{"id":"9a246d85-33bd-4f40-b28a-7471a93bd6ee","arxiv_id":"1909.02483","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"This paper derives a class of gradient controllers with conditional robustness guarantees for signal temporal logic tasks, extends them to unicycle dynamics by adding an auxiliary non-degeneracy task, and demonstrates a heuristic controller-combination scheme in simulation.","lead":"Researchers developed gradient-based controllers that steer robots to satisfy temporal logic tasks such as reaching a goal region or avoiding an obstacle, and extended them to unicycle-like robots. The work matters because these controllers are cheap to compute and could guide reinforcement learning explorations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 silently requires ρψ to be C^2: control law (19) uses ∂v/∂x in (16)-(17), but Assumption 1(i) only gives a locally Lipschitz gradient, so the augmented controller can be undefined for systems satisfying the stated assumptions.","rationale":"I read the paper's central claim as the conditional Theorem 3: under Assumptions 1 and 3, the two control laws jointly achieve local robustness satisfaction in D. The theorem is plausible for the unicycle example, where all quantities are smooth away from eg=0. The reader's weakest assumption, the non-degeneracy of v and vaug, is an explicitly stated hypothesis and mainly limits applicability; it does not reveal an internal gap in the theorem's proof for the systems it covers. My check found a more load-bearing issue: the proof and controller construction require ∂v/∂x1 and ∂v/∂x2, i.e., second derivatives of ρψ, while Assumption 1 only guarantees the first gradient is locally Lipschitz. A locally Lipschitz gradient does not imply differentiability, and one can satisfy every stated assumption yet have an undefined control law at a point inside the region of interest. This is a genuine proof gap in the general statement, though it is easily repaired by strengthening Assumption 1 to require ρψ ∈ C^2 and g11 sufficiently smooth. Because the gap is fixable and does not affect the simulation study, it reinforces the conditional verdict rather than overturning it. I therefore recommend keeping the reader's CONDITIONAL verdict and adding the smoothness requirement to the list of conditions to be settled.","tokens_in":14190,"tokens_out":24831,"duration_ms":271858,"concrete_test":"Construct the 2D system ẋ1 = x2 u1, ẋ2 = u2 with w=0, ρψ(x1) = x1|x1|/2 + x1, γ(t)=-1, Γ(t)=3, γ_aug=0, Γ_aug=2, vmin=0.1, and initial state x0=(0,1). Verify that ρψ(x0)=0 ≥ γ(0), ρψ_aug(x0)=1-0.1=0.9 lies in X_aug, and vaug(x0) ≠ 0, so the stated Assumptions 1 and 3 are satisfied. Then evaluate G(x0) from equation (16): it contains x2 · d(|x1|+1)/dx1 at x1=0, which is undefined; likewise F(x0,w) in Lemma 5 is undefined. This shows the control law (19) and the κ_aug condition are not well-defined under the stated assumptions. Re-run the check after adding ρψ ∈ C^2 to Assumption 1; the obstruction disappears, confirming that the missing smoothness is the issue.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5 and Theorem 3 define u2 through G(x) and vaug(x) in (16)-(17), which involve ∂v/∂x1 and ∂v/∂x2, with v(x)^T = (∂ρψ/∂x1) g11(x2). Assumption 1(i) only asserts that ρψ and ∂ρψ/∂x are locally Lipschitz; this does not imply the needed second derivatives exist. Concretely, take x1,x2 ∈ R, dynamics ẋ1 = x2 u1, ẋ2 = u2, and ρψ(x1) = x1|x1|/2 + x1. Then ∂ρψ/∂x1 = |x1| + 1, which is locally Lipschitz everywhere, so Assumption 1(i) is satisfied, but the second derivative is undefined at x1=0. With g11(x2)=x2 and g22=1 we have v(x) = (|x1|+1)x2. At x0=(0,1), v=1, ρψ=0, and with γ_aug=0, Γ_aug=2, vmin=0.1 the point lies in X_aug, and vaug(x0) = sign(v)(|x1|+1) = 1 ≠ 0, so Assumption 3 can also hold locally. Nevertheless, G(x) in (16) and the terms in F(x,w) require ∂v/∂x1 = x2 · d(|x1|+1)/dx1 at x1=0, which does not exist. Hence the κ_aug condition in Lemma 5 is not well-defined, u2 in (19) need not be locally Lipschitz, and Lemma 1 cannot be applied. The unicycle example is smooth away from eg=0, so the simulation is not affected, but the theorem as stated for general system (10) needs an explicit C^2 