Skip to main navigation menu Skip to main content Skip to site footer

An aspect of the turnpike property. Long time horizon behavior

Abstract

The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval.

In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

2010 Mathematics Subject Classification:

49N10, 90C31, 46N10

Keywords

turnpike property, linear-quadratic optimal control problem, linear systems, dynamic optimal control problem

Full text

Author Details

Martin Gugat

Friedrich-Alexander-Universität Erlangen-Nürnberg
Department of Mathematics
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics
(Alexander von Humboldt-Professur)
Cauerstr. 11, 91058 Erlangen, Germany
martin.gugat@fau.de

Jan Sokolowski

Institut Élie Cartan de Lorraine, CNRS
UMR 7502, Université de Lorraine, B.P. 70239
54506 Vandoeuvre-lés-Nancy Cedex, France
jan.sokolowski@univ-lorraine.fr
and
Systems Research Institute of the Polish Academy of Sciences
ul. Newelska 6, 01-447 Warszawa, Poland
and
Department of Scientific Computing, Informatics Center
Federal University of Paraiba, Brazil


References

  1. S. J. Bernau. The square root of a positive self-adjoint operator. J. Austral. Math. Soc. 8 (1968), 17–36.
  2. T. Breiten, L. Pfeiffer. On the turnpike property and the receding-horizon method for linear-quadratic optimal control problems. SIAM J. Control Optim. 58, 2 (2020), 1077–1102, doi: 10.1137/18M1225811.
  3. T. Faulwasser, K. Flasskamp, S. Ober-Blöbaum, M. Schaller, K. Worthmann. Manifold turnpikes, trims, and symmetries. Math. Control Signals Systems 34, 4 (2022), 759–788, doi: 10.1007/s00498-022-00321-6.
  4. T. Faulwasser, L. Grüne, J.-P. Humaloja, M. Schaller. Inferring the adjoint turnpike property from the primal turnpike property, 2021 60th IEEE Conference on Decision and Control (CDC), Austin, TX, USA, 2021, 2578–2583, doi: 10.1109/CDC45484.2021.9683079.
  5. T. Faulwasser, L. Grüne. Turnