator in $\mathcal{L}(U,X_{-1/2})$; the linear system is therefore $L^q$-ISS for every $q>4/3$, and the semilinear theorem applies to reactions such as $f(x)=-x^3$ in dimensions up to three.
For Dirichlet boundary control with state space $X=L^2(\Omega)$, the same trace calculation gives only $\mathcal{L}(U,X_{-3/4})$; the linear threshold degrades to $q>4$, and the semilinear theorem's $q\ge 2$ conclusion is not available, matching the known failure of $L^2$-ISS.
If the state is measured in $H^{-1}(\Omega)$ instead of $L^2(\Omega)$, Dirichlet control becomes $L^2$-ISS, because the same boundary operator is smoother relative to the weaker state norm.
For linear systems the paper's reformulation shows that $L^q$-ISS is equivalent to exponential stability plus the classical admissibility condition that inputs of finite $L^q$ norm produce bounded states; hence existing admissibility results become ISS results directly.
Boundary input systems that fail $L^q$-ISS for every finite $q$ can still be $L^\infty$-ISS with an Orlicz-type estimate, so failure of finite-$q$ stability does not mean instability in every input norm.

Reading between the lines

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

A natural extension, not pursued in the paper, is a semilinear analogue of the linear regularity ladder: allow $\mathcal{B}\in\mathcal{L}(U,X_{-1+\alpha})$ with $\alpha\in(0,1/2)$ and expect $L^q$-ISS for $q>1/\alpha$ rather than only $q\ge 2$; the energy estimate in the proof would need an interpolation step replacing the $X_{-1/2}$ bound.
The $H^{-1}$ workaround for Dirichlet control points to a systematic trade-off: spatial regularity of the state norm can be exchanged for temporal integrability of the input norm. One could interpolate between $L^2$ and $H^{-1}$ state spaces to get a family of ISS exponents for the same boundary system.
For semilinear Dirichlet boundary control the paper leaves the $L^\infty$ case partially open; the cited monotonicity and maximum-principle methods suggest that the right tool there is not a Lyapunov estimate but comparison arguments, so the two approaches may cover complementary classes of nonlinearities.

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

Here's my read. The linear half of this paper is the better half: the Fattorini-trick recap, Proposition 2.13 on Lq-ISS for analytic semigroups, and the equivalence of solution concepts in Proposition 2.11 are clean and will save people time. The L∞-ISS result for finite-rank boundary operators is a fair repackaging of prior work. The semilinear part is where I'd push back.

Theorem 3.3 is a routine Lipschitz-perturbation result and fine. Theorem 3.4 is the genuine new item: it generalizes Zheng–Zhu to n-dimensional domains and general self-adjoint A, and the Lyapunov estimate (3.9) is standard. But the statement runs ahead of the proof. It claims Lq-ISS for all q≥2, while the argument is carried out only for u∈W^{2,1} with a compatibility condition; Remark 3.5 acknowledges this and says a density argument should fill the gap, but no such argument appears.

The bigger problem is Example 3.6. The paper says conditions (1) and (2) of Theorem 3.4 follow from W^{1,2}⊂L^6 for the cubic nonlinearity f(x)=−x^3. That is not true. Condition (2) requires ‖f(t,x)‖_X ≤ k(t)(1+‖x‖_{X_{1/2}}); for f(x)=−x^3 on L^2, the left side grows like ‖x‖_{H^1}^3, not linearly. The Sobolev embedding gives exactly the cubic bound, not the linear one. So the headline example doesn't satisfy the theorem's hypothesis. The theorem may still be true for sublinear f, but the paper's advertised generalization to polynomial dissipative nonlinearities is not established.

Minor: Example 2.14's threshold q>4/3 is fine but would be easier to check if the chosen right-inverse B0 were specified.

The citation pattern looks honest; self-citations are to published work and not inflated. The survey is genuinely useful for someone entering ISS for parabolic boundary control. But as refereed work, Theorem 3.4 and Example 3.6 need to be reconciled. I'd send it to a referee, not desk reject, because the linear parts are solid and the semilinear theorem is plausible. The author should either replace the example with one that satisfies (2) or relax condition (2) to allow polynomial growth with a matching proof.

If I were a journal editor, I'd request major revision before acceptance.

