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Control of Moving Domains, Shape Stabilization and Variational Tube Formulations

  • Jean-Paul Zolésio
Part of the International Series of Numerical Mathematics book series (ISNM, volume 155)

Abstract

This paper deals with the control of a moving dynamical domain in which a non cylindrical dynamical boundary value problem is considered. We consider weak Eulerian evolution of domains through the convection of a measurable set by (non necessarily smooth) vector field V. We introduce the concept of tubes by “product space” and we show a closure result leading to existence results for a variational shape principle. We illustrate this by new results: heat equation and wave equation in moving domains with various boundary conditions and also the geodesic characterisation for two Eulerian shape metrics leading to the Euler equation through the transverse field considerations. We consider the non linear Hamilton-Jacobi like equation associated with level set parametrization of the moving domain and give new existence result of possible topological change in finite time in the solution.

Keywords

Wave Equation Lateral Boundary Boundary Control Topological Change Free Boundary Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M.C. Delfour and J.P. Zolésio. Oriented distance function and its evolution equation for initial sets with thin boundary. SIAM J. Control Optim. 42 (2004), no. 6, 2286–2304MATHCrossRefGoogle Scholar
  2. [2]
    M. Cuer and J.P. Zolésio. Control of singular problem via differentiation of a minmax. Systems Control Lett. 11 (1988), no. 2, 151–158.MATHCrossRefGoogle Scholar
  3. [3]
    D. Bucur and J.P. Zolésio. Free Boundary Problems and Density Perimeter. J. Differential. Equations 126(1996), 224–243.MATHCrossRefGoogle Scholar
  4. [4]
    D. Bucur and J.P. Zolésio. Boundary Optimization under Pseudo Curvature Constraint. Annali della Scuola Normale Superiore di Pisa, IV, XXIII(4), 681–699, 1996.LGoogle Scholar
  5. [5]
    R. Dziri and J.P. Zolésio. Dynamical Shape Control in Non-cylindrical Navier-Stokes Equations. J. convex analysis, vol. 6,2, 293–318, 1999.MATHGoogle Scholar
  6. [6]
    N. Gomez and J.P. Zolésio. Shape sensitivity and large deformation of the domain for norton-hoff flow. Volume 133 of Int. Series of Num. Math., pages 167–176, 1999.Google Scholar
  7. [7]
    J.P. Zolésio. Identification de domaine par déformations. Thèse de doctorat d’état, Université de Nice, 1979.Google Scholar
  8. [8]
    J.P. Zolésio. In Optimization of Distributed Parameter structures, vol. II, (E. Haug and J. Céa eds.), Adv. Study Inst. Ser. E: Appl. Sci., 50, Sijthoff and Nordhoff, Alphen aan den Rijn, 1981. i) The speed method for Shape Optimization. 1089–1151. ii) Domain Variational Formulation for Free Boundary Problems, 1152–1194. iii) Semiderivative of repeated eigenvalues, 1457–1473.Google Scholar
  9. [9]
    J. Sokolowski and J.P. Zolésio. Introduction to shape optimization, sci, 16, Springer Verlag, Heidelberg, N.Y., 1991Google Scholar
  10. [10]
    B. Kawohl, O. Pironneau, L. Tartar, J.P. Zolésio. Optimal Shape Design, (yellow) Lecture Notes in Mathematics, 1740, Springer Verlag, Heidelberg, N.Y., 1998.Google Scholar
  11. [11]
    J.P. Zolésio. Variational Principle in the Euler Flow. In G. Leugering, editor, Proceedings of the IFIP-WG7.2 conference, Chemnitz, volume 133 of Int. Series of Num. Math., 1999.Google Scholar
  12. [12]
