{"id":"2bf7caf2-44b9-4679-9e50-e35df7165360","arxiv_id":"1908.08317","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Input-to-state stability estimates for parabolic boundary control systems are reviewed and extended to semilinear equations on general domains via semigroup and energy methods.","lead":"This math paper studies when heat-like equations with control applied at the boundary stay stable under disturbances, and extends known results to more general equations in higher dimensions. It gives a unified framework for proving input-to-state stability estimates and works through examples like the heat equation with Neumann or Dirichlet boundary control.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4's linear-growth hypothesis (2) is incompatible with its headline semilinear example: f(x)=-x^3 has cubic growth in X_{1/2}=H^1, so Example 3.6 is not covered.","rationale":"Reading in good faith, I did not find a fatal error in the linear sections; Proposition 2.13 and the Lq thresholds (4/3 for Neumann control and 4 for Dirichlet control) are standard and correctly explained. The main new result is Theorem 3.4, and the most load-bearing problem is that its flagship example does not satisfy the theorem's stated growth condition (2). This is more concrete than the input-class caveat: even with maximally smooth W^{2,1} inputs, Example 3.6 cannot be obtained from Theorem 3.4 as written. The reader's weakest_assumption concentrated on B∈L(U,X_{-1/2}); that is a real scope restriction, but it is an explicit assumption and the paper correctly notes that Dirichlet control cannot give L2-ISS. The example mismatch, by contrast, is an internal inconsistency between the theorem and its application. The conditional verdict remains appropriate: the manuscript should either weaken or repair assumption (2), for instance by using the dissipative structure of f to prove global existence for polynomial nonlinearities, or replace Example 3.6. The Lq-ISS density issue noted by the reader also remains open, so no change from the existing CONDITIONAL verdict is needed.","tokens_in":28138,"tokens_out":20205,"duration_ms":199088,"concrete_test":"Take Ω⊂R^3 a bounded domain and x≡c in Example 3.6. Compute ||f(x)||_{L^2}=c^3|Ω|^{1/2} and ||x||_{H^1}=c|Ω|^{1/2}; let c→∞ to see that the ratio ||f(x)||_{L^2}/(1+||x||_{H^1}) is unbounded, contradicting condition (2). Alternatively, recompute the Sobolev bound stated in Example 3.6: ||x^3||_{L^2}=||x||_{L^6}^3≤C||x||_{H^1}^3, which is cubic, not linear. If condition (2) is relaxed to a polynomial growth bound, the proof's global-existence step, which invokes [38, Thm. 6.3.3] with linear growth, must be replaced by an energy/regularity argument, and the theorem's hypotheses and example need to be revised accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Example 3.6 is offered as the main semilinear application, and the text states that conditions (1) and (2) of Theorem 3.4 follow from the Sobolev embedding W^{1,2}(Ω)⊂L^6(Ω). This is not correct. Condition (2) requires ||f(t,x)||_X ≤ k(t)(1+||x||_{X_{1/2}}) for all x, that is, linear growth in the X_{1/2}-norm. With f(x)=-x^3, X=L^2(Ω), and X_{1/2}=H^1(Ω), take x≡c for large c>0: ||f(x)||_{L^2}=c^3|Ω|^{1/2}, while 1+||x||_{H^1}=1+c|Ω|^{1/2}; no finite k(t) can dominate the quotient c^2. The Sobolev embedding gives ||x^3||_{L^2}=||x||_{L^6}^3≤C||x||_{H^1}^3, a cubic bound, not a linear one. Thus the paper's assertion that condition (2) follows from the stated embedding is false, and the advertised generalization to the cubic heat equation is not an instance of Theorem 3.4 as written. The theorem may still be valid for functions satisfying (2), but the central example and the claimed extension to polynomial dissipative nonlinearities rest on an invalid verification. This is an internal inconsistency, distinct from the acknowledged limitation that the proof treats only W^{2,1} inputs.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28406,"tokens_out":18477,"duration_ms":165417,"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":[{"comment":"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.","section":"Section 3, Example 3.6"},{"comment":"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.","section":"Theorem 3.4 and Remark 3.5"},{"comment":"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.","section":"Example 3.6"}],"minor_comments":[{"comment":"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}.","section":"Theorem 3.4"},{"comment":"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.","section":"Theorem 3.4, condition (1)"},{"comment":"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.","section":"Example 2.14"},{"comment":"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.","section":"Theorem 2.18"}],"recommendation":"major_revision","confidential_remarks":"The survey parts are solid and will be useful. The main concern is the mismatch between Theorem 3.4 and Example 3.6; this is not a matter of style but of the central claim. Given the author's prior work [19,21], the linear part is largely known; the semilinear part should be revised either by proving a version that covers polynomial nonlinearities or by explicitly limiting the claims to linear-growth nonlinearities. If the latter, the contribution relative to [48] needs to be reassessed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nTheorem 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.\n\nThe 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.