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Regularization of Neutral Delay Differential Equations with Several Delays

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Abstract

For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system is presented, which characterizes the kind of generalized solution that is approximated. For the case that the solution of the regularized problem has high frequency oscillations around a codimension-2 weak solution of the original problem, a new stabilizing regularization is proposed and analyzed.

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Notes

  1. With the word “generic” we mean that both, \(g_{1}(\theta _{1},\theta _{2}) \) and \(g_{2}(\theta _{1},\theta _{2}) \) are non-zero at the corners of the unit square, and that \(g_{1}(\theta _{1},\theta _{2}) = g_{2}(\theta _{1},\theta _{2}) =0\) can occur only inside the unit square.

References

  1. Dieci, L., Guglielmi, N.: Regularizing piecewise smooth differential systems: co-dimension \(2\) discontinuity surface. JDDE (2013) (in press)

  2. Dieci, L., Lopez, L.: Sliding motion in Filippov differential systems: theoretical results and a computational approach. SIAM J. Numer. Anal. 47(3), 2023–2051 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dieci, L., Lopez, L.: Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field. Numer. Math. 117(4), 779–811 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications. Soviet Series. Kluwer, Dordrecht (1988) (Translated from the Russian)

  5. Guglielmi, N., Hairer, E.: Implementing Radau IIA methods for stiff delay differential equations. Computing 67(1), 1–12 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29(3), 229–247 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guglielmi, N., Hairer, E.: Asymptotic expansions for regularized state-dependent neutral delay equations. SIAM J. Math. Anal. 44(4), 2428–2458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics 14, 2nd edn. Springer, Berlin (1996)

  9. O’Malley Jr, R.E.: Singular Perturbation Methods for Ordinary Differential Equations, vol. 89 of Applied Mathematical Sciences. Springer, New York (1991)

    Book  Google Scholar 

  10. Utkin, V.I.: Sliding mode control: mathematical tools, design and applications. In: Nonlinear and Optimal Control Theory, vol. 1932 of Lecture Notes in Mathematics, pp. 289–347. Springer, Berlin (2008)

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Acknowledgments

The authors wish to thank the Referees for their careful reading of the paper and their useful remarks. This work was partially supported by the Fonds National Suisse, Project No. 200020-126638.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, via Vetoio (Coppito), 67010,  L’Aquila, Italy

    Nicola Guglielmi

  2. Section de Mathématiques, Université de Genève, 1211, Geneva 24, Switzerland

    Ernst Hairer

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  1. Nicola Guglielmi
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  2. Ernst Hairer
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Correspondence to Nicola Guglielmi.

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Guglielmi, N., Hairer, E. Regularization of Neutral Delay Differential Equations with Several Delays. J Dyn Diff Equat 25, 173–192 (2013). https://doi.org/10.1007/s10884-013-9288-3

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  • DOI: https://doi.org/10.1007/s10884-013-9288-3

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