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Numerical Schemes for Discontinuous Value Functions of Optimal Control

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Abstract

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.

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References

  1. Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Basel, 1991.

    Google Scholar 

  2. Aubin, J.-P. and Frankowska, H.: The viability kernel algorithm for computing value functions of infinite horizon optimal control problems, J. Math. Anal. Appl. 201 (1996), 555–576.

    Google Scholar 

  3. Aubin, J.-P.: Viability Theory, Birkhäuser, Basel, 1992.

    Google Scholar 

  4. Bardi, M., Bottacin, S. and Falcone, M.: Convergence of discrete schemes for discontinuous value functions of pursuit evasion games, In: New Trends in Dynamic Games and Applications, Ann. Int. Soc. Diff. Games 3, Birkhäuser, Basel, 1995, pp. 273–304.

    Google Scholar 

  5. Bardi, M. and Falcone, M.: An approximation scheme for the minimal time function, SIAM J. Control Optim. 28 (1990), 950–965.

    Google Scholar 

  6. Bardi, M. and Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhäuser, Basel, 1996.

    Google Scholar 

  7. Barles, G. and Souganidis, P. E.: Convergence of approximation schemes for fully nonlinear systems, Asymptot. Anal. 4 (1991), 271–283.

    Google Scholar 

  8. Capuzzo-Dolcetta, I. and Falcone, M.: Discrete dynamic programming and viscosity solutions of the Bellman Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 161–183.

    Google Scholar 

  9. Capuzzo-Dolcetta, I. and Ishii: Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim. 11 (1984), 161–181.

    Google Scholar 

  10. Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P.: Optimal times for constrained nonlinear control problems without local controllability, Appl. Math. Optim. 35 (1997), 1–22.

    Google Scholar 

  11. Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P.: In: Bardi, Parthasaranthy, Raghavan (eds), Numerical Methods for Optimal Control and Numerical Games, Ann. Int. Soc. Dynam. Games, Birkhäauser, Basel, 1999, pp. 127–247.

  12. Cardaliaguet, P.: On the regularity of semi-permeable surfaces in control theory and application to the optimal time problem, SIAM J. Control Optim. 5 (1997), 1638–1671.

    Google Scholar 

  13. Crandall, M. C. and Lions P. L.: Two approximations of solutions of Hamilton–Jacobi equations, Math. Comp. 43 (1984), 1–19.

    Google Scholar 

  14. Falcone, M. and Saint-Pierre, P.: Algorithms for constrained optimal control problem: A link between viability and dynamic programming, Preprint, Université di Roma I, 1995.

  15. Gonzalez, R. L. V. and Rofman, M. E.: On deterministic control problems: An approximation procedure for the optimal cost, part 1 and 2, SIAM J. Control Optim. 23 (1985), 242–285.

    Google Scholar 

  16. Lempio, F. and Veliov, M. V.: Discrete approximations of differential inclusions, Bayreuther Math. Schr. 54 (1998), 149–232.

    Google Scholar 

  17. Pourtallier O., Tidball, M. and Altman, E.: Approximation in dynamic zero-sum games, SIAM J. Control Optim. 35 (1997), 2101–2117.

    Google Scholar 

  18. Quincampoix, M. and Saint-Pierre, P.: An Algorithm for viability kernels in Hoölderian case: Approximation by discrete viability kernels, J. Math. Syst. Estim. Control 8(1) (1998), 17–29.

    Google Scholar 

  19. Rozyev, I. and Subbotin, A. I.: Semicontinuous solutions of Hamilton–Jacobi equations, J. Appl. Math. Optim. 52(2) (1988), 141–146.

    Google Scholar 

  20. Saint-Pierre, P.: Approximation of the viability kernel, Appl. Math. Optim. 29 (1994), 187–209.

    Google Scholar 

  21. Visintin, A.: Strong convergence results related to strict convexity, Comm. Partial Differential Equations 9 (1984), 439–466.

    Google Scholar 

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Authors and Affiliations

  1. Centre de Recherche Viabilité, Jeux, Contrôle - CNRS E.R.S. 2064, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775, Paris cedex 16, France

    Pierre Cardaliaguet & Patrick Saint-Pierre

  2. Département de Mathématiques, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, BP 809, 29285, Brest, France

    Marc Quincampoix

Authors
  1. Pierre Cardaliaguet
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  2. Marc Quincampoix
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  3. Patrick Saint-Pierre
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Cardaliaguet, P., Quincampoix, M. & Saint-Pierre, P. Numerical Schemes for Discontinuous Value Functions of Optimal Control. Set-Valued Analysis 8, 111–126 (2000). https://doi.org/10.1023/A:1008774508738

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