Referee Report

3 major / 4 minor

Summary. This paper surveys input-to-state stability (ISS) for parabolic boundary control systems and adds new results for semilinear systems. The linear part recalls the boundary-control-system formalism via the Fattorini trick, mild solutions, and the equivalence of solution concepts; Proposition 2.13 gives a sufficient condition for Lq-ISS using interpolation spaces, recovering the sharp ranges q>4/3 for Neumann and q>4 for Dirichlet heat control. Theorem 2.18 gives L-infinity-ISS with an Orlicz-type estimate for finite-rank boundary operators. The semilinear part contains Theorem 3.3 for globally Lipschitz nonlinearities and Theorem 3.4 for dissipative nonlinearities with linear growth in X_{1/2}, proved by a Lyapunov energy estimate; Example 3.6 claims to apply this to the cubic heat equation on domains of dimension n<=3.

Significance. The survey portions are a useful, clearly written synthesis of known operator-theoretic facts and are likely to be a helpful reference. Proposition 2.13 is simple and effective, and the examples showing sharp Lq thresholds for Neumann and Dirichlet control are instructive. The main new semilinear result, Theorem 3.4, is a plausible abstract extension of the one-dimensional results in [48] to self-adjoint operators in Hilbert space, under linear growth and dissipativity assumptions. However, as discussed below, the paper's headline application to the cubic heat equation is not covered by the theorem as stated, and the Lq-ISS conclusion is only proved for a restricted class of smooth inputs. These issues materially affect the claimed contribution, though they appear fixable within the scope of a revision.

major comments (3)

[Section 3, Example 3.6] The assertion that condition (2) of Theorem 3.4 follows from the Sobolev embedding W^{1,2}(Omega) subset L^6(Omega) is incorrect. For X=L^2(Omega), X_{1/2}=H^1(Omega), and f(x)=-x^3, take x identical to a constant c>0. Then ||f(x)||_{L^2}=c^3|Omega|^{1/2} while 1+||x||_{H^1}=1+c|Omega|^{1/2}, so the quotient is unbounded as c tends to infinity. The Sobolev embedding only yields ||x^3||_{L^2}=||x||_{L^6}^3 <= C||x||_{H^1}^3, which is cubic, not linear, growth in the X_{1/2}-norm. Hence Example 3.6 is not an instance of Theorem 3.4, and the claimed extension to cubic dissipative nonlinearities is not proved.
[Theorem 3.4 and Remark 3.5] The theorem claims Lq-ISS for any q>=2, but the proof establishes the estimate only for inputs u in W^{2,1}(R+;U) with the compatibility condition A_{-1}x0+Bu(0) in X. No density or continuity argument is provided to extend the result to arbitrary u in Lq_loc(R+;U), and Remark 3.5 explicitly defers such an extension. As stated, the theorem therefore overclaims: it proves an ISS estimate for a class of smooth inputs, not Lq-ISS for the dynamical system with Lq input functions. The statement should be restricted accordingly, or the missing extension should be proved.
[Example 3.6] The displayed PDE in Example 3.6 contains an in-domain disturbance d(xi,t), but the abstract semilinear system (A,B,f) in Theorem 3.4 has no distributed input term. If d is meant to be absorbed into f(t,x), then condition (2) of Theorem 3.4 will in general fail because ||d(t)||_{L^2} is not controlled by 1+||x||_{H^1}. Thus the example either does not match the theorem's framework or introduces an additional input not covered by the theorem.

minor comments (4)

[Theorem 3.4] In the final display of the theorem, the quantifier is over 'R+ x X_alpha x W^{2,1}(R+;U)', but alpha is not defined in the statement; it should be X_{1/2}.
[Theorem 3.4, condition (1)] The local Holder/Lipschitz condition is written as holding 'for all (s,t) in the ball B_rho(t,x)', but the second point should be a different space variable, for example (s,y); the current phrasing is a typo.
[Example 2.14] The sentence 'we obtain the ISS estimates for any q > 3/4 and q_tilde >= 1' is confusing in light of the preceding condition q>4/3; the roles of q and q_tilde for the boundary and distributed inputs should be clarified.
[Theorem 2.18] The dissipativity assumption should state explicitly that Re<Ax,x>_{new} <= 0 holds for all x in D(A), not only implicitly for the operator A as a whole.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.4's ISS estimate is derived from an explicit Lyapunov/Gronwall argument, and the linear-survey material rests on published, non-fitted results.

