Optimization and Engineering

, Volume 5, Issue 4, pp 395–415 | Cite as

Parametric Optimization of Hybrid Car Engines

  • Joseph Frédéric Bonnans
  • Thérèse Guilbaud
  • Ahmed Ketfi-Cherif
  • Dirk von Wissel
  • Claudia Sagastizábal
  • Housnaa Zidani


We consider the problem of optimal design of hybrid car engines which combine thermal and electric power. The optimal configuration of the different motors composing the hybrid system involves the choice of certain design parameters. For a given configuration, the goal is to minimize the fuel consumption along a trajectory. This is an optimal control problem with one state variable.

The simultaneous optimization of design parameters and trajectories can be formulated as a bilevel optimization problem. The lower level computes the optimal control for a given architecture. The higher level seeks for the optimal design parameters by solving a nonconvex nonsmooth optimization problem with a bundle method.

hybrid car engines bilevel optimization nonsmooth optimization bundle method optimal control problem Hamilton-Jacobi-Bellman equation 


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  1. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equa-tions, Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997.Google Scholar
  2. G. Barles, Solutions de viscositédes équations de Hamilton-Jacobi,vol. 17 of Mathématiques et applications. Springer, Paris, 1994.Google Scholar
  3. G. Barles and P. E. Souganidis, “Convergence of approximation schemes for fully nonlinear second order equations,” Asymptotic Analysis vol. 4, pp. 271–283, 1991.Google Scholar
  4. J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.Google Scholar
  5. G. Blanchon, J. C. Dodu, and J. F. Bonnans, “Optimisation des réseaux électriques de grande taille,” in Analysis and Optimization of Systems, A. Bensoussan and J. L. Lions (Eds.), vol. 144 of Lecture Notes in Information and Control Sciences, Springer-Verlag: Berlin, 1990.Google Scholar
  6. J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimization, Universitext. Springer-Verlag: Berlin, 2002.Google Scholar
  7. J. F. Bonnans, Th. Guilbaud, and H. Zidani, “Bilevel optimization for parametric optimal control problems,” Working paper. F. H. Clarke, Optimization and Nonsmooth Analysis, J.Wiley: New York, 1983.Google Scholar
  8. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag: Berlin, 1990.Google Scholar
  9. J. Gauvin, “The generalized gradient of a marginal function in mathematical programming,” Mathematics of Operations Research vol. 4, pp. 458–463, 1979.Google Scholar
  10. W. W. Hager, “Runge-Kutta methods in optimal control and the transformed adjoint system,” Numerische Mathematik vol. 87, pp. 247–282, 2000.Google Scholar
  11. H. Ishii and S. Koike, “A new formulation of state constraint problems for first-order PDEs,” SIAM Journal on Control and Optimization vol. 34, no. 2, pp. 554–571, 1996.Google Scholar
  12. H. J. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edition, vol. 24 of Applications of Mathematics. Springer: New York. 2001.Google Scholar
  13. C. Lemaréchal and C. Sagastizábal, “Variable metric bundle methods: From conceptual to implementable forms,” Mathematical Programming vol. 76, pp. 393–410, 1997.Google Scholar
  14. C. Lemaréchal, J.-J. Strodiot, and A. Bihain, “On a bundle method for nonsmooth optimization,” in Nonlinear Programming,O. Mangasarian, R. R. Meyer, and S. M. Robinson (Eds.), vol. 4, Academic Press, 1981, pp. 245–282.Google Scholar
  15. P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations,vol. 69 of Research Notes in Mathematics. Pitman, Boston, 1982.Google Scholar
  16. L. Lukšan and J. Vlček, “A bundle-Newton method for nonsmooth unconstrained minimization,” Math. Programming vol. 83, no. 3, Ser. A, pp. 373–391, 1998.Google Scholar
  17. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica J. IFAC,vol. 36, no. 6, pp. 789–814, 2000. See also the correction in Automatica vol. 37, no. 3, p. 483, 2001.Google Scholar
  18. R. Mifflin, “Semismooth and semiconvex functions in constrained optimization,” SIAM J. Control and Optimization vol. 15, pp. 959–972, 1977.Google Scholar
  19. R. Mifflin, “A modification and extension of Lemarechal's algorithm for nonsmooth minimization,” Math. Programming Stud. vol. 17, pp. 77–90, 1982.Google Scholar
  20. J. Outrata, M. Kočvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers: Boston, 1998.Google Scholar
  21. H. Schramm and J. Zowe, “A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results,” SIAM J. on Optimization vol. 2, no. 1, pp. 121–152, 1992.Google Scholar
  22. H. Sonnet, Dictionnaire des Mathématiques Appliquées, Librairie de L. Hachette et Cie, Paris, 1867.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Joseph Frédéric Bonnans
    • 1
  • Thérèse Guilbaud
    • 2
  • Ahmed Ketfi-Cherif
    • 3
  • Dirk von Wissel
    • 3
  • Claudia Sagastizábal
    • 4
  • Housnaa Zidani
    • 5
  1. 1.Projet Sydoco, INRIA, B.P. 105RocquencourtFrance
  2. 2.Université Paris VIII and Cermics, ENPCMarne la Vallée Cx 2France
  3. 3.Direction de la RechercheTechnocentre RenaultGuyancourtFrance
  4. 4.IMPA, Estrada Dona CastorinaRio de Janeiro RJBrazil
  5. 5.ENSTAINRIAFrance

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