assumption on ρψ (and corresponding differentiability of g11) that is absent from Assumption 1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes gradient-based controllers for signal temporal logic (STL) specifications. For systems of the form ẋ = f(x)+g(x)u+w, it defines a control law (Theorem 1) that guarantees local robustness satisfaction ρψ(x(t)) ≥ γ(t) by enforcing Nagumo-type boundary conditions, assuming the gradient of the robustness metric is controllable (Assumption 2). It then extends the approach to unicycle-like systems (10) by introducing an augmented robustness specification ρψ_aug = ||v(x)|| - v_min whose purpose is to keep the controllability term v(x) away from zero. The main theoretical result, Theorem 3, states that the two control laws (15) and (19) jointly achieve local robustness satisfaction of the conjoined specification within a domain D. The paper also discusses how elementary controllers can be combined (Section V) and illustrates the approach on a unicycle navigation example and a case study with obstacle avoidance.","tokens_in":14715,"tokens_out":8008,"duration_ms":78308,"significance":"If the technical gaps identified below are repaired, the paper offers a computationally inexpensive, local-satisfaction controller for a class of nonholonomic systems with STL tasks, complementing MPC and barrier-function approaches. The explicit construction of an augmented task to avoid singular controllability directions is a useful idea, and the paper is honest about the loss of guarantees when controllers are combined. The theoretical claims are, however, conditional on regularity assumptions that are not fully stated, so the main theorem as written is not established for the general system class claimed.","major_comments":[{"comment":"Assumption 1(i) only requires ρψ and its gradient to be locally Lipschitz, but the augmented controller (19) uses ∂v/∂x1 and ∂v/∂x2, i.e., second derivatives of ρψ. This is not implied by the stated assumption. For example, take x1,x2 ∈ R, dynamics ẋ1 = x2 u1, ẋ2 = u2, and ρψ(x1) = x1|x1|/2 + x1 with g11(x2)=x2. Then ∂ρψ/∂x1 = |x1|+1 is locally Lipschitz, so Assumption 1(i) is satisfied, but the second derivative is undefined at x1=0. With v(x) = (|x1|+1)x2, the point x0=(0,1) lies in the appropriate domain for suitable γ_aug, Γ_aug, and vmin, and vaug(x0) ≠ 0, yet ∂v/∂x1 does not exist at x0. Consequently, the κ_aug condition in Lemma 5 is not well-defined, u2 in (19) need not be locally Lipschitz, and Lemma 1 cannot be applied. The unicycle example is smooth away from eg = 0, so the simulations are unaffected, but Theorem 3 as stated for general system (10) requires an explicit C^2 assumption on ρψ (and corresponding differentiability of g11, g21, g22) that is absent from Assumption 1.","section":"§IV, Lemma 5 (application of Lemma 3)"},{"comment":"The proof of Lemma 5 invokes Lemma 3 (Nagumo) for the time-varying set S_aug(t) = {x : γ_aug(t) − ρψ_aug(x) ≤ 0}. Lemma 3 requires ∂ζ/∂x ≠ 0 at the boundary, i.e., ∂ρψ_aug/∂x ≠ 0 on ∂X_aug. This is not established from Assumption 3. Assumption 3 only states that vaug(x) ≠ 0 in the augmented region of interest; it does not imply that the gradient of ρψ_aug is nonzero on the boundary. Without this condition, the invariance argument for the augmented specification is incomplete. Adding a C^2 regularity assumption would make ∂ρψ_aug/∂x well-defined, but nonvanishing still needs to be checked or imposed.","section":"§IV, Theorem 3 (proof)"},{"comment":"The proof of Theorem 3 proceeds by taking δ = min(δ1, δ2) from Lemmas 4 and 5, but each lemma assumes that the other control input is already Lipschitz continuous. In the coupled closed loop, u1 and u2 are both determined by the same state, so one must verify that the joint right-hand side is locally Lipschitz before applying Lemma 1. This is a fixable issue once the missing C^2 assumption is added, but as written the existence argument for the combined system is not fully rigorous.","section":"§IV, Theorem 3 (proof)"}],"minor_comments":[{"comment":"The phrasing 'ρψ(x) and its gradient ∂ρψ/∂x are locally Lipschitz continuous' is ambiguous; it should explicitly state that ρψ is C^1 with a locally Lipschitz gradient, or