Pub Date : 2025-10-16DOI: 10.1016/j.nahs.2025.101652
Yifu Feng , Zonghai Hu , Jiancun Wu , Engang Tian
Remote state estimation systems are vulnerable to malicious attacks, particularly when operating over unreliable communication networks with nonlinear dynamics. This paper proposes a stochastic critical-packet-based (SCPB) attack strategy that selectively targets packets according to their impact on the estimation process. A criticality indicator is defined as the difference between prior and posterior local estimates. Based on this indicator, a probabilistic attack mechanism is designed in which the attack success probability (ASP) increases with both packet importance and the assigned attack strength. To enhance the effectiveness of the attack, an adaptive energy adjustment scheme is introduced, assigning higher attack intensity to more critical packets. Theoretical analysis establishes explicit relationships among packet criticality, attack strength, ASP, and estimation performance degradation. Simulation results demonstrate that the proposed strategy induces significantly larger estimation errors than baseline attacks under the same average attack probability.
{"title":"Stochastic critical-packet-based attack on remote state estimation: From attackers’ perspective","authors":"Yifu Feng , Zonghai Hu , Jiancun Wu , Engang Tian","doi":"10.1016/j.nahs.2025.101652","DOIUrl":"10.1016/j.nahs.2025.101652","url":null,"abstract":"<div><div>Remote state estimation systems are vulnerable to malicious attacks, particularly when operating over unreliable communication networks with nonlinear dynamics. This paper proposes a stochastic critical-packet-based (SCPB) attack strategy that selectively targets packets according to their impact on the estimation process. A criticality indicator is defined as the difference between prior and posterior local estimates. Based on this indicator, a probabilistic attack mechanism is designed in which the attack success probability (ASP) increases with both packet importance and the assigned attack strength. To enhance the effectiveness of the attack, an adaptive energy adjustment scheme is introduced, assigning higher attack intensity to more critical packets. Theoretical analysis establishes explicit relationships among packet criticality, attack strength, ASP, and estimation performance degradation. Simulation results demonstrate that the proposed strategy induces significantly larger estimation errors than baseline attacks under the same average attack probability.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101652"},"PeriodicalIF":3.7,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-11DOI: 10.1016/j.nahs.2025.101651
Roberto M. Fuentes , Gabriela W. Gabriel , André M. de Oliveira , Jonathan M. Palma
This paper studies the reference-tracking control problem for uncertain sampled-data systems with partial observation of the states. We show that, by robustly stabilizing the closed-loop system, the proposed digital controller ensures that the controlled output tracks constant set points. Moreover, new design conditions given in terms of Differential Linear Matrix Inequalities are presented for obtaining controllers with partial and complete measurements of the states. The design conditions are illustrated by numerical examples.
{"title":"Constant reference tracking control for uncertain linear sampled-data systems","authors":"Roberto M. Fuentes , Gabriela W. Gabriel , André M. de Oliveira , Jonathan M. Palma","doi":"10.1016/j.nahs.2025.101651","DOIUrl":"10.1016/j.nahs.2025.101651","url":null,"abstract":"<div><div>This paper studies the reference-tracking control problem for uncertain sampled-data systems with partial observation of the states. We show that, by robustly stabilizing the closed-loop system, the proposed digital controller ensures that the controlled output tracks constant set points. Moreover, new design conditions given in terms of Differential Linear Matrix Inequalities are presented for obtaining controllers with partial and complete measurements of the states. The design conditions are illustrated by numerical examples.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101651"},"PeriodicalIF":3.7,"publicationDate":"2025-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-11DOI: 10.1016/j.nahs.2025.101648
Wei Ren
This paper addresses the state estimation problem of multi-agent systems under multiple networks and event-triggered mechanisms. Multiple networks are to accommodate physical separation of different agents, while event-triggered mechanisms are to address the resource constraints of networks. This setting results in the distributed structure and even the simultaneous design of distributed observers and event-triggered mechanisms. We follow the emulation-based approach to propose a novel hybrid model for the state estimation of multi-agent systems. Both Lyapunov-based conditions and distributed event-triggered mechanisms are established such that the estimation errors are convergent asymptotically. The proposed approach is further implemented to deal with the distributed observer-based controller design problem of multi-agent systems. Both the general framework is established and the conditions in the form of linear matrix inequalities are derived. Finally, the derived results are illustrated via two numerical examples from inverted pendulum robots and power systems.
