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The Euler Method for Linear Control Systems Revisited

  • Josef L. Haunschmied
  • Alain Pietrus
  • Vladimir M. VeliovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8353)

Abstract

Although optimal control problems for linear systems have been profoundly investigated in the past more than 50 years, the issue of numerical approximations and precise error analyses remains challenging due the bang-bang structure of the optimal controls. Based on a recent paper by M. Quincampoix and V.M. Veliov on metric regularity of the optimality conditions for control problems of linear systems the paper presents new error estimates for the Euler discretization scheme applied to such problems. It turns out that the accuracy of the Euler method depends on the “controllability index” associated with the optimal solution, and a sharp error estimate is given in terms of this index. The result extends and strengthens in several directions some recently published ones.

References

  1. 1.
    Alt, W., Baier, R., Gerdts, M., Lempio, F.: Error bounds for Euler approximations of linear-quadratic control problems with bang-bang solutions. Numer. Algebra Control Optim. 2(3), 547–570 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alt, W., Seydenschwanz, M.: An implicit discretization scheme for linear-quadratic control problems with bang-bang solutions. Optim. Method Softw. 29(3), 535–560 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Alt, W., Baier, R., Lempio, F., Gerdts, M.: Approximations of linear control problems with bang-bang solutions. Optimization 62(1), 9–32 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Felgenhauer, U.: On stability of bang-bang type controls. SIAM J. Control Optim. 41(6), 1843–1867 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Felgenhauer, U., Poggolini, L., Stefani, G.: Optimality and stability result for bang-bang optimal controls with simple and double switch behavior. Control Cybern. 38(4B), 1305–1325 (2009)zbMATHGoogle Scholar
  6. 6.
    Osmolovskii, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. Part 1: main results. Control Cybern. 34, 927–950 (2005)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Osmolovskii, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. Part 2: proofs, variational derivatives and representations. Control Cybern. 36, 5–45 (2007)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Pontryagin, L.S., Boltyanskij, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes, Fizmatgiz, Moscow, 1961. Pergamon, Oxford (1964)Google Scholar
  9. 9.
    Quincampoix, M., Veliov, V.M.: Metric regularity and stability of optimal control problems for linear systems. SIAM J. Control Optim. 51(5), 4118–4137 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Veliov, V.M.: Error analysis of discrete approximation to bang-bang optimal control problems: the linear case. Control Cybern. 34(3), 967–982 (2005)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Josef L. Haunschmied
    • 1
  • Alain Pietrus
    • 2
  • Vladimir M. Veliov
    • 1
    Email author
  1. 1.Institute of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustralia
  2. 2.Laboratoire LAMIA, Dépt. de MathématiquesUniversité des Antilles et de la GuyanePointe-à-PitreGuadeloupe

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