otherwise clarify the differentiability class.","section":"Definition 1 and §IV (Eq. (14))"},{"comment":"The definition D := {x : ∃t, x ∈ (X(t) ∪ A(t)) ∩ (A_aug(t) ∪ X_aug(t))} uses an existential quantifier inside a set builder, which is confusing. It should be written as a union over t, and the subsequent condition (14) should be stated for all x ∈ D, not with an additional '∃t' quantifier.","section":"Assumption 2 and Assumption 3"},{"comment":"The quantifier structure in (5) and (18), 'v(x) ≠ 0, ∀x : ∃t s.t. x ∈ X(t)', is nonstandard. It should be 'for all x and all t such that x ∈ X(t), v(x) ≠ 0' to be unambiguous.","section":"Theorem 1 and Lemma 4 proofs"},{"comment":"In the proofs, the noise bound uses 'max_w' without specifying the set B over which w ranges; this should be defined explicitly, e.g., as a supremum over the bounded set B.","section":"Definition 1"},{"comment":"The notation for the upper boundary ¯X(t) and lower boundary ¯X(t) appears identical in the typeset version; the authors should use visually distinct symbols (e.g., overbar and underline) to avoid confusion.","section":"Example 2 and Section V-C"},{"comment":"The paper should state clearly that the main guarantee requires the initial state to satisfy ρψ_aug(x(0)) ≥ γ_aug(0); if the system starts at a configuration where v = 0 or vaug = 0, the controller (19) is not well-defined and the theorem does not apply. This limitation is acknowledged in Example 2 but should appear as a remark after Theorem 3.","section":"Example 2 and Section V-C"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is promising, but the main theorem requires additional regularity assumptions that are not stated. The missing C^2 condition on ρψ is a genuine gap, as the augmented controller explicitly uses second derivatives. The other issues (Nagumo gradient condition, joint Lipschitz proof) are readily fixable once the regularity assumption is added. The authors should also clarify the domain D and the initial-condition limitations. I do not see evidence of circularity or data-fitting concerns; the guarantees are conditional theorems with design parameters. The companion links [9] and [10] are appropriate. Overall, the paper merits a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is worth reading for the unicycle extension. The augmented task of keeping ||v|| away from zero is a neat idea that addresses a real failure mode of PPC-style controllers for nonholonomic systems. Theorem 1 also generalizes the prior PPC and barrier-function results by dropping concavity and full-row-rank assumptions, which is a modest but real contribution.\n\nThat said, the main theorem has a load-bearing gap that the stress-test note identifies correctly. Theorem 3 relies on Assumption 1(i), which only gives locally Lipschitz gradients of rho_psi. But the augmented controller (19) uses ∂v/∂x in (16)-(17), and v contains ∂rho_psi/∂x. That requires second derivatives. The concrete counterexample in the note works: rho_psi = x1|x1|/2 + x1 satisfies Assumption 1(i) but the augmented control is undefined at x1=0. So the statement of Theorem 3 is not true as written for all systems satisfying its assumptions. This is fixable by adding an explicit C^2 assumption on rho_psi and corresponding differentiability of g11; the proof structure then holds. It's a technical gap, not a conceptual disaster, but it should be fixed before the guarantees are used.\n\nThe non-degeneracy assumptions also exclude some practically relevant states. The paper acknowledges the perpendicular case is excluded from D and the parallel case by parameter choice, but there is no controller driving such states into D. That is a limitation worth stating explicitly.\n\nThe combination scheme in Section V is explicitly heuristic, and the case study is a single qualitative simulation. That is fine given the paper frames it as initial thoughts, but it shouldn't be overread.\n\nThe citations look appropriate, including the self-citations to the companion learning papers, since those are used for motivation rather than to claim novelty.\n\nOverall, this is a solid, honest paper for the STL-control community. It deserves a serious referee; the smoothness gap is likely to be caught and fixed in revision. I would bring it to a reading group and would cite the unicycle extension after the C^2 fix lands.