Convex Computations for Controlled Safety Invariant Sets of Black-box Discrete-time Dynamical Systems

Arvind Easwaran; Bai Xue; Dejin Ren; Jingduo Pan; Taoran Wu; Yiling Xue

arxiv: 2504.01638 · v4 · submitted 2025-04-02 · 📡 eess.SY · cs.SY

Convex Computations for Controlled Safety Invariant Sets of Black-box Discrete-time Dynamical Systems

Taoran Wu , Yiling Xue , Jingduo Pan , Dejin Ren , Arvind Easwaran , Bai Xue This is my paper

Pith reviewed 2026-05-22 22:00 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords controlled safety invariant setsblack-box dynamical systemsPAC guaranteesbarrier functionsscenario optimizationlinear programmingdiscrete-time systemssafety-critical control

0 comments

The pith

Barrier functions and scenario optimization reduce computation of Probably Approximately Correct controlled safety invariant sets to a linear program for unknown discrete-time dynamics.

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

The paper seeks tractable ways to identify controlled safety invariant sets when the system dynamics are black-box and only simulation samples are available. Classical controlled invariance demands that every state in the set admits a control keeping the next state inside, but this universal requirement proves impractical for data-driven settings. The authors replace it with a Probably Approximately Correct (PAC) version: with user-specified , a chosen fraction of states must admit suitable controls. They encode the relaxed invariance condition via barrier functions and apply scenario optimization to obtain a convex linear program whose solution certifies the PAC property on the sampled data.

Core claim

A PAC CSIS exists when, with prescribed probability, at least a specified fraction of states inside the set admit a control input that keeps the next state inside the set; barrier functions convert this condition into polynomial inequalities, and scenario optimization converts the resulting semi-infinite program into a finite linear program solvable from finite simulation data alone.

What carries the argument

Barrier-function encoding of the PAC controlled-invariance condition together with scenario optimization that produces a linear program from sampled transitions.

If this is right

Safety verification becomes possible for systems whose equations are unavailable but whose trajectories can be simulated.
The resulting sets come with explicit probabilistic guarantees rather than requiring deterministic invariance for every state.
The linear-program formulation scales with the number of samples rather than the dimension of the state space in a combinatorial way.
The same pipeline can be rerun whenever new simulation data arrive, yielding updated PAC sets without re-deriving analytic models.

Where Pith is reading between the lines

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

The method could be combined with model-predictive control by treating the computed PAC set as a terminal constraint.
If the barrier degree is increased, the same scenario-optimization template may produce tighter PAC sets at the cost of larger LPs.
The framework suggests a route to PAC versions of other set-valued properties such as reachability or stabilizability for black-box plants.

Load-bearing premise

The finite samples drawn from the unknown dynamics are representative enough that a solution certified on those samples transfers to the true system with the stated PAC probability.

What would settle it

Generate a large independent test set of state-control pairs from the true dynamics and count the fraction of states inside the computed set for which no control keeps the successor inside; if that fraction exceeds the allowed violation rate by more than the PAC tolerance, the claim is falsified.

Figures

Figures reproduced from arXiv: 2504.01638 by Arvind Easwaran, Bai Xue, Dejin Ren, Jingduo Pan, Taoran Wu, Yiling Xue.

**Figure 1.** Figure 1: Red curves: the boundary of X. Blue region: the calculated PAC SIS, and gray region: the SIS obtained using the Monte Carlo method. TABLE I PARAMETERS AND RESULTS EX α N M d ϵ Time Px[x ∈ eS] 1 − α Px[x∈eS] 1 0.3 921 − 12 − 0.52 0.6646 0.5486 0.1 2762 − 12 − 0.76 0.6231 0.8395 0.05 5523 − 12 − 1.16 0.5629 0.9112 0.01 27611 − 12 − 2.91 0.5352 0.9813 2 0.01 13611 9 5 0.7 3.49 0.5984 0.9833 0.01 13611 9 5 0.5… view at source ↗

read the original abstract

Identifying controlled safety invariant sets (CSISs) is essential for safety-critical systems. This paper addresses the problem of computing CSISs for black-box discrete-time systems, where the dynamics are unknown and only limited simulation data are available. Traditionally, a CSIS requires that for every state in the set, there exists a control input that keeps the system within the set at the next step. However, enforcing such universal invariance, i.e., requiring the set to remain controlled invariant for all states, is often overly restrictive or impractical for black-box systems. To address this, we introduce the notion of a Probably Approximately Correct (PAC) CSIS, in which, with prescribed confidence, there exists a suitable control input to keep the system within the set at the next step for at least a specified fraction of the states. Our approach leverages barrier functions and scenario optimization, yielding a tractable linear programming method for estimating PAC CSISs. Several illustrative examples demonstrate the effectiveness of the proposed framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a PAC relaxation of controlled safety invariant sets for black-box dynamics and computes it via barrier functions turned into scenario-optimized LPs, but the transfer guarantee from samples to the full state space is not supported by any visible bounds or sampling assumptions.