\n\nMinor: Example 2.14's threshold q>4/3 is fine but would be easier to check if the chosen right-inverse B0 were specified.\n\nThe 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.\n\nIf I were a journal editor, I'd request major revision before acceptance.","headline":"A useful ISS survey whose central semilinear theorem is not supported by its own headline example: cubic f fails the linear-growth hypothesis.","tokens_in":28999,"tokens_out":5342,"would_cite":false,"duration_ms":51537,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C20","35K58","47D06","93D20","93C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["input-to-state stability","boundary control","parabolic equations","semilinear systems","analytic semigroups","interpolation spaces","admissibility","Lyapunov estimates"],"falsifier":"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.","tokens_in":27869,"feed_emoji":"🔥","tokens_out":22593,"duration_ms":183456,"temperature":0.7,"pith_summary":"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.","feed_headline":"Boundary regularity decides when parabolic control systems are stable","feed_subtitle":"Semilinear heat-type systems are Lq-stable for q≥2 when boundary inputs smooth half a derivative.","key_machinery":"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.","core_discovery":"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 $q\\ge 2$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The one-dimensional Robin/Neumann semilinear result that Theorem 3.4 explicitly generalizes to higher-dimensional domains.","marker":"[48]"},{"why":"Supplies the local existence, continuation, and global existence theory for semilinear parabolic equations used in the proof of Theorem 3.4.","marker":"[38]"},{"why":"Provides the abstract boundary-control-system framework, the Fattorini-trick representation of mild solutions, and the classical solution regularity results used throughout.","marker":"[44]"},{"why":"Establishes the equivalence between $L^q$-admissibility and ISS for linear systems and supplies the Orlicz-space example that is not $L^q$-ISS for any finite $q$.","marker":"[19]"},{"why":"Proves the $L^\\infty$-ISS and Orlicz-estimate result for parabolic boundary control systems used as Theorem 2.18.","marker":"[21]"},{"why":"Supplies the identification of the abstract spaces $X_\\alpha$ with fractional Sobolev spaces and the trace embeddings used in the Neumann and Dirichlet examples.","marker":"[28]"},{"why":"Provides the classical counterexample that Dirichlet boundary input is not $L^2$-admissible, motivating the $q>4$ threshold and the $H^{-1}$ workaround.","marker":"[31]"},{"why":"Shows $L^\\infty$-ISS for semilinear parabolic equations with Dirichlet boundary control by monotonicity principles, the case the semilinear theorem does not cover.","marker":"[33]"}],"fun_headline_variants":["Boundary regularity decides ISS for parabolic control","Semilinear parabolic ISS also set by boundary input regularity","Input regularity: the key to L^q ISS in parabolic boundary control","Boundary smoothness condition unifies linear and semilinear parabolic ISS","Parabolic ISS: input smoothness sets the L^q threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Boundary regularity decides ISS for parabolic control","Semilinear parabolic ISS also set by boundary input regularity","Input regularity: the key to L^q ISS in parabolic boundary control","Boundary smoothness condition unifies linear and semilinear parabolic ISS","Parabolic ISS: input smoothness sets the L^q threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001192,"raw_usage":{"total_tokens":4918,"prompt_tokens":945,"completion_tokens":3973,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":3889}},"tokens_in":561,"tokens_out":3973,"duration_ms":24753,"temperature":1.0,"reasoning_tokens":3889,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:43:29.999450+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Zheng and G","cited_arxiv_id":null,"evidence_quote":"The one-dimensional Robin/Neumann semilinear result that Theorem 3.4 explicitly generalizes to higher-dimensional domains."},{"cited_title":"Tucsnak and G","cited_arxiv_id":null,"evidence_quote":"Provides the abstract boundary-control-system framework, the Fattorini-trick representation of mild solutions, and the classical solution regularity results used throughout."},{"cited_title":"Jacob, R","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between $L^q$-admissibility and ISS for linear systems and supplies the Orlicz-space example that is not $L^q$-ISS for any finite $q$."},{"cited_title":"Jacob, F","cited_arxiv_id":null,"evidence_quote":"Proves the $L^\\infty$-ISS and Orlicz-estimate result for parabolic boundary control systems used as Theorem 2.18."},{"cited_title":"Lasiecka and R","cited_arxiv_id":null,"evidence_quote":"Supplies the identification of the abstract spaces $X_\\alpha$ with fractional Sobolev spaces and the trace embeddings used in the Neumann and Dirichlet examples."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical counterexample that Dirichlet boundary input is not $L^2$-admissible, motivating the $q>4$ threshold and the $H^{-1}$ workaround."},{"cited_title":"Mironchenko, I","cited_arxiv_id":null,"evidence_quote":"Shows $L^\\infty$-ISS for semilinear parabolic equations with Dirichlet boundary control by monotonicity principles, the case the semilinear theorem does not cover."}],"review_version":1}