{"title":"Distributed event-triggered observer design for networked multi-agent systems","authors":"Wei Ren","doi":"10.1016/j.nahs.2025.101648","DOIUrl":"10.1016/j.nahs.2025.101648","url":null,"abstract":"<div><div>This paper addresses the state estimation problem of multi-agent systems under multiple networks and event-triggered mechanisms. Multiple networks are to accommodate physical separation of different agents, while event-triggered mechanisms are to address the resource constraints of networks. This setting results in the distributed structure and even the simultaneous design of distributed observers and event-triggered mechanisms. We follow the emulation-based approach to propose a novel hybrid model for the state estimation of multi-agent systems. Both Lyapunov-based conditions and distributed event-triggered mechanisms are established such that the estimation errors are convergent asymptotically. The proposed approach is further implemented to deal with the distributed observer-based controller design problem of multi-agent systems. Both the general framework is established and the conditions in the form of linear matrix inequalities are derived. Finally, the derived results are illustrated via two numerical examples from inverted pendulum robots and power systems.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101648"},"PeriodicalIF":3.7,"publicationDate":"2025-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-07DOI: 10.1016/j.nahs.2025.101629
Eric Goubault, Sylvie Putot
We propose an approach for computing inner and outer-approximations of the sets of values that satisfy constraints expressed as arbitrarily quantified formulas. Such formulas arise for example when specifying important problems in control such as robustness, motion planning and controller comparison. We propose an interval-based method which allows for tight but tractable approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation. Finally, we develop higher-order methods, particularly tractable order one methods which provide even tighter results.
{"title":"Inner and outer approximations of arbitrarily quantified reachability problems","authors":"Eric Goubault, Sylvie Putot","doi":"10.1016/j.nahs.2025.101629","DOIUrl":"10.1016/j.nahs.2025.101629","url":null,"abstract":"<div><div>We propose an approach for computing inner and outer-approximations of the sets of values that satisfy constraints expressed as arbitrarily quantified formulas. Such formulas arise for example when specifying important problems in control such as robustness, motion planning and controller comparison. We propose an interval-based method which allows for tight but tractable approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation. Finally, we develop higher-order methods, particularly tractable order one methods which provide even tighter results.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101629"},"PeriodicalIF":3.7,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-30DOI: 10.1016/j.nahs.2025.101644
Henk A.P. Blom
In MJLS literature the separation principle between filtering and control has been established in case the Markov mode switching process is fully observed, and the Euclidean state process is partially observed. In case the exact remains hidden, the separation principle has only been established under a linear filtering restriction. Since nonlinear filters can provide significant better estimates, the desire to extend the separation principle to MJLS with hidden is a long-standing challenge. The objective of this paper is to resolve this long-standing challenge in three steps. The first step is to transform the MJLS stochastic control problem into control under a quadratic performance criterion of a linear system driven by a martingale which is influenced by the control. The certainty equivalence (CE) condition known in literature applies to stochastic control of a linear system that is driven by a control independent martingale. Therefore, the second step is to relax this known CE condition such that it allows this control influence on the martingale. The third step is to prove that the relaxed CE condition is satisfied for the general MJLS control problem considered. The overall achievement is a CE control law for a partially observed MJLS, which assures the Separation Principle between filtering and control. The paper also shows that for the case that is fully observed and the exactremains hidden, that the novel CE control law differs significantly from the in literature well-developed Averaging MJLS control policy.
