On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems

  • Lars GrüneEmail author
  • Christopher M. Kellett
  • Steven R. Weller


We show that, under appropriate regularity conditions, a finite horizon optimal control problem exhibits the turnpike property, if and only if its infinite horizon counterpart does. We prove the result for both undiscounted and discounted problems and also provide a version which incorporates quantitative information about the convergence rates.


Finite horizon optimal control Infinite horizon optimal control Optimal equilibrium 

Mathematics Subject Classification

49N60 49J21 


  1. 1.
    von Neumann, J.: A model of general economic equilibrium. Rev. Econ. Stud. 13(1), 1–9 (1945)CrossRefGoogle Scholar
  2. 2.
    McKenzie, L.W.: Optimal economic growth, turnpike theorems and comparative dynamics. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of Mathematical Economics, vol. III, pp. 1281–1355. North-Holland, Amsterdam (1986)Google Scholar
  3. 3.
    Dorfman, R., Samuelson, P.A., Solow, R.M.: Linear Programming and Economic Analysis. Dover Publications, New York (1987). Reprint of the 1958 originalzbMATHGoogle Scholar
  4. 4.
    Faulwasser, T., Korda, M., Jones, C.N., Bonvin, D.: Turnpike and dissipativity properties in dynamic real-time optimization and economic MPC. In: Proceedings of the 53rd IEEE Conference on Decision and Control—CDC 2014, pp. 2734–2739. Los Angeles, CA, USA (2014)Google Scholar
  5. 5.
    Zaslavski, A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Springer, New York (2006)zbMATHGoogle Scholar
  6. 6.
    Zaslavski, A.J.: Turnpike properties of approximate solutions of autonomous variational problems. Control Cybern. 37(2), 491–512 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Zaslavski, A.J.: Turnpike Phenomenon and Infinite Horizon Optimal Control. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  8. 8.
    Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258(1), 81–114 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grüne, L., Müller, M.A.: On the relation between strict dissipativity and the turnpike property. Syst. Control Lett. 90, 45–53 (2016)CrossRefzbMATHGoogle Scholar
  10. 10.
    Anderson, B.D.O., Kokotović, P.V.: Optimal control problems over large time intervals. Automatica 23(3), 355–363 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grüne, L.: Economic receding horizon control without terminal constraints. Automatica 49(3), 725–734 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grüne, L.: Approximation properties of receding horizon optimal control. Jahresber. DMV 118(1), 3–37 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Porretta, A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control Optim. 51(6), 4242–4273 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Angeli, D., Amrit, R., Rawlings, J.B.: On average performance and stability of economic model predictive control. IEEE Trans. Autom. Control 57(7), 1615–1626 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Damm, T., Grüne, L., Stieler, M., Worthmann, K.: An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J. Control Optim. 52(3), 1935–1957 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Müller, M.A., Angeli, D., Allgöwer, F.: On necessity and robustness of dissipativity in economic model predictive control. IEEE Trans. Autom. Control 60(6), 1671–1676 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Müller, M.A., Grüne, L., Allgöwer, F.: On the role of dissipativity in economic model predictive control. In: Proceedings of the 5th IFAC Conference on Nonlinear Model Predictive Control—NMPC’15, pp. 110–116. Seville, Spain (2015)Google Scholar
  18. 18.
    Gaitsgory, V., Grüne, L., Thatcher, N.: Stabilization with discounted optimal control. Syst. Control Lett. 82, 91–98 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Scheinkman, J.: On optimal steady states of \(n\)-sector growth models when utility is discounted. J. Econ. Theory 12(1), 11–30 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Brock, W.A., Mirman, L.: Optimal economic growth and uncertainty: the discounted case. J. Econ. Theory 4(3), 479–513 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Santos, M.S., Vigo-Aguiar, J.: Analysis of a numerical dynamic programming algorithm applied to economic models. Econometrica 66(2), 409–426 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kellett, C.M.: A compendium of comparison function results. Math. Control Signals Syst. 26(3), 339–374 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zabczyk, J.: Remarks on the control of discrete-time distributed parameter systems. SIAM J. Control 12(4), 721–735 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gibson, J.S., Rosen, I.G.: Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem. SIAM J. Control Optim. 26(2), 428–451 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Lars Grüne
    • 1
    Email author
  • Christopher M. Kellett
    • 2
  • Steven R. Weller
    • 2
  1. 1.Mathematical InstituteUniversity of BayreuthBayreuthGermany
  2. 2.School of Electrical Engineering and ComputingUniversity of NewcastleCallaghanAustralia

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