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Numerically-Robust Inductive Proof Rules for Continuous Dynamical Systems [chapter]

Sicun Gao, James Kapinski, Jyotirmoy Deshmukh, Nima Roohi, Armando Solar-Lezama, Nikos Arechiga, Soonho Kong
2019 Lecture Notes in Computer Science  
We formulate numerically-robust inductive proof rules for unbounded stability and safety properties of continuous dynamical systems.  ...  In this way, numerically-driven decision procedures can establish a sound and relative-complete proof system for unbounded properties of very general nonlinear systems.  ...  In this paper, we formulate new inductive proof rules for continuous dynamical systems for establishing robust notions of stability and safety.  ... 
doi:10.1007/978-3-030-25543-5_9 fatcat:33dacwjf7jgonlcsjxdezghrbi

Controller Synthesis with Inductive Proofs for Piecewise Linear Systems: an SMT-based Algorithm [article]

Zhenqi Huang, Yu Wang, Sayan Mitra, Geir E. Dullerud, Swarat Chaudhuri
2015 arXiv   pre-print
We present a controller synthesis algorithm for reach-avoid problems for piecewise linear discrete-time systems.  ...  Our algorithm generates feedback control laws together with inductive proofs of unbounded time safety and progress properties with respect to the reach-avoid sets.  ...  The evolution of the continuous state of the system is governed by the dynamic function of the location invariant it is currently in.  ... 
arXiv:1509.04623v1 fatcat:5ygga4qnt5cgzbpal6cohmyidi

Computing differential invariants of hybrid systems as fixedpoints

André Platzer, Edmund M. Clarke
2009 Formal methods in system design  
In order to verify non-trivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required  ...  By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure.  ...  We thank Silke Wagner and Alex Donzé for their helpful proofreading remarks.  ... 
doi:10.1007/s10703-009-0079-8 fatcat:s3lrcobqn5dgbkxnh43e6o5ddq

Continuity and robustness of programs

Swarat Chaudhuri, Sumit Gulwani, Roberto Lublinerman
2012 Communications of the ACM  
Computer scientists have long believed that software is different from physical systems in one fundamental way: while the latter have continuous dynamics, the former do not.  ...  In this paper, we argue that notions of continuity from mathematical analysis are relevant and interesting even for software.  ...  Acknowledgments This research was supported by NSF CAREER Award #1156059 ("Robustness Analysis of Uncertain Programs: Theory, Algorithms, and Tools").  ... 
doi:10.1145/2240236.2240262 fatcat:wujtqll5v5gn5h3h7oc4h7itte

On Numerical Integration in Neural Ordinary Differential Equations [article]

Aiqing Zhu, Pengzhan Jin, Beibei Zhu, Yifa Tang
2022 arXiv   pre-print
However, deciphering the numerical integration in Neural ODE is still an open challenge, as many researches demonstrated that numerical integration significantly affects the performance of the model.  ...  Several experiments are performed to numerically verify our theoretical analysis.  ...  Acknowledgements The authors thank the anonymous reviewers for their valuable comments.  ... 
arXiv:2206.07335v1 fatcat:ecbplu3otngb3ccv55e5xmosma

Differential Dynamic Logic for Hybrid Systems

André Platzer
2008 Journal of automated reasoning  
Hybrid systems are models for complex physical systems and are defined as dynamical systems with interacting discrete transitions and continuous evolutions along differential equations.  ...  As a common mathematical model for complex physical systems, hybrid systems are dynamical systems [46] where the system state evolves over time according to interacting laws of discrete and continuous  ...  Acknowledgements I am grateful to the anonymous referees for their most valuable and helpful comments. I also thank Ernst-Rüdiger Olderog and Andreas Schäfer for their proofreading remarks.  ... 
doi:10.1007/s10817-008-9103-8 fatcat:rr2zob6u6jdnzftqjysgybtvyq

ROBUST ASYMPTOTIC CONTROLLABILITY UNDER TIME-VARYING PERTURBATIONS

LARS GRÜNE
2004 Stochastics and Dynamics  
We investigate the effect of time-varying perturbations on the dynamical behavior of nonlinear control systems.  ...  In the second part we introduce appropriate robustness properties and prove that these are inherent properties for asymptotically controllable sets under these classes of perturbations.  ...  Acknowledgment: I would like to thank Tony Michel for information about the history of K and KL functions.  ... 
doi:10.1142/s0219493704001085 fatcat:j6g3jkafbfhkbdeqyu3435hrmy

Modular deductive verification of sampled-data systems

Daniel Ricketts, Gregory Malecha, Sorin Lerner
2016 Proceedings of the 13th International Conference on Embedded Software - EMSOFT '16  
systems for which full automation is possible, and (3) provides a platform for proving powerful, higher-order modularity theorems that are crucial for scaling verification to complex systems.  ...  Unsafe behavior of cyber-physical systems can have disastrous consequences, motivating the need for formal verification of these kinds of systems.  ...  In between executions of the controller, the system evolves according to continuous physical dynamics.  ... 
doi:10.1145/2968478.2968495 dblp:conf/emsoft/RickettsML16 fatcat:ok5hflbneza2nnljxhfpqobk4y

