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1. Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints

   https://doi.org/10.3934/jgm.2023011  

   Clark, William; Bloch, Anthony (January 2023, Journal of Geometric Mechanics)

   We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these conditions dictate that a basic invariant density exists if and only if a certain 1-form is exact and a certain function vanishes (this function automatically vanishes for linear constraints). Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. Examples of nonlinear/affine/linear constraints are considered. 

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2. Output-only identification of self-excited systems using discrete-time Lur'e models with application to a gas-turbine combustor

   https://doi.org/10.1080/00207179.2022.2137702  

   Paredes, Juan A.; Yang, Yulong; Bernstein, Dennis S. (November 2022, International Journal of Control)

   A self-excited system is a nonlinear system with the property that a constant input yields a bounded, nonconvergent response. Nonlinear identification of self-excited systems is considered using a Lur'e model structure, where a linear model is connected in feedback with a nonlinear feedback function. To facilitate identification, the nonlinear feedback function is assumed to be continuous and piecewise affine (CPA). The present paper uses least-squares optimization to estimate the coefficients of the linear dynamics and the slope vector of the CPA nonlinearity, as well as mixed-integer optimization to estimate the order of the linear dynamics and the breakpoints of the CPA function. The proposed identification technique requires only output data, and thus no measurement of the constant input is required. This technique is illustrated on a diverse collection of low-dimensional numerical examples as well as data from a gas-turbine combustor. 

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3. Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno

   W. Clark and A. Bloch (January 2023, Archive for rational mechanics and analysis)

   Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time).We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the “boundary identity property” along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top. 

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4. Persistence landscapes of affine fractals

   https://doi.org/10.1515/dema-2022-0015  

   Catanzaro, Michael J.; Przybylski, Lee; Weber, Eric S. (January 2022, Demonstratio Mathematica)

   Abstract We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations. 

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5. Stabilization Under Arbitrary Tight and One Sided Control Constraints: A Variational Equations Approach

   https://doi.org/10.23919/ACC60939.2024.10644733  

   Kolmanovsky, Ilya; Garone, Emanuele (July 2024, IEEE)

   Stabilization of a linear system under control constraints is approached by combining the classical variation of parameters method for solving ODEs and a straightforward construction of a feedback law for the variational system based on a quadratic Lyapunov function. Sufficient conditions for global closed-loop stability under control constraints with zero in the interior and zero on the boundary of the control set are derived, and several examples are reported. The extension of the method to nonlinear systems with control constraints is described. 

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