read the letter

The one thing to know is that this work relaxes the usual all-states invariance requirement for controlled safety invariant sets into a PAC version that only needs to hold for most states with high probability, then solves for it with linear programming on barrier functions from finite data. That combination looks new relative to the standard controlled-invariant literature they cite. The framing as a practical tool for black-box discrete-time systems is reasonable, and using scenario optimization to get convexity is a direct move that fits the setting. The examples are presented as evidence that the method runs and produces sets, which is at least a start for an applied paper. The soft spot is the one the stress-test flags. Scenario optimization gives PAC-style guarantees only under i.i.d. sampling from a known distribution, an explicit sample-size formula tying N to the violation probability and failure probability, and a barrier class fixed before seeing data. The abstract states only that prescribed is achieved and supplies none of those three ingredients. Without them the computed set satisfies the invariance condition on the observed points but not necessarily on the continuous state space as claimed. If the full paper contains a theorem with the missing bound and distribution, the concern disappears; on the provided text it stands. This is for the data-driven verification corner of control theory. A reader already working on approximate safety certification from simulation traces would get a concrete method to try. It deserves peer review because the core construction is coherent on its own terms and the computational claim is testable, even if the PAC transfer needs tightening.

Referee Report

1 major / 0 minor

Summary. The paper introduces Probably Approximately Correct (PAC) Controlled Safety Invariant Sets (CSISs) for black-box discrete-time dynamical systems with unknown dynamics, where only simulation data are available. It proposes using barrier functions together with scenario optimization to obtain a linear programming formulation that computes sets satisfying the PAC invariance property: with prescribed confidence, a control input exists that keeps the system inside the set for at least a specified fraction of states.

Significance. If the PAC transfer is rigorously justified with explicit bounds, the work would supply a convex, data-driven method for safety verification of unknown systems, extending barrier-function techniques to the black-box setting while retaining probabilistic guarantees. The explicit use of scenario optimization for invariance is a constructive strength.

major comments (1)

[Abstract and scenario-optimization section] The central claim that the LP solution yields a PAC CSIS for the unknown continuous dynamics rests on scenario optimization transferring from finite samples to the true map. The manuscript must supply the explicit sample-complexity inequality N(ε,δ) (with the sampling measure stated) that justifies the PAC statement; without it the guarantee holds only on the observed points rather than on the state space as asserted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestion regarding the PAC guarantees. We address the single major comment below.

read point-by-point responses

Referee: [Abstract and scenario-optimization section] The central claim that the LP solution yields a PAC CSIS for the unknown continuous dynamics rests on scenario optimization transferring from finite samples to the true map. The manuscript must supply the explicit sample-complexity inequality N(ε,δ) (with the sampling measure stated) that justifies the PAC statement; without it the guarantee holds only on the observed points rather than on the state space as asserted.

Authors: We agree that the manuscript should explicitly state the sample-complexity bound to make the transfer from finite samples to the unknown dynamics rigorous. The current version invokes scenario optimization for the PAC property but omits the concrete inequality N(ε,δ) and the underlying sampling measure. In the revised manuscript we will insert the standard scenario-optimization bound (e.g., N ≥ (1/ε)(ln(1/δ) + m) where m is the number of decision variables, or the precise form appropriate to the linear program) together with the statement that samples are drawn uniformly from a compact state-space set X. This addition will confirm that the computed barrier function yields a PAC CSIS with the claimed (ε,δ) guarantees for the black-box discrete-time system. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation constructs PAC CSIS via independent scenario optimization on barrier functions

full rationale

The paper defines PAC CSIS as a probabilistic relaxation of controlled invariance, then applies standard barrier-function encoding and scenario optimization to obtain an LP whose solution is claimed to satisfy the PAC property. No equation reduces a fitted quantity to a prediction by construction, no load-bearing premise rests on self-citation, and no ansatz or uniqueness result is imported from the authors' prior work. The method is presented as a direct construction from sampled data and chosen barrier class; the central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Abstract-only; ledger populated from stated elements only. The central claim rests on the existence of suitable barrier functions and on the validity of scenario optimization for PAC guarantees, neither of which is derived here.

