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Optimal control of systems with discontinuous differential equations

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Abstract

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of “optimize the discretization” will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler’s method is used where the step size is “sufficiently small” in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    David E. Stewart

  2. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USA

    Mihai Anitescu

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  1. David E. Stewart
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  2. Mihai Anitescu
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Correspondence to David E. Stewart.

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The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory (“Argonne”) under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy.

The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

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Stewart, D.E., Anitescu, M. Optimal control of systems with discontinuous differential equations. Numer. Math. 114, 653–695 (2010). https://doi.org/10.1007/s00211-009-0262-2

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