    J.P. Zolésio. Shape Differential with Non Smooth Field. In Computational Methods for Optimal Design and Control. J. Borggard, J. Burns, E. Cliff and S. Schreck eds., volume 24 of Progress in Systems and Control Theory, pp. 426–460, Birkhäuser, 1998.Google Scholar
  13. [13]
    J.P. Zolésio. Weak set evolution and variational applications in Shape optimization and optimal design, lecture notes in pure and applied mathematics, vol. 216, pp. 415–442, Marcel Dekker, N.Y., 2001.Google Scholar
  14. [14]
    M.C. Delfour and J.P. Zolésio. Structure of shape derivatives for non smooth domains, Journal of Functional Analysis, 1992, 104.Google Scholar
  15. [15]
    P. Cannarsa, G. Da Prato and J.P. Zolésio. The damped wave equation in a moving domain, Journal of Differential Equations, 1990, 85, 1–16.MATHCrossRefGoogle Scholar
  16. [16]
    M.C. Delfour and J.P. Zolésio. Shape analysis via oriented distance functions, J. Funct. Anal., 1994, 123, 1–56.CrossRefGoogle Scholar
  17. [17]
    R. Dziri and J.P. Zolésio. Dynamical shape control in non-cylindrical hydrodynamics, Inverse Problem, 1999, 15,1, 113–122.MATHCrossRefGoogle Scholar
  18. [18]
    M.C. Delfour and J.P. Zolésio. Shape sensitivity analysis via min max differentiability, SIAM J. Control Optim., 1988, 26,4, 834–862.MATHCrossRefGoogle Scholar
  19. [19]
    G. Da Prato and J.P. Zolésio. Dynamical Programming for non Cylindrical Parabolic Equation, Sys. Control Lett., 11, 1988.Google Scholar
  20. [20]
    G. Da Prato and J.P. Zolésio. Existence and Control for wave equation in moving domain, L.N.C.I.S Springer Verlag, 144, 1988.Google Scholar
  21. [21]
    P. Acquistapace. Boundary control for non-autonomous parabolic equations in noncylindrical domains in Boundary control and variation (Sophia Antipolis, 1992), 1–12, L. N. P. A. Math., 163, Dekker, New York, 1994.Google Scholar
  22. [22]
    L. Ambrosio. Lecture notes on optimal transport problems. Mathematical aspects of evolving interfaces (Funchal, 2000), 1–52, Lecture Notes in Math., 1812, Springer, Berlin, 2003. 49Q20 (49-02)Google Scholar
  23. [23]
    K. Burdzy, Z. Chen and J. Sylvester. The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 32 (2004), no. 1B, 775–804.MATHCrossRefGoogle Scholar
  24. [24]
    M. Delfour and J.P. Zolésio. Shape and Geometry Advances in Design and Control, 04, SIAM, 2001.Google Scholar
  25. [25]
    J.P. Zolésio. Set Weak Evolution and Transverse Field, Variational Applications and Shape Differential Equation INRIA report RR-464, 2002. (http://www-sop.inria.fr/rapports/sophia/RR-464)
  26. [26]
    M. Moubachir and J.P. Zolésio. Moving Shape Analysis and Control: application to fluid structure interaction. Pure and Applied Mathematics series, CRC, 2006.Google Scholar
  27. [27]
    M.C. Delfour and J.P. Zolésio. Structure of shape derivatives for non smooth domains. Journal of Functional Analysis, 104, 1992.Google Scholar
  28. [28]
    M.C. Delfour and J.P. Zolésio. Shape analysis via oriented distance functions. Journal of Functional Analysis, 123, 1994.Google Scholar
  29. [29]
    F.R. Desaint and J.P. Zolésio. Manifold derivative in the laplace-beltrami equation. Journal of Functionnal Analysis, 151(1): 234, 269, 1997.MATHCrossRefGoogle Scholar
  30. [30]
    J.P. Zolésio. Introduction to shape optimization and free boundary problems. In Michel C. Delfour, editor, Shape Optimization and Free Boundaries, volume 380 of NATO ASI, Series C: Mathematical and Physical Sciences, pages 397, 457, 1992.Google Scholar
  31. [31]
    J.P. Zolésio. Shape Topology by Tube Geodesic. In Information Processing: Recent Mathematical Advances in Optimization and Control. Presses de l’Ecole des Mines de Paris, pages 185–204, 2004.Google Scholar