free parameters (1)

PAC confidence level and fraction
Prescribed by user; directly affects the LP constraints and the meaning of the output set.

axioms (1)

domain assumption Existence of a barrier function that certifies the PAC invariance property for the sampled data
Invoked when the method is said to 'leverage barrier functions' without further justification in the abstract.

invented entities (1)

PAC CSIS no independent evidence
purpose: Relaxed controlled safety invariant set that holds for a fraction of states with high probability
New definition introduced to make the problem tractable for black-box systems.

pith-pipeline@v0.9.0 · 5721 in / 1299 out tokens · 16348 ms · 2026-05-22T22:00:06.224440+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Our approach leverages barrier functions and scenario optimization, yielding a tractable linear programming method for estimating PAC CSISs.

What do these tags mean?

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

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Automatica 44(11), 2724–2734 (2008)

Abate, A., Prandini, M., Lygeros, J., Sastry, S.: Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica 44(11), 2724–2734 (2008)

work page 2008
[2]

In: Robotics: Science and Systems

Agrawal, A., Sreenath, K.: Discrete control barrier functions for safety-critical control of discrete systems with application to bipedal robot navigation. In: Robotics: Science and Systems. vol. 13, pp. 1–10. Cambridge, MA, USA (2017)

work page 2017
[3]

In: 2019 18th European control conference (ECC)

Ames, A.D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., Tabuada, P.: Control barrier functions: Theory and applications. In: 2019 18th European control conference (ECC). pp. 3420–3431. IEEE (2019)

work page 2019
[4]

Automatica35(11), 1747–1767 (1999)

Blanchini, F.: Set invariance in control. Automatica35(11), 1747–1767 (1999)

work page 1999
[5]

Automatica41(9), 1583–1589 (2005)

Bravo, J.M., Limón, D., Alamo, T., Camacho, E.F.: On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach. Automatica41(9), 1583–1589 (2005)

work page 2005
[6]

IEEE Transactions on automatic control51(5), 742–753 (2006)

Calafiore, G.C., Campi, M.C.: The scenario approach to robust control design. IEEE Transactions on automatic control51(5), 742–753 (2006)

work page 2006
[7]

Annual Reviews in Control33(2), 149–157 (2009)

Campi, M.C., Garatti, S., Prandini, M.: The scenario approach for systems and control design. Annual Reviews in Control33(2), 149–157 (2009)

work page 2009
[8]

In: 2018 IEEE Conference on Decision and Control (CDC)

Chakrabarty, A., Raghunathan, A., Di Cairano, S., Danielson, C.: Data-driven estimation of backward reachable and invariant sets for unmodeled systems via active learning. In: 2018 IEEE Conference on Decision and Control (CDC). pp. 372–377. IEEE (2018)

work page 2018
[9]

In: 2018 IEEE Conference on Decision and Control (CDC)

Chen, Y ., Peng, H., Grizzle, J., Ozay, N.: Data-driven computation of minimal robust control invariant set. In: 2018 IEEE Conference on Decision and Control (CDC). pp. 4052–4058. IEEE (2018)

work page 2018
[10]

In: Learning for dynamics and control

Devonport, A., Arcak, M.: Estimating reachable sets with scenario optimization. In: Learning for dynamics and control. pp. 75–84. PMLR (2020)

work page 2020
[11]

In: 6th Annual Learning for Dynamics & Control Conference

Dietrich, E., Devonport, A., Arcak, M.: Nonconvex scenario optimization for data-driven reachability. In: 6th Annual Learning for Dynamics & Control Conference. pp. 514–527. PMLR (2024)

work page 2024
[12]

In: Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control

Edwards, A., Peruffo, A., Abate, A.: Fossil 2.0: Formal certificate synthesis for the verification and control of dynamical models. In: Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control. pp. 1–10 (2024)

work page 2024
[13]

IEEE Transactions on Automatic Control59(2), 297–312 (2013)

Henrion, D., Korda, M.: Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control59(2), 297–312 (2013)

work page 2013
[14]

Advances in neural information processing systems32(2019)