\n\nRecommendation: send to peer review with a request for revision.","headline":"Useful unicycle extension of gradient-based STL control, but Theorem 3 as stated misses a C^2 requirement on rho_psi; fixable, worth refereeing.","tokens_in":15207,"tokens_out":2248,"would_cite":true,"duration_ms":22296,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that, for a class of unicycle-like systems, a temporal-logic task can be enforced by two stacked gradient controllers—one that keeps the task-robustness measure above a designed curve and one that keeps the…","keywords":["signal temporal logic","gradient-based control","prescribed performance control","nonholonomic systems","unicycle dynamics","robustness satisfaction","control barrier functions","temporal logic task specifications"],"falsifier":"Simulate the unicycle system (9)-(10) with additive process noise, starting from an initial state inside the domain $D$ with $\\rho_\\psi(x_1(0))\\ge\\gamma(0)$ and $\\rho_{\\psi_{\\mathrm{aug}}}(x(0))\\ge\\gamma_{\\mathrm{aug}}(0)$, and apply (15)-(19). If along any such trajectory the solution exists beyond the claimed interval but either $\\rho_\\psi$ or $\\rho_{\\psi_{\\mathrm{aug}}}$ falls below its curve before the state leaves $D$, then Theorem 3 is false. A direct check of the boundary condition $\\dot{\\rho}_\\psi\\ge\\dot{\\gamma}$ at $\\rho_\\psi=\\gamma(t)$ with the formulas (15) and (19) would likewise settle the claim.","tokens_in":13989,"feed_emoji":"🤖","tokens_out":12919,"duration_ms":125217,"temperature":0.7,"pith_summary":"The paper turns a signal temporal logic (STL) task—such as 'reach a goal within ten seconds and then stay near it'—into feedback laws that only act when a robustness measure approaches a user-chosen lower curve. For simple dynamics a single controller keeps the measure above the curve and thereby guarantees the task, provided the system state remains bounded. The paper's main extension handles unicycle-like nonholonomic systems: a naive gradient controller fails because the direction in which it can push the robustness metric may vanish, so a second, auxiliary controller is added to keep that direction away from zero. The two controllers together give what the paper calls local robustness satisfaction, and the intention is to use such inexpensive controllers to guide exploration inside learning algorithms.","feed_headline":"Two gradient controls certify STL tasks for unicycle-like robots","feed_subtitle":"A robustness curve plus a steering-safety input turns temporal-logic specs into cheap controls","key_machinery":"The machinery is a Nagumo-type invariance argument over time-varying sets defined by the robustness measure. Each task is packaged as a region of interest $X(t)=\\{x:\\gamma(t)\\le \\rho_\\psi(x)\\le\\Gamma(t)\\}$ with $\\Gamma(t)\\ge\\gamma(t)+\\epsilon$, and the control is zero in the uncontrolled region $A(t)=\\{\\rho_\\psi(x)>\\Gamma(t)\\}$, so the controller is active only when the measure is near its lower bound. On the boundary $\\rho_\\psi=\\gamma(t)$, the control law (6), $u=\\kappa(x,t)\\frac{K v(x)}{\\|v(x)\\|_2^2+\\Delta}$, is chosen so that $\\dot{\\rho}_\\psi\\ge\\dot{\\gamma}$, which by the generalized Nagumo theorem prevents the trajectory from crossing below the curve. For unicycle-like systems the same construction is applied to the augmented robustness $\\rho_{\\psi_{\\mathrm{aug}}}=\\|v(x)\\|_2-v_{\\min}$, with the second input acting through the coefficient $v_{\\mathrm{aug}}(x)^T=\\frac{v(x)^T}{\\|v(x)\\|_2}\\frac{\\partial v}{\\partial x_2}g_{22}(x)$; Theorem 3 combines the two invariance conditions.","core_discovery":"The central claim is Theorem 3: for the triangular system (10), under Assumptions 1 and 3, the control laws (15) and (19) achieve local robustness satisfaction of the conjoined specification $\\rho_\\psi(x_1(t)) \\ge \\gamma(t)$ and $\\rho_{\\psi_{\\mathrm{aug}}}(x(t)) \\ge \\gamma_{\\mathrm{aug}}(t)$ within the domain $D$. Here $\\rho_\\psi$ is the spatial robustness of a non-temporal formula $\\psi$, and $\\gamma(t)$ is a designed curve whose domination by $\\rho_\\psi$ guarantees the temporal task $\\varphi$ by Assumption 1(iii). Because a direct gradient step on $\\rho_\\psi$ may be impossible when the direction $v(x)^T = \\frac{\\partial \\rho_\\psi}{\\partial x_1} g_{11}(x_2)$ vanishes, the paper defines an augmented task $\\rho_{\\psi_{\\mathrm{aug}}}(x) = \\|v(x)\\|_2 - v_{\\min}$ and uses the second input to keep $v$ bounded away from zero. Theorem 3 says these two invariance conditions hold together for finite time; if the state also remains bounded, Theorem 2 upgrades local robustness satisfaction to satisfaction of the original STL task.","pith_inferences":["The authors leave implicit that the same augmented-task construction could extend inductively to systems with several nonholonomic constraints: after $v_1\\neq 0$ is enforced, a second augmented task can enforce $v_2\\neq 0$, and so on.","Not addressed in the paper is a wrapper controller that keeps the state inside $D$, or a mode-switching controller that first rotates the heading away from the excluded perpendicular and parallel configurations; such a wrapper would turn the local guarantee into a global one for unicycle tasks.","A tunable consequence not drawn by the paper is that the regularization parameter $\\Delta$ could serve as an exploration knob in a learning loop, since larger $\\Delta$ changes how strongly the steering input responds to the second derivative of robustness.","The case study suggests a testable extension: once exact guarantees are relaxed, steering by the second derivative (the 'practical' controller) may outperform maintaining the augmented invariant set, and a sweep over obstacle placements and initial headings would quantify the trade-off."],"forward_implications":["For any system fitting the triangular form (10) and satisfying the two non-vanishing-direction assumptions, a temporal-logic task can be certified by two independent, minimally invasive feedback laws—one per input channel—that are active only near the robustness lower bound.","If the state remains bounded, or if the conjoined specifications themselves enforce boundedness, the local guarantee of Theorem 3 becomes global satisfaction of the STL task by Theorem 2.","A conjunction of several such tasks can be controlled by a weighted consensus (20) that gives priority to whichever robustness measure is closest to its lower bound; individual guarantees are traded for cheap, combined steering, which is the intended use in policy-improvement learning.","With a nonzero regularization parameter $\\Delta>0$, the second input can influence the second derivative of the robustness metric; the paper's 'practical' controller (25) uses that mechanism to steer a unicycle away from an obstacle while still heading toward the goal.","Configurations where the controllability direction vanishes are outside the theorem: such states are excluded from the domain $D$ by the augmented task, so the guarantee is explicitly local and conditional."],"supporting_citations":[{"why":"Defines signal temporal logic over continuous-time signals and its satisfaction relation, the formalism in which the tasks are written.","marker":"[2]"},{"why":"Introduces prescribed performance control for STL and the earlier assumption set that Assumption 2 generalizes, including the aggressive-controller limit discussed in Remark 3.","marker":"[4]"},{"why":"Shows that prescribed-performance-guided exploration improves policy improvement, the learning application that motivates combining elementary controllers.","marker":"[9]"},{"why":"Provides the prescribed-performance idea of prescribing the evolution of a performance metric, which underlies the design of the curve $\\gamma(t)$.","marker":"[11]"},{"why":"Gives the spatial robustness metric used throughout the paper to express the extent to which a task is satisfied.","marker":"[12]"},{"why":"Supplies the local and global existence-and-uniqueness theorems (Lemmas 1 and 2) on which the local robustness proofs rely.","marker":"[13]"},{"why":"Supplies the generalized Nagumo theorem (Lemma 3), the invariance condition used to keep the robustness measure above its lower curve.","marker":"[14]"},{"why":"Presents the control-barrier-function approach for STL that the