{"title":"Separation principle for stochastic control of continuous-time Markov jump linear systems under partial observations","authors":"Henk A.P. Blom","doi":"10.1016/j.nahs.2025.101644","DOIUrl":"10.1016/j.nahs.2025.101644","url":null,"abstract":"<div><div>In MJLS literature the separation principle between filtering and control has been established in case the Markov mode switching process <span><math><mrow><mo>{</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span> is fully observed, and the Euclidean state process <span><math><mrow><mo>{</mo><msub><mi>x</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span>is partially observed. In case the exact <span><math><mrow><mo>{</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span>remains hidden, the separation principle has only been established under a linear filtering restriction. Since nonlinear filters can provide significant better estimates, the desire to extend the separation principle to MJLS with hidden <span><math><mrow><mo>{</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span>is a long-standing challenge. The objective of this paper is to resolve this long-standing challenge in three steps. The first step is to transform the MJLS stochastic control problem into control under a quadratic performance criterion of a linear system driven by a martingale which is influenced by the control. The certainty equivalence (CE) condition known in literature applies to stochastic control of a linear system that is driven by a control independent martingale. Therefore, the second step is to relax this known CE condition such that it allows this control influence on the martingale. The third step is to prove that the relaxed CE condition is satisfied for the general MJLS control problem considered. The overall achievement is a CE control law for a partially observed MJLS, which assures the Separation Principle between filtering and control. The paper also shows that for the case that <span><math><mrow><mo>{</mo><msub><mi>x</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span>is fully observed and the exact<span><math><mrow><mo>{</mo><msub><mi>θ</mi><mi>t</mi></msub><mo>}</mo></mrow></math></span>remains hidden, that the novel CE control law differs significantly from the in literature well-developed Averaging MJLS control policy.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101644"},"PeriodicalIF":3.7,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-26DOI: 10.1016/j.nahs.2025.101647
Hoang Thi Duyen , Ky Quan Tran
This paper investigates -exponential stability – also known as exponential stability in the mean square – of neutral stochastic delay differential equations with impulsive perturbations. Our primary objective is to stabilize an impulsive-free system by appropriately designing impulsive controls. Unlike previous studies, we introduce new and verifiable criteria for -exponential stability. We further demonstrate that Euler–Maruyama-type approximations preserve -exponential stability provided that the step sizes are sufficiently small; explicit conditions on these step sizes are derived. Moreover, we detail the design of impulsive perturbations that achieve -exponential stabilization. Two examples are presented to validate the effectiveness of our criteria.
{"title":"L2-exponential stability and impulsive stabilization of neutral stochastic delay differential equations","authors":"Hoang Thi Duyen , Ky Quan Tran","doi":"10.1016/j.nahs.2025.101647","DOIUrl":"10.1016/j.nahs.2025.101647","url":null,"abstract":"<div><div>This paper investigates <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-exponential stability – also known as exponential stability in the mean square – of neutral stochastic delay differential equations with impulsive perturbations. Our primary objective is to stabilize an impulsive-free system by appropriately designing impulsive controls. Unlike previous studies, we introduce new and verifiable criteria for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-exponential stability. We further demonstrate that Euler–Maruyama-type approximations preserve <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-exponential stability provided that the step sizes are sufficiently small; explicit conditions on these step sizes are derived. Moreover, we detail the design of impulsive perturbations that achieve <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-exponential stabilization. Two examples are presented to validate the effectiveness of our criteria.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101647"},"PeriodicalIF":3.7,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-18DOI: 10.1016/j.nahs.2025.101642
Xiaoyun Wei , Xingwen Liu , Jun Yang , Tingjin Liu
This paper focuses on exponential stability analysis for discrete-time almost periodic piecewise nonlinear systems (APPNSs) with uncertain dwell time of subsystems. A discrete-time APPNS has a fundamental period, during which a finite number of subsystems that constitute the system are cyclically activated. Such systems can be modeled as switched systems with cyclically switching signals. With the assumption that the vector field of each subsystem of discrete-time APPNSs is continuously differentiable, a Lyapunov theorem is presented first to verify the exponential stability of discrete-time APPNSs. Then, a linearization method is employed and a mixed-mode time-varying homogeneous Lyapunov function is constructed to derive specific stability conditions expressed by linear matrix inequalities (LMIs). Note that this condition can verify the exponential stability of the considered nonlinear systems, as well as that of the corresponding linearized systems. Furthermore, the linearization method used here can be applied to general switched systems.