  32. [32]
    Raja Dziri, J.P. Zolésio Tube Derivative of Non-Cylindrical Shape Functionals and Variational Formulations in proc. ifip7.2 conf. Houston, Dec. 2004, R. Glowinski, J.P. Zolésio eds., CRC press book, 2006.Google Scholar
  33. [33]
    J.P. Zolésio. Tubes Analysis in proc. ifip7.2 conf. Houston, Dec. 2004, R. Glowinski, J.P. Zolésio eds., CRC press book, 2006.Google Scholar
  34. [34]
    J.P. Zolésio Shape Stabilization of Flexible Structures. Lecture notes in Control and Information Sciences, vol. 75 Distributed Parameter Systems ed. F. Kappel, K. Kunish, W. Schapacher, proc. of the 2nd Inter. conf. in Vorau (Austria), Springer-Verlag, Berlin-Heidelberg-N.Y., 1985.Google Scholar
  35. [35]
    C. Truchi, J.P. Zolésio Shape stabilization of wave equation (read) Lecture Notes in Control and Information Sciences, vol. 100 Boundary Control and Boundary VariationsM, ed. J.P. Zolésio, proc. ifip conf. June 86 Nice (France), Springer-Verlag, Berlin-Heidelberg-N.Y., 1987.Google Scholar
  36. [36]
    J.P. Zolésio. Shape Formulation for Free Boundary Problem with Non Linearized Bernoulli Condition (read) Lecture Notes in Control and Information Sciences, vol 178 Boundary Control and Boundary VariationsM, ed. J.P. Zolésio, proc. ifip conf. June 1991, Sophia Antipolis (France), Springer-Verlag, Berlin-Heidelberg-N.Y., pp. 362–392, 1992.Google Scholar
  37. [37]
    J.P. Zolésio. Weak shape formulation of free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21(1):11–44, 1994.MATHGoogle Scholar
  38. [38]
    J.P. Zolésio. Numerical algorithms and existence result for a Bernoulli-like steady free boundary problem. Large Scale Systems, 6(3):263–278, 1984.MATHGoogle Scholar
  39. [39]
    J.P. Zolésio. Some results concerning free boundary problems solved by domain (or shape) optimization of energy. In Modelling and inverse problems of control for distributed parameter systems (Laxenburg, 1989), volume 154 of Lecture Notes in Control and Inform. Sci., pages 161–170. Springer, Berlin, 1991.CrossRefGoogle Scholar
  40. [40]
    J.P. Zolésio. Shape differential equation with a non-smooth field. In Computational methods for optimal design and control (Arlington, VA, 1997), volume 24 of Progr. Systems Control Theory, pages 427–460. Birkhäuser Boston, Boston, MA, 1998.Google Scholar
  41. [41]
    Dorin Bucur and Jean-Paul Zolésio. Flat cone condition and shape analysis. In Control of partial differential equations (Trento, 1993), volume 165 of Lecture Notes in Pure and Appl. Math., pages 37–49. Dekker, New York, 1994.Google Scholar
  42. [42]
    Dorin Bucur and Jean-Paul Zolésio. Optimisation de forme sous contrainte capacitaire. C. R. Acad. Sci. Paris Sér. I Math., 318(9):795–800, 1994.MATHGoogle Scholar
  43. [43]
    Dorin Bucur and Jean-Paul Zolésio. N-dimensional shape optimization under capacitary constraint. J. Differential Equations, 123(2):504–522, 1995.MATHCrossRefGoogle Scholar
  44. [44]
    V. Sverak. On optimal shape design. J. Math. Pures Appl. (9) 72 (1993), no. 6, 537–551.MATHGoogle Scholar
  45. [45]
    D. Bucur, INLN preprint, 1994, Sophia Antipolis, France.Google Scholar
  46. [46]
    J. Cooper and W.A. Strauss. Energy boundedness and decay of waves reflecting off a moving obstacle. Indiana Univ. Math. J. 25, 671–690 (1976).MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Jean-Paul Zolésio
    • 1
  1. 1.CNRS/INRIASophia AntipolisFrance

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