full rationale

The central semilinear result, Theorem 3.4, is not circular. Its proof constructs local solutions by adapting Pazy's fixed-point argument to include the boundary-input term, then derives the ISS estimate from the energy identity, the dissipativity condition (3) on f, and Gronwall's inequality, leading to inequality (3.9). No parameter is fitted to the target ISS estimate, and no hypothesis is a restatement of the conclusion. The linear results in Section 2 are either proved directly, as in Proposition 2.13, or cited from prior published work; the self-citations [19,21] support survey material and a quoted L∞-ISS theorem, not the new semilinear generalization, so they are not load-bearing in a circular sense. The paper also explicitly flags the technical restriction to W^{2,1} inputs and the A_{-1}x0+Bu(0)∈X compatibility condition in Remark 3.5, which shows that these assumptions are acknowledged regularity conditions rather than hidden encodings of the result. The only substantive concern is in Example 3.6: the text asserts that condition (2) of Theorem 3.4 follows from the Sobolev embedding W^{1,2}(Ω)⊂L^6(Ω), but for f(x)=-x^3 the L^2-norm of f(x) grows cubically in the H^1-norm, whereas condition (2) requires linear growth in ||x||_{1/2}. This is an internal correctness or consistency issue, not a circularity, because the theorem itself does not depend on that example and the derivation of the ISS estimate does not require the example's verification to be valid.

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

The paper introduces no new physical entities, no fitted parameters, and no ad hoc mathematical constants. It relies on standard semigroup theory, interpolation spaces, and trace theorems, plus domain assumptions on the generator and boundary operator that are stated explicitly in Theorem 3.4.

assumptions (5)

domain assumption A = A|ker B generates an exponentially stable analytic semigroup (Proposition 2.13, Theorem 3.4).
This structural assumption on the uncontrolled linear system is needed for smoothing estimates and for the interpolation space setup.
domain assumption B in L(U, X_{-1/2}) (Theorem 3.4 condition (ii)).
This regularity of the boundary input operator enables the energy estimate (3.9); it fails for Dirichlet control on L^2(Omega).
domain assumption The nonlinearity f satisfies the sign condition <f(t,x),x> <= -m1<Ax,x> + m2||x||^2 with (1-m1)omega_A + m2 < 0 (Theorem 3.4 conditions (3)-(4)).
Guarantees global existence and the decay estimate; it is the nonlinear analog of a sector condition.
standard math Pazy's local and global existence theorems for semilinear parabolic equations [38, Thm 6.3.1 and 6.3.3] extend to the controlled case (Theorem 3.4 proof).
The author adapts these classical results by adding the boundary input term; the extension is plausible but only sketched.
standard math Sobolev trace and interpolation space properties, e.g. gamma0 in L(H^{1/2+epsilon}, L^2(dOmega)) (Examples 2.14 and 2.16).
Used to verify the abstract regularity conditions on the boundary operator B.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Input-to-state stability for parabolic boundary control: Linear and semi-linear systems." pith.science (2026). https://pith.science/paper/SQB66URV

@misc{pith2026190808317,
  author       = {Pith},
  title        = {Pith review of: Input-to-state stability for parabolic boundary control: Linear and semi-linear systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQB66URV}},
  note         = {Machine review of arXiv:1908.08317}
}

abstract

Input-to-state stability (ISS) for systems described by partial differential equations has seen intensified research activity recently, and in particular the class of boundary control systems, for which truly infinite-dimensional effects enter the situation. This note reviews input-to-state stability for parabolic equations with respect to general $L^{p}$-input-norms in the linear case and includes extensions of recent results on semilinear equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 46 canonical work pages

[48]

Zheng and G

J. Zheng and G. Zhu. Input-to-state stability with resp ect to boundary disturbances for a class of semi-linear parabolic equations. Automatica J. IF AC, 97:271–277, 2018

work page 2018
[1]

R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathe- matics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 200 3. 25

work page
[2]