{"title":"Exponential stability analysis of discrete-time almost periodic piecewise nonlinear systems","authors":"Xiaoyun Wei , Xingwen Liu , Jun Yang , Tingjin Liu","doi":"10.1016/j.nahs.2025.101642","DOIUrl":"10.1016/j.nahs.2025.101642","url":null,"abstract":"<div><div>This paper focuses on exponential stability analysis for discrete-time almost periodic piecewise nonlinear systems (APPNSs) with uncertain dwell time of subsystems. A discrete-time APPNS has a fundamental period, during which a finite number of subsystems that constitute the system are cyclically activated. Such systems can be modeled as switched systems with cyclically switching signals. With the assumption that the vector field of each subsystem of discrete-time APPNSs is continuously differentiable, a Lyapunov theorem is presented first to verify the exponential stability of discrete-time APPNSs. Then, a linearization method is employed and a mixed-mode time-varying homogeneous Lyapunov function is constructed to derive specific stability conditions expressed by linear matrix inequalities (LMIs). Note that this condition can verify the exponential stability of the considered nonlinear systems, as well as that of the corresponding linearized systems. Furthermore, the linearization method used here can be applied to general switched systems.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101642"},"PeriodicalIF":3.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-18DOI: 10.1016/j.nahs.2025.101643
Bhim Kumar, Muslim Malik
This paper investigates the finite-time stability and stabilization of impulsive positive switched delay systems defined on arbitrary time scales, using Lyapunov function-based methods. The finite-time stabilization problem is solved using state feedback controllers on time scales. Furthermore, the applicability of the proposed finite-time stability criteria is demonstrated for positive switched multi-agent delay systems by designing an event-triggered controller on time scales. The analysis extends the existing results to incorporate hybrid time domains with time-dependent switching and provide sufficient conditions for the system’s stability and stabilization within the specified time period. To demonstrate the results of this paper, several numerical and simulation-based examples are presented by considering continuous, discrete, and hybrid time domains simultaneously.
{"title":"Finite-time stability and stabilization of positive switched delay systems with non-instantaneous impulses on time scales and applications to multi-agent systems","authors":"Bhim Kumar, Muslim Malik","doi":"10.1016/j.nahs.2025.101643","DOIUrl":"10.1016/j.nahs.2025.101643","url":null,"abstract":"<div><div>This paper investigates the finite-time stability and stabilization of impulsive positive switched delay systems defined on arbitrary time scales, using Lyapunov function-based methods. The finite-time stabilization problem is solved using state feedback controllers on time scales. Furthermore, the applicability of the proposed finite-time stability criteria is demonstrated for positive switched multi-agent delay systems by designing an event-triggered controller on time scales. The analysis extends the existing results to incorporate hybrid time domains with time-dependent switching and provide sufficient conditions for the system’s stability and stabilization within the specified time period. To demonstrate the results of this paper, several numerical and simulation-based examples are presented by considering continuous, discrete, and hybrid time domains simultaneously.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101643"},"PeriodicalIF":3.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-12DOI: 10.1016/j.nahs.2025.101632
Mengyao Lu, Jingyuan Zhan, Liguo Zhang
This study addresses the boundary consensus problem of multi-agent systems, in which each agent is governed by hyperbolic partial differential equations. To reduce the controller update frequency, the event-triggered boundary controller is proposed for each agent based on the boundary state information of its neighbors and itself. Firstly, we present the well-posedness and consensus analysis under undirected communication graphs. By using Lyapunov approach and graph theory, we obtain the sufficient conditions associated with triggering parameters, Laplacian eigenvalues and boundary coefficient matrices for guaranteeing the asymptotic consensus and excluding Zeno behaviors. Next, we further consider the consensus problem under the directed communication topologies, and we also establish the sufficient conditions for ensuring the asymptotic consensus. Finally, we present a three-lane freeway traffic flow system described by Aw–Rascle–Zhang Equations as an example to demonstrate the effectiveness of the designed event-triggered boundary consensus controllers.