Janner, M., Fu, J., Zhang, M., Levine, S.: When to trust your model: Model-based policy optimization. Advances in neural information processing systems32(2019)

work page 2019
[15]

In: Proceedings of the 39th IEEE conference on decision and control (Cat

Kerrigan, E.C., Maciejowski, J.M.: Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In: Proceedings of the 39th IEEE conference on decision and control (Cat. No. 00CH37187). vol. 5, pp. 4951–4956. IEEE (2000)

work page 2000
[16]

SIAM Journal on Control and Optimization58(5), 2871–2899 (2020)

Korda, M.: Computing controlled invariant sets from data using convex optimization. SIAM Journal on Control and Optimization58(5), 2871–2899 (2020)

work page 2020
[17]

In: 52nd IEEE Conference on Decision and Control

Korda, M., Henrion, D., Jones, C.N.: Convex computation of the maximum controlled invariant set for discrete-time polynomial control systems. In: 52nd IEEE Conference on Decision and Control. pp. 7107–7112. IEEE (2013)

work page 2013
[18]

IEEE Transactions on Automatic Control62(8), 4236–4242 (2017)

Le Mézo, T., Jaulin, L., Zerr, B.: An interval approach to compute invariant sets. IEEE Transactions on Automatic Control62(8), 4236–4242 (2017)

work page 2017
[19]

In: Proc

Lorenz, E.N.: Predictability: A problem partly solved. In: Proc. Seminar on predictability. vol. 1. Reading (1996)

work page 1996
[20]

IEEE Transactions on Automatic Control68(5), 3011–3024 (2023)

Nejati, A., Lavaei, A., Jagtap, P., Soudjani, S., Zamani, M.: Formal verification of unknown discrete-and continuous-time systems: A data-driven approach. IEEE Transactions on Automatic Control68(5), 3011–3024 (2023)

work page 2023
[21]

Automatica159, 111323 (2024)

Salamati, A., Lavaei, A., Soudjani, S., Zamani, M.: Data-driven verification and synthesis of stochastic systems via barrier certificates. Automatica159, 111323 (2024)

work page 2024
[22]

In: Learning for Dynamics and Control Conference

Salamati, A., Zamani, M.: Data-driven safety verification of stochastic systems via barrier certificates: A wait-and-judge approach. In: Learning for Dynamics and Control Conference. pp. 441–452. PMLR (2022)

work page 2022
[23]

IEEE Control Systems Letters7, 448–453 (2022)

Salamati, A., Zamani, M.: Safety verification of stochastic systems: A repetitive scenario approach. IEEE Control Systems Letters7, 448–453 (2022)

work page 2022
[24]

Automatica48(12), 3114– 3121 (2012)

Sassi, M.A.B., Girard, A.: Computation of polytopic invariants for polynomial dynamical systems using linear programming. Automatica48(12), 3114– 3121 (2012)

work page 2012
[25]

IFAC-PapersOnLine 56(2), 11229–11234 (2023)

Schmid, N., Lygeros, J.: Probabilistic reachability and invariance computation of stochastic systems using linear programming. IFAC-PapersOnLine 56(2), 11229–11234 (2023)

work page 2023
[26]

IEEE Transactions on Automatic Control53(2), 565–571 (2008)

Tan, W., Packard, A.: Stability region analysis using polynomial and composite polynomial lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control53(2), 565–571 (2008)

work page 2008
[27]

Communications of the ACM27(11), 1134–1142 (1984)

Valiant, L.G.: A theory of the learnable. Communications of the ACM27(11), 1134–1142 (1984)

work page 1984
[28]

IEEE control systems letters5(1), 193–198 (2020)

Wang, Z., Jungers, R.M.: Scenario-based set invariance verification for black-box nonlinear systems. IEEE control systems letters5(1), 193–198 (2020)

work page 2020
[29]

arXiv preprint arXiv:2408.15572 (2024) 11

Xue, B.: Sufficient and necessary barrier-like conditions for safety and reach-avoid verification of stochastic discrete-time systems. arXiv preprint arXiv:2408.15572 (2024) 11

work page arXiv 2024
[30]

IEEE Transactions on Automatic Control67(2), 1053–1060 (2021)

Xue, B., Zhan, N.: Robust invariant sets computation for discrete-time perturbed nonlinear systems. IEEE Transactions on Automatic Control67(2), 1053–1060 (2021)

work page 2021
[31]

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems39(11), 3944–3955 (2020) APPENDIX In the comparative experiment, each run was limited to 1 hour