A. A. Agrachev, A. S. Morse, E. D. Sontag, H. J. Sussmann, a nd V. I. Utkin. Nonlinear and optimal control theory , volume 1932 of Lecture Notes in Mathematics . Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures giv en at the C.I.M.E. Summer School held in Cetraro, June 19–29, 2004, Edited by Paolo Nistri and Gianna Stefani

work page 1932
[3]

Arendt, C

W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. Vector-valued Laplace transforms and Cauchy problems , volume 96 of Monographs in Mathematics . Birkh¨ auser/Springer Basel AG, Basel, second edition, 2011

2011
[4]

Baillon and P

J.-B. Baillon and P. Cl´ ement. Examples of unbounded ima ginary powers of operators. J. Funct. Anal., 100(2):419–434, 1991

work page 1991
[5]

Bribiesca Argomedo, E

F. Bribiesca Argomedo, E. Witrant, and C. Prieur. D1-inp ut-to-state stability of a time- varying nonhomogeneous diﬀusive equation subject to bound ary disturbances. In 2012 Amer- ican Control Conference (ACC) , pages 2978–2983, June 2012

work page 2012
[6]

C. I. Byrnes, D. S. Gilliam, V. I. Shubov, and G. W eiss. Reg ular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Systems , 8(3):341–370, 2002

work page 2002
[7]

R. F. Curtain and H. Zwart. An introduction to inﬁnite-dimensional linear systems the ory, volume 21 of Texts in Applied Mathematics . Springer-Verlag, New York, 1995

work page 1995

Show all 50 references

[8]

Dashkovskiy and A

S. Dashkovskiy and A. Mironchenko. On the uniform input- to-state stability of reaction- diﬀusion systems. In Proceedings of the 49th IEEE Conference on Decision and Cont rol (CDC), 2010 , pages 6547–6552. IEEE, 2010

2010
[9]

Dashkovskiy and A

S. Dashkovskiy and A. Mironchenko. Input-to-state stab ility of inﬁnite-dimensional control systems. Mathematics of Control, Signals, and Systems , 25(1):1–35, Aug. 2013

2013
[10]

Engel and R

K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations , volume 194 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2000

2000
[11]

Fabre, J.-P

C. Fabre, J.-P. Puel, and E. Zuazua. Approximate contro llability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A , 125(1):31–61, 1995

1995
[12]

G. Greiner. Perturbing the boundary conditions of a gen erator. Houston J. Math. , 13(2):213– 229, 1987

1987
[13]

Guiver, H

C. Guiver, H. Logemann, and M. R. Opmeer. Inﬁnite-dimen sional Lur’e systems: input-to- state stability and convergence properties. SIAM J. Control Optim. , 57(1):334–365, 2019

2019
[14]

B. H. Haak and P. C. Kunstmann. Admissibility of unbound ed operators and wellposedness of linear systems in Banach spaces. Integral Equations Operator Theory, 55(4):497–533, 2006

2006
[15]

M. Haase. The Functional Calculus for Sectorial Operators , volume 169 of Operator Theory: Advances and Applications . Birkh¨ auser Verlag, Basel, 2006

2006
[16]

D. Henry. Geometric theory of semilinear parabolic equations , volume 840 of Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1981

1981
[17]

Jacob, A

B. Jacob, A. Mironchenko, J. R. Partington, and F. Wirth . Remarks on input-to-state stabil- ity and non-coercive Lyapunov functions. In Proc. of the 57th IEEE Conference on Decision and Control, Miami Beach, USA , pages 4803–4808, 2018

2018
[18]

Jacob, R

B. Jacob, R. Nabiullin, J. Partington, and F. Schwennin ger. On input-to-state-stability and integral input-to-state-stability for parabolic boundar y control systems. In 2016 IEEE 55th Conference on Decision and Control (CDC) , pages 2265–2269, Dec 2016

2016
[19]

Jacob, R

B. Jacob, R. Nabiullin, J. R. Partington, and F. Schwenn inger. Inﬁnite-dimensional input- to-state stability and Orlicz spaces. SIAM J. Control Optim. , 56(2):868–889, 2018

2018
[20]

Jacob, J

B. Jacob, J. R. Partington, and S. Pott. Applications of Laplace-Carleson embeddings to admissibility and controllability. SIAM J. Control Optim. , 52(2):1299–1313, 2014

2014
[21]

