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1. Coordinating distributed MPC efficiently on a plantwide scale: The Lyapunov envelope algorithm

   https://doi.org/10.1016/j.compchemeng.2021.107532  

   Tang, Wentao ; Daoutidis, Prodromos ( December 2021 , Computers & Chemical Engineering)
2. Deep Incremental RNN for Learning Sequential Data: A Lyapunov Stable Dynamical System

   https://doi.org/10.1109/ICDM51629.2021.00108  

   Zhang, Ziming ; Wu, Guojun ; Li, Yanhua ; Yue, Yun ; Zhou, Xun ( December 2021 , 2020 IEEE International Conference on Data Mining (ICDM))
3. Out-of-time-order correlators and Lyapunov exponents in sparse SYK

   https://doi.org/10.1007/JHEP11(2023)088  

   Cáceres, Elena ; Guglielmo, Tyler ; Kent, Brian ; Misobuchi, Anderson ( November 2023 , Journal of High Energy Physics)

   We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order ofN, the standard result for theq-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter,k. We present an algorithm to sum the diagrams at any given order of 1/(kq)ⁿ. We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finiteNbecomes more significant for larger values ofN.

    

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4. Counterexample-guided computation of polyhedral Lyapunov functions for piecewise linear systems

   https://doi.org/10.1016/j.automatica.2023.111165  

   Berger, Guillaume O. ; Sankaranarayanan, Sriram ( September 2023 , Automatica)

   This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for continuous-time piecewise linear systems. Polyhedral Lyapunov functions provide an alternative to commonly used polynomial Lyapunov functions. Our approach first characterizes intrinsic properties of a polyhedral Lyapunov function including its “eccentricity” and “robustness” to perturbations. We then derive an algorithm that either computes a polyhedral Lyapunov function proving that the system is asymptotically stable, or concludes that no polyhedral Lyapunov function exists whose eccentricity and robustness parameters satisfy some user-provided limits. Significantly, our approach places no a-priori bound on the number of linear pieces that make up the desired polyhedral Lyapunov function. The algorithm alternates between a learning step and a verification step, always maintaining a finite set of witness states. The learning step solves a linear program to compute a candidate Lyapunov function compatible with a finite set of witness states. In the verification step, our approach verifies whether the candidate Lyapunov function is a valid Lyapunov function for the system. If verification fails, we obtain a new witness. We prove a theoretical bound on the maximum number of iterations needed by our algorithm. We demonstrate the applicability of the algorithm on numerical examples. 

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5. Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

   https://doi.org/10.23919/ACC55779.2023.10155996  

   Jones, Morgan ; Peet, Matthew M. ( May 2023 , Proceedings of the American Control Conference)

   — This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions. 

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