Xue, B., Zhang, M., Easwaran, A., Li, Q.: Pac model checking of black-box continuous-time dynamical systems. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems39(11), 3944–3955 (2020) APPENDIX In the comparative experiment, each run was limited to 1 hour. The PCSIS parameters are given in Tab. III. For each example, the SOS meth...

work page 2020

[1] [1]

Automatica 44(11), 2724–2734 (2008)

Abate, A., Prandini, M., Lygeros, J., Sastry, S.: Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica 44(11), 2724–2734 (2008)

work page 2008

[2] [2]

In: Robotics: Science and Systems

Agrawal, A., Sreenath, K.: Discrete control barrier functions for safety-critical control of discrete systems with application to bipedal robot navigation. In: Robotics: Science and Systems. vol. 13, pp. 1–10. Cambridge, MA, USA (2017)

work page 2017

[3] [3]

In: 2019 18th European control conference (ECC)

Ames, A.D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., Tabuada, P.: Control barrier functions: Theory and applications. In: 2019 18th European control conference (ECC). pp. 3420–3431. IEEE (2019)

work page 2019

[4] [4]

Automatica35(11), 1747–1767 (1999)

Blanchini, F.: Set invariance in control. Automatica35(11), 1747–1767 (1999)

work page 1999

[5] [5]

Automatica41(9), 1583–1589 (2005)

Bravo, J.M., Limón, D., Alamo, T., Camacho, E.F.: On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach. Automatica41(9), 1583–1589 (2005)

work page 2005

[6] [6]

IEEE Transactions on automatic control51(5), 742–753 (2006)

Calafiore, G.C., Campi, M.C.: The scenario approach to robust control design. IEEE Transactions on automatic control51(5), 742–753 (2006)

work page 2006

[7] [7]

Annual Reviews in Control33(2), 149–157 (2009)

Campi, M.C., Garatti, S., Prandini, M.: The scenario approach for systems and control design. Annual Reviews in Control33(2), 149–157 (2009)

work page 2009

[8] [8]

In: 2018 IEEE Conference on Decision and Control (CDC)

Chakrabarty, A., Raghunathan, A., Di Cairano, S., Danielson, C.: Data-driven estimation of backward reachable and invariant sets for unmodeled systems via active learning. In: 2018 IEEE Conference on Decision and Control (CDC). pp. 372–377. IEEE (2018)

work page 2018

[9] [9]

In: 2018 IEEE Conference on Decision and Control (CDC)

Chen, Y ., Peng, H., Grizzle, J., Ozay, N.: Data-driven computation of minimal robust control invariant set. In: 2018 IEEE Conference on Decision and Control (CDC). pp. 4052–4058. IEEE (2018)

work page 2018

[10] [10]

In: Learning for dynamics and control

Devonport, A., Arcak, M.: Estimating reachable sets with scenario optimization. In: Learning for dynamics and control. pp. 75–84. PMLR (2020)

work page 2020

[11] [11]

In: 6th Annual Learning for Dynamics & Control Conference

Dietrich, E., Devonport, A., Arcak, M.: Nonconvex scenario optimization for data-driven reachability. In: 6th Annual Learning for Dynamics & Control Conference. pp. 514–527. PMLR (2024)

work page 2024

[12] [12]

In: Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control

Edwards, A., Peruffo, A., Abate, A.: Fossil 2.0: Formal certificate synthesis for the verification and control of dynamical models. In: Proceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control. pp. 1–10 (2024)

work page 2024

[13] [13]

IEEE Transactions on Automatic Control59(2), 297–312 (2013)

Henrion, D., Korda, M.: Convex computation of the region of attraction of polynomial control systems. IEEE Transactions on Automatic Control59(2), 297–312 (2013)

work page 2013

[14] [14]

Advances in neural information processing systems32(2019)

Janner, M., Fu, J., Zhang, M., Levine, S.: When to trust your model: Model-based policy optimization. Advances in neural information processing systems32(2019)

work page 2019

[15] [15]

In: Proceedings of the 39th IEEE conference on decision and control (Cat

Kerrigan, E.C., Maciejowski, J.M.: Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In: Proceedings of the 39th IEEE conference on decision and control (Cat. No. 00CH37187). vol. 5, pp. 4951–4956. IEEE (2000)

work page 2000

[16] [16]

SIAM Journal on Control and Optimization58(5), 2871–2899 (2020)