Jacob, F

B. Jacob, F. L. Schwenninger, and H. Zwart. On continuit y of solutions for parabolic control systems and input-to-state stability. J. Diﬀerential Equations , 266(10):6284–6306, 2019

2019
[22]

Jayawardhana, H

B. Jayawardhana, H. Logemann, and E. Ryan. Inﬁnite-dim ensional feedback systems: the circle criterion and input-to-state stability. Commun. Inf. Syst. , 8(4):413–414, 2008

2008
[23]

Karafyllis and Z.-P

I. Karafyllis and Z.-P. Jiang. Stability and stabilization of nonlinear systems . Communications and Control Engineering Series. Springer-Verlag London, L td., London, 2011. With a foreword by E. Sontag

2011
[24]

Karafyllis and M

I. Karafyllis and M. Krstic. ISS with respect to boundar y disturbances for 1-D parabolic PDEs. IEEE Trans. Automat. Control , 61:3712–3724, 2016

2016
[25]

Karafyllis and M

I. Karafyllis and M. Krstic. ISS in diﬀerent norms for 1- D parabolic PDEs with boundary disturbances. SIAM Journal on Control and Optimization , 55:1716–1751, 2017

2017
[26]

Karafyllis and M

I. Karafyllis and M. Krstic. Input-to-state stability for PDEs . Communications and Control Engineering Series. Springer, Cham, 2019

2019
[27]

P. C. Kunstmann and L. W eis. Maximal Lp-regularity for parabolic equations, Fourier mul- tiplier theorems and H ∞ -functional calculus. In Functional analytic methods for evolution equations, volume 1855 of Lecture Notes in Math. , pages 65–311. Springer, Berlin, 2004. 26 F.L...

2004
[28]

Lasiecka and R

I. Lasiecka and R. Triggiani. Control theory for partial diﬀerential equations: continu ous and approximation theories. I , volume 74 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2000. Abstract par abolic systems

2000
[29]

Lhachemi, D

H. Lhachemi, D. Saussi´ e, G. Zhu, and R. Shorten. Input- to-State Stability of a Clamped- Free Damped String in the Presence of Distributed and Bounda ry Disturbances. Available at arXiv: 1807.11696, 2019

2019 arXiv
[30]

Lhachemi and R

H. Lhachemi and R. Shorten. ISS Property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems. Automatica J. IF AC, 109:108504, 2019

2019
[31]

J.-L. Lions. Contrˆ ole optimal de syst` emes gouvern´ es par des ´ equations aux d´ eriv´ ees partielles. Avant propos de P. Lelong. Dunod, Paris; Gauthier-Villars, Paris, 1968

1968
[32]

Mironchenko and H

A. Mironchenko and H. Ito. Construction of Lyapunov Fun ctions for Interconnected Parabolic Systems: An iISS Approach. SIAM Journal on Control and Optimization , 53(6):3364–3382, 2015

2015
[33]

Mironchenko, I

A. Mironchenko, I. Karafyllis, and M. Krstic. Monotoni city methods for input-to-state sta- bility of nonlinear parabolic PDEs with boundary disturban ces. SIAM J. Control Optim. , 57(1):510–532, 2019

2019
[34]

Mironchenko and F

A. Mironchenko and F. Wirth. Restatements of input-to- state stability in inﬁnite dimensions: what goes wrong? In Proc. of the 22th International Symposium on Mathematical T heory of Networks and Systems , pages 667–674, 2016

2016
[35]

Mironchenko and F

A. Mironchenko and F. Wirth. Characterizations of inpu t-to-state stability for inﬁnite- dimensional systems. IEEE Trans. Automat. Control , 63(6):1602–1617, 2018

2018
[36]

Mironchenko and F

A. Mironchenko and F. Wirth. Lyapunov characterizatio n of input-to-state stability for semi- linear control systems over Banach spaces. Systems Control Lett. , 119:64–70, 2018

2018
[37]

Nabiullin and F

R. Nabiullin and F. L. Schwenninger. Strong input-to-s tate stability for inﬁnite-dimensional linear systems. Math. Control Signals Systems , 30(1):Art. 4, 15, 2018

2018
[38]

A. Pazy. Semigroups of linear operators and applications to partial diﬀerential equations , volume 44 of Applied Mathematical Sciences . Springer-Verlag, New York, 1983

