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Moving Constraints as Stabilizing Controls in Classical Mechanics

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Abstract

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motion contain terms which are linear or quadratic with respect to time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point by means of oscillating controls. This problem is first reduced to theweak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alberto Bressan

  2. Dipartimento di Matematica Pura ed Applicata, Universitá di Padova, 35141, Padua, Italy

    Franco Rampazzo

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  1. Alberto Bressan
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  2. Franco Rampazzo
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Correspondence to Alberto Bressan.

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Bressan, A., Rampazzo, F. Moving Constraints as Stabilizing Controls in Classical Mechanics. Arch Rational Mech Anal 196, 97–141 (2010). https://doi.org/10.1007/s00205-009-0237-6

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