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Abstract

We propose a new numerical method for the computation of the optimal value function of perturbed control systems and associated globally stabilizing optimal feedback controllers. The method is based on a set-oriented discretization of the state space in combination with a new algorithm for the computation of shortest paths in weighted directed hypergraphs. Using the concept of multivalued game, we prove the convergence of the scheme as the discretization parameter goes to zero.

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References

  1. Junge, O., Osinga, H.M.: A set oriented approach to global optimal control. ESAIM: Control Optim. Calc. Var. 10(2), 259–270 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Grüne, L., Junge, O.: A set oriented approach to optimal feedback stabilization. Syst. Control Lett. 54(2), 169–180 (2005)

    Article  MATH  Google Scholar 

  3. Grüne, L.: An Adaptive Grid scheme for the discrete Hamilton-Jacobi-Bellman equation. Numer. Math. 75(3), 319–337 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Nešić, D., Teel, A.R.: A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE. Trans. Autom. Control 49(7), 1103–1122 (2004)

    Article  Google Scholar 

  5. Huang, S., James, M.R., Nešić, D., Dower, P.M.: A unified approach to controller design for achieving ISS and related properties. IEEE Trans. Autom. Control 50(11), 1681–1697 (2005)

    Article  Google Scholar 

  6. Grüne, L., Junge, O.: Global optimal control of perturbed systems. Technical Report, arXived at math.OC/0703874 (2006)

  7. Fleming, W.: The convergence problem for differential games. J. Math. Anal. Appl. 3, 102–116 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 2. Athena Scientific, Belmont (1995)

    MATH  Google Scholar 

  9. Dijkstra, E.: A note on two problems in connection with graphs. Numer. Math. 5, 269–271 (1959)

    Article  MathSciNet  Google Scholar 

  10. Gallo, G., Longo, G., Nguyen, S., Pallottino, S.: Directed hypergraphs and applications. Discrete Appl. Math. 40, 177–201 (1992)

    MathSciNet  Google Scholar 

  11. Dellnitz, M., Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75(3), 293–317 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. von Lossow, M.: A min-man version of Dijkstra’s algorithm with application to perturbed optimal control problems. In: Proceedings of the GAMM Annual Meeting, Zürich, Switzerland, 2007 (to appear)

  13. Junge, O.: Rigorous discretization of subdivision techniques. In: Fiedler, B., Gröger, K., Sprekels, J. (eds.) Proceedings of EQUADIFF 99, pp. 916–918. World Scientific, Singapore (2000)

    Google Scholar 

  14. Grüne, L.: Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization. Lecture Notes in Mathematics, vol. 1783. Springer, New York (2002)

    MATH  Google Scholar 

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Correspondence to O. Junge.

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Communicated by H.J. Pesch.

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Grüne, L., Junge, O. Global Optimal Control of Perturbed Systems. J Optim Theory Appl 136, 411–429 (2008). https://doi.org/10.1007/s10957-007-9312-z

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  • DOI: https://doi.org/10.1007/s10957-007-9312-z

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