1983
[39]

Prieur and F

C. Prieur and F. Mazenc. ISS-Lyapunov functions for tim e-varying hyperbolic systems of balance laws. Math. Control Signals Systems , 24(1-2):111–134, 2012

2012
[40]

D. Salamon. Inﬁnite-dimensional linear systems with u nbounded control and observation: a functional analytic approach. Trans. Amer. Math. Soc. , 300(2):383–431, 1987

1987
[41]

Schmid and H

J. Schmid and H. Zwart. Stabilization of port-Hamilton ian systems by nonlinear boundary control in the presence of disturbances. Available at arXiv : 1804.10598, 2018

2018 arXiv
[42]

E. Sontag. Input to state stability: basic concepts and results. In Nonlinear and optimal control theory, volume 1932 of Lecture Notes in Math. , pages 163–220. Springer Berlin, 2008

1932
[43]

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

2005
[44]

Tucsnak and G

M. Tucsnak and G. W eiss. Observation and control for operator semigroups . Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. Birkh¨ auser Verlag, Basel, 2009

2009
[45]

Tucsnak and G

M. Tucsnak and G. W eiss. W ell-posed systems—the LTI cas e and beyond. Automatica J. IF AC, 50(7):1757–1779, 2014

2014
[46]

G. W eiss. Admissibility of unbounded control operator s. SIAM J. Control Optim. , 27(3):527– 545, 1989

1989
[47]

Zheng, H

J. Zheng, H. Lhachemi, G. Zhu, and D. Saussi´ e. ISS with r espect to boundary and in-domain disturbances for a coupled beam-string system. Math. Control Signals Systems , 30(4):Art. 21, 25, 2018

2018
[49]

Zheng and G

J. Zheng and G. Zhu. ISS with Respect to In-domain and Bou ndary Disturbances for a Generalized Burgers’ Equation. In IEEE Conference on Decision and Control (CDC) , pages 3758–3764, Miami Beach, FL, USA, Dec. 17-19, 2018

2018
[50]

Zheng and G

J. Zheng and G. Zhu. A De Giorgi Iteration-Based Approac h for the Establishment of ISS Properties for Burgers’ Equation with Boundary and In-doma in Disturbances. IEEE Trans. Automat. Control, 64(8):3476–3483, 2019

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [48]

Zheng and G

J. Zheng and G. Zhu. Input-to-state stability with resp ect to boundary disturbances for a class of semi-linear parabolic equations. Automatica J. IF AC, 97:271–277, 2018

work page 2018

[2] [1]

R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathe- matics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 200 3. 25

work page

[3] [2]

A. A. Agrachev, A. S. Morse, E. D. Sontag, H. J. Sussmann, a nd V. I. Utkin. Nonlinear and optimal control theory , volume 1932 of Lecture Notes in Mathematics . Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures giv en at the C.I.M.E. Summer School held in Cetraro, June 19–29, 2004, Edited by Paolo Nistri and Gianna Stefani

work page 1932

[4] [3]

Arendt, C

W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. Vector-valued Laplace transforms and Cauchy problems , volume 96 of Monographs in Mathematics . Birkh¨ auser/Springer Basel AG, Basel, second edition, 2011

2011

[5] [4]

Baillon and P

J.-B. Baillon and P. Cl´ ement. Examples of unbounded ima ginary powers of operators. J. Funct. Anal., 100(2):419–434, 1991

work page 1991

[6] [5]

Bribiesca Argomedo, E

F. Bribiesca Argomedo, E. Witrant, and C. Prieur. D1-inp ut-to-state stability of a time- varying nonhomogeneous diﬀusive equation subject to bound ary disturbances. In 2012 Amer- ican Control Conference (ACC) , pages 2978–2983, June 2012

work page 2012

[7] [6]

C. I. Byrnes, D. S. Gilliam, V. I. Shubov, and G. W eiss. Reg ular linear systems governed by a boundary controlled heat equation. J. Dynam. Control Systems , 8(3):341–370, 2002

work page 2002

[8] [7]

R. F. Curtain and H. Zwart. An introduction to inﬁnite-dimensional linear systems the ory, volume 21 of Texts in Applied Mathematics . Springer-Verlag, New York, 1995

work page 1995