gradient controller relates to as a barrier-style boundary enforcement method.","marker":"[15]"}],"fun_headline_variants":["Gradient controls certify STL tasks for unicycle robots","Two gradient controls yield STL satisfaction for unicycles","Local robustness for STL via dual gradient controls","Gradient STL method guarantees for nonholonomic systems","Two-input control proves STL specs on unicycle dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantee only holds while the robot's heading is neither exactly perpendicular nor exactly parallel to the direction of the target error; at either of those two headings one of the two control directions vanishes, and the theorem's assumptions stop applying.","fun_headline_variants_meta":{"raw":{"variants":["Gradient controls certify STL tasks for unicycle robots","Two gradient controls yield STL satisfaction for unicycles","Local robustness for STL via dual gradient controls","Gradient STL method guarantees for nonholonomic systems","Two-input control proves STL specs on unicycle dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000494,"raw_usage":{"total_tokens":2406,"prompt_tokens":907,"completion_tokens":1499,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":1420}},"tokens_in":523,"tokens_out":1499,"duration_ms":10934,"temperature":1.0,"reasoning_tokens":1420,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:50:30.580428+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the unicycle system (9)-(10) with additive process noise, starting from an initial state inside the domain $D$ with $\\rho_\\psi(x_1(0))\\ge\\gamma(0)$ and $\\rho_{\\psi_{\\mathrm{aug}}}(x(0))\\ge\\gamma_{\\mathrm{aug}}(0)$, and apply (15)-(19). If along any such trajectory the solution exists beyond the claimed interval but either $\\rho_\\psi$ or $\\rho_{\\psi_{\\mathrm{aug}}}$ falls below its curve before the state leaves $D$, then Theorem 3 is false. A direct check of the boundary condition $\\dot{\\rho}_\\psi\\ge\\dot{\\gamma}$ at $\\rho_\\psi=\\gamma(t)$ with the formulas (15) and (19) would likewise settle the claim.","supporting_citations":[{"cited_title":"Monitoring temporal properti es of contin- uous signals,","cited_arxiv_id":null,"evidence_quote":"Defines signal temporal logic over continuous-time signals and its satisfaction relation, the formalism in which the tasks are written."},{"cited_title":"Pre scribed performance control for signal temporal logic specificatio ns,","cited_arxiv_id":null,"evidence_quote":"Introduces prescribed performance control for STL and the earlier assumption set that Assumption 2 generalizes, including the aggressive-controller limit discussed in Remark 3."},{"cited_title":"Prescribed Performance Control Guided Policy Improvement for Satisfying Signal Temporal Logic Tasks","cited_arxiv_id":"1903.04340","evidence_quote":"Shows that prescribed-performance-guided exploration improves policy improvement, the learning application that motivates combining elementary controllers."},{"cited_title":"Robust adaptiv e control of feedback linearizable MIMO nonlinear systems with presc ribed performance,","cited_arxiv_id":null,"evidence_quote":"Provides the prescribed-performance idea of prescribing the evolution of a performance metric, which underlies the design of the curve $\\gamma(t)$."},{"cited_title":"Robust satisfaction of tempora l logic over real-valued signals,","cited_arxiv_id":null,"evidence_quote":"Gives the spatial robustness metric used throughout the paper to express the extent to which a task is satisfied."},{"cited_title":"Nonlinear systems, vol. 3,","cited_arxiv_id":null,"evidence_quote":"Supplies the local and global existence-and-uniqueness theorems (Lemmas 1 and 2) on which the local robustness proofs rely."},{"cited_title":"Blanchini and S","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized Nagumo theorem (Lemma 3), the invariance condition used to keep the robustness measure above its lower curve."},{"cited_title":"Control barrier fu nctions for signal temporal logic tasks,","cited_arxiv_id":null,"evidence_quote":"Presents the control-barrier-function approach for STL that the gradient controller relates to as a barrier-style boundary enforcement method."}],"review_version":1}