Korda, M.: Computing controlled invariant sets from data using convex optimization. SIAM Journal on Control and Optimization58(5), 2871–2899 (2020)

work page 2020

[17] [17]

In: 52nd IEEE Conference on Decision and Control

Korda, M., Henrion, D., Jones, C.N.: Convex computation of the maximum controlled invariant set for discrete-time polynomial control systems. In: 52nd IEEE Conference on Decision and Control. pp. 7107–7112. IEEE (2013)

work page 2013

[18] [18]

IEEE Transactions on Automatic Control62(8), 4236–4242 (2017)

Le Mézo, T., Jaulin, L., Zerr, B.: An interval approach to compute invariant sets. IEEE Transactions on Automatic Control62(8), 4236–4242 (2017)

work page 2017

[19] [19]

In: Proc

Lorenz, E.N.: Predictability: A problem partly solved. In: Proc. Seminar on predictability. vol. 1. Reading (1996)

work page 1996

[20] [20]

IEEE Transactions on Automatic Control68(5), 3011–3024 (2023)

Nejati, A., Lavaei, A., Jagtap, P., Soudjani, S., Zamani, M.: Formal verification of unknown discrete-and continuous-time systems: A data-driven approach. IEEE Transactions on Automatic Control68(5), 3011–3024 (2023)

work page 2023

[21] [21]

Automatica159, 111323 (2024)

Salamati, A., Lavaei, A., Soudjani, S., Zamani, M.: Data-driven verification and synthesis of stochastic systems via barrier certificates. Automatica159, 111323 (2024)

work page 2024

[22] [22]

In: Learning for Dynamics and Control Conference

Salamati, A., Zamani, M.: Data-driven safety verification of stochastic systems via barrier certificates: A wait-and-judge approach. In: Learning for Dynamics and Control Conference. pp. 441–452. PMLR (2022)

work page 2022

[23] [23]

IEEE Control Systems Letters7, 448–453 (2022)

Salamati, A., Zamani, M.: Safety verification of stochastic systems: A repetitive scenario approach. IEEE Control Systems Letters7, 448–453 (2022)

work page 2022

[24] [24]

Automatica48(12), 3114– 3121 (2012)

Sassi, M.A.B., Girard, A.: Computation of polytopic invariants for polynomial dynamical systems using linear programming. Automatica48(12), 3114– 3121 (2012)

work page 2012

[25] [25]

IFAC-PapersOnLine 56(2), 11229–11234 (2023)

Schmid, N., Lygeros, J.: Probabilistic reachability and invariance computation of stochastic systems using linear programming. IFAC-PapersOnLine 56(2), 11229–11234 (2023)

work page 2023

[26] [26]

IEEE Transactions on Automatic Control53(2), 565–571 (2008)

Tan, W., Packard, A.: Stability region analysis using polynomial and composite polynomial lyapunov functions and sum-of-squares programming. IEEE Transactions on Automatic Control53(2), 565–571 (2008)

work page 2008

[27] [27]

Communications of the ACM27(11), 1134–1142 (1984)

Valiant, L.G.: A theory of the learnable. Communications of the ACM27(11), 1134–1142 (1984)

work page 1984

[28] [28]

IEEE control systems letters5(1), 193–198 (2020)

Wang, Z., Jungers, R.M.: Scenario-based set invariance verification for black-box nonlinear systems. IEEE control systems letters5(1), 193–198 (2020)

work page 2020

[29] [29]

arXiv preprint arXiv:2408.15572 (2024) 11

Xue, B.: Sufficient and necessary barrier-like conditions for safety and reach-avoid verification of stochastic discrete-time systems. arXiv preprint arXiv:2408.15572 (2024) 11

work page arXiv 2024

[30] [30]

IEEE Transactions on Automatic Control67(2), 1053–1060 (2021)

Xue, B., Zhan, N.: Robust invariant sets computation for discrete-time perturbed nonlinear systems. IEEE Transactions on Automatic Control67(2), 1053–1060 (2021)

work page 2021

[31] [31]

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems39(11), 3944–3955 (2020) APPENDIX In the comparative experiment, each run was limited to 1 hour

Xue, B., Zhang, M., Easwaran, A., Li, Q.: Pac model checking of black-box continuous-time dynamical systems. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems39(11), 3944–3955 (2020) APPENDIX In the comparative experiment, each run was limited to 1 hour. The PCSIS parameters are given in Tab. III. For each example, the SOS meth...

work page 2020