Continuity analysis of programs

Swarat Chaudhuri, Sumit Gulwani, Roberto Lublinerman
2010 SIGPLAN notices  
Our proof rules identify appropriate "synchronization points" between executions and their perturbed counterparts, and establish that values of certain variables converge back to the original results in  ...  However, it is difficult to apply continuity proofs from real analysis to functions that are coded as imperative programs, especially when they use diverse data types and features such as assignments,  ...  Questions of continuity and robustness appear routinely in the literature on dynamical and hybrid systems [16, 17] .  ... 
doi:10.1145/1707801.1706308 fatcat:6atnk6elzrelrd5ueh3hvvmubm

Continuity analysis of programs

Swarat Chaudhuri, Sumit Gulwani, Roberto Lublinerman
2010 Proceedings of the 37th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '10  
Our proof rules identify appropriate "synchronization points" between executions and their perturbed counterparts, and establish that values of certain variables converge back to the original results in  ...  However, it is difficult to apply continuity proofs from real analysis to functions that are coded as imperative programs, especially when they use diverse data types and features such as assignments,  ...  Questions of continuity and robustness appear routinely in the literature on dynamical and hybrid systems [16, 17] .  ... 
doi:10.1145/1706299.1706308 dblp:conf/popl/ChaudhuriGL10 fatcat:2dm66k6vpjglbcnsok63cwh4xq

Continuous opinions and discrete actions in social networks: a multi-agent system approach [article]

N. R. Chowdhury, I.-C. Morarescu, S. Martin, S. Srikant
2016 arXiv   pre-print
Numerical examples illustrate the main features of this model.  ...  This paper proposes and analyzes a novel multi-agent opinion dynamics model in which agents have access to actions which are quantized version of the opinions of their neighbors.  ...  Martin was funded by the ANR project COMPACS -"Computation Aware Control Systems", ANR-13-BS03-004. I.-C. Morȃrescu and S.  ... 
arXiv:1602.02098v1 fatcat:bjyw7ho4bvdlbjhc42dk7ahdjy

Continuous opinions and discrete actions in social networks: A multi-agent system approach

N. R. Chowdhury, I.-C. Morarescu, S. Martin, S. Srikant
2016 2016 IEEE 55th Conference on Decision and Control (CDC)  
Numerical examples illustrate the main features of this model.  ...  This paper proposes and analyzes a novel multiagent opinion dynamics model in which agents have access to actions which are quantized version of the opinions of their neighbors.  ...  Proof: The proof will be done by induction. Let us remark first that following the first item of Definition 6 we have q i (0) = 0, ∀i ∈ A.  ... 
doi:10.1109/cdc.2016.7798516 dblp:conf/cdc/ChowdhuryMMS16 fatcat:gjw4wvo6bjfthoi5ezoz4yorsm

Best experienced payoff dynamics and cooperation in the centipede game

William H. Sandholm, Segismundo S. Izquierdo, Luis R. Izquierdo
2019 Theoretical Economics  
In the centipede game, these best experienced payoff dynamics lead to cooperative play.  ...  We study population game dynamics under which each revising agent tests each of his strategies a fixed number of times, with each play of each strategy being against a newly drawn opponent, and chooses  ...  Combining these facts with fundamental results from dynamical systems theory lets us complete the proof that ξ † is a repellor.  ... 
doi:10.3982/te3565 fatcat:thsaqxs2v5ge3mgfctiwugizq4

Stability of Controllers for Gaussian Process Dynamics

Julia Vinogradska, Bastian Bischoff, Duy Nguyen-Tuong, Jan Peters
2017 Journal of machine learning research  
We consider differentiable Markovian control policies and system dynamics given as (i) the mean of a GP, and (ii) the full GP distribution.  ...  As a step towards widespread deployment of learning control, we provide stability analysis tools for controllers acting on dynamics represented by Gaussian processes (GPs).  ...  By induction, x (t) ∈ X c for all t > 0, if x (0) ∈ X c and Equation ( 6 ) holds for all t.  ... 
dblp:journals/jmlr/VinogradskaBNP17 fatcat:cqv7hatfmze57czalrh66s4jge

Switching Rules for Stabilization of Linear Systems of Conservation Laws

Pierre-Olivier Lamare, Antoine Girard, Christophe Prieur
2015 SIAM Journal of Control and Optimization  
Moreover, we consider the robustness issues for these switching rules in presence of measurement noise. Some numerical examples illustrate our approach and show the merits of the proposed strategies.  ...  In this paper, the exponential convergence in L 2 -norm is analyzed for a class of switched linear systems of conservation laws. The boundary conditions are subject to switches.  ...  In these cases, a switched system occurs: a system where there is both a continuous dynamics (given by partial differential equation) and a discrete dynamics (given by a switching rule).  ... 
doi:10.1137/140953952 fatcat:xv3txvs7dffclndxlqf63lo5ly
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