{"title":"Event-triggered boundary consensus for multi-agent systems of hyperbolic balance laws","authors":"Mengyao Lu, Jingyuan Zhan, Liguo Zhang","doi":"10.1016/j.nahs.2025.101632","DOIUrl":"10.1016/j.nahs.2025.101632","url":null,"abstract":"<div><div>This study addresses the boundary consensus problem of multi-agent systems, in which each agent is governed by hyperbolic partial differential equations. To reduce the controller update frequency, the event-triggered boundary controller is proposed for each agent based on the boundary state information of its neighbors and itself. Firstly, we present the well-posedness and consensus analysis under undirected communication graphs. By using Lyapunov approach and graph theory, we obtain the sufficient conditions associated with triggering parameters, Laplacian eigenvalues and boundary coefficient matrices for guaranteeing the asymptotic consensus and excluding Zeno behaviors. Next, we further consider the consensus problem under the directed communication topologies, and we also establish the sufficient conditions for ensuring the asymptotic consensus. Finally, we present a three-lane freeway traffic flow system described by Aw–Rascle–Zhang Equations as an example to demonstrate the effectiveness of the designed event-triggered boundary consensus controllers.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101632"},"PeriodicalIF":3.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Deep neural networks (DNN) verification primarily focuses on qualitative verification, determining whether a DNN violates safety or robustness properties. This paper introduces a novel approach for quantitative verification of Feedforward Neural Networks (FFNN), transforming qualitative assessments into probabilistic evaluations. The resulting quantitative verification method not only can answer YES or NO questions but also can compute the probability of a property being violated. To do that, we introduce the concept of a probabilistic star (or shortly ProbStar), a new variant of the well-known star set, in which the predicate variables belong to a Gaussian distribution. We further propose an approach to compute the probability of a probabilistic star in high-dimensional space. Unlike existing works dealing with constrained input sets, our work considers the input set as a truncated multivariate normal (Gaussian) distribution, i.e., besides the constraints on the input variables, the input set has a probability of the constraints being satisfied. The input distribution is represented as a probabilistic star set and propagates through a network to construct the output reachable set containing multiple ProbStars, which are used to verify the safety or robustness properties of the network. In case a property is violated, the violation probability can be computed precisely by an exact verification algorithm or approximately by an over-approximate verification algorithm. Building on this foundation, we extend our quantitative verification framework to Learning-Enabled Cyber-Physical Systems (Le-CPS), where a piecewise linear FFNN controls a linear physical plant model. Our approach enables the construction of probabilistic reachable sets for Le-CPS, allowing for both qualitative Safe/Unsafe assessments and quantitative probability computations of property violations. We have implemented our verification framework in a tool named StarV and evaluated its effectiveness on benchmarks including HorizontalCAS and ACASXu networks, a rocket landing system, as well as advanced emergency braking and adaptive cruise control systems within the Le-CPS domain.
{"title":"Quantitative verification of learning-enabled systems using ProbStar reachability","authors":"Yuntao Li , Sung Woo Choi , Hideki Okamoto , Bardh Hoxha , Georgios Fainekos , Hoang-Dung Tran","doi":"10.1016/j.nahs.2025.101623","DOIUrl":"10.1016/j.nahs.2025.101623","url":null,"abstract":"<div><div>Deep neural networks (DNN) verification primarily focuses on <em>qualitative verification</em>, determining whether a DNN violates safety or robustness properties. This paper introduces a novel approach for <em>quantitative verification</em> of Feedforward Neural Networks (FFNN), transforming qualitative assessments into probabilistic evaluations. The resulting quantitative verification method not only can answer YES or NO questions but also can compute the probability of a property being violated. To do that, we introduce the concept of a probabilistic star (or shortly ProbStar), a new variant of the well-known star set, in which the predicate variables belong to a Gaussian distribution. We further propose an approach to compute the probability of a probabilistic star in high-dimensional space. Unlike existing works dealing with constrained input sets, our work considers the input set as a truncated multivariate normal (Gaussian) distribution, i.e., besides the constraints on the input variables, the input set has a probability of the constraints being satisfied. The input distribution is represented as a probabilistic star set and propagates through a network to construct the output reachable set containing multiple ProbStars, which are used to verify the safety or robustness properties of the network. In case a property is violated, the violation probability can be computed precisely by an exact verification algorithm or approximately by an over-approximate verification algorithm. Building on this foundation, we extend our quantitative verification framework to Learning-Enabled Cyber-Physical Systems (Le-CPS), where a piecewise linear FFNN controls a linear physical plant model. Our approach enables the construction of probabilistic reachable sets for Le-CPS, allowing for both qualitative Safe/Unsafe assessments and quantitative probability computations of property violations. We have implemented our verification framework in a tool named <em>StarV</em> and evaluated its effectiveness on benchmarks including HorizontalCAS and ACASXu networks, a rocket landing system, as well as advanced emergency braking and adaptive cruise control systems within the Le-CPS domain.</div></div>","PeriodicalId":49011,"journal":{"name":"Nonlinear Analysis-Hybrid Systems","volume":"59 ","pages":"Article 101623"},"PeriodicalIF":3.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145046682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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