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Shape Differential Equation with a Non Smooth Field

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Computational Methods for Optimal Design and Control

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 24))

Abstract

This paper is concerned with shape dynamical evolution and global existence results for the shape differential equation. In order to construct large evolutions of the domain we adopt the flow transformation method. We recall the strong theory results initiated in 1976 and present new results on the weak evolution of characteristic functions governed by non-Lipschitzian vector fields. This weak shape differential equation is well suited for topological identification (including level curves approach) as well as for non cylindrical large evolution of domains involving the shape active control setting. In these directions we give here the example of the shape Liapounov trajectory for the first eigen value of a Shrodinger problem built by a weak shape differential equation. Also we give a new example of a strong solution concerning the Dirichlet-Laplace and the Dirichlet-Wave problems using the extractor identities in order to get the boundedness of the gradients in some negative Sobolev spaces. In order to illustrate the use of the shape differential equation for solving non cylindrical domains we consider the Tube derivative and we characterize the new field Z which built the transverse tube \( s \to {{T}_{1}}\left( {V + sW} \right)\left( \Omega \right). \) This new “transverse” field evolution is governed by the Lie brackets of the two non autonomous vector fields V, W. We illustrate the Tube derivative by characterizing the Eulerian large evolution of an elastic body (under large displacement with small deformation assumptions) describing the field extremality of the action integral governing the problem.

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Author information

Authors and Affiliations

  1. CNRS Institut Non Linéaire de Nice, 1361 route des lucioles, Sophia Antipolis Cedex, 06904, France

    Jean-Paul Zolésio

  2. Centre de Mathématiques Appliquées (CMA/MEIJE, INRIA), 2004 route des Lucioles, Sophia Antipolis Cedex, 06902, France

    Jean-Paul Zolésio

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  1. Jean-Paul Zolésio
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Editor information

Editors and Affiliations

  1. Ctr. for Optimal Design and Control Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, 14853, USA

    Jeff Borggaard

  2. Ctr. for Optimal Design and Control Interdisciplinary Ctr. for Applied Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    John Burns  & Eugene Cliff  & 

  3. Air Force Office of Scientific Research, Bolling Air Force Base, Washington, DC, 20332, USA

    Scott Schreck (Directorate of Mathematics and Geosciences) (Directorate of Mathematics and Geosciences)

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Zolésio, JP. (1998). Shape Differential Equation with a Non Smooth Field. In: Borggaard, J., Burns, J., Cliff, E., Schreck, S. (eds) Computational Methods for Optimal Design and Control. Progress in Systems and Control Theory, vol 24. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1780-0_22

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  • DOI: https://doi.org/10.1007/978-1-4612-1780-0_22

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7279-3

  • Online ISBN: 978-1-4612-1780-0

  • eBook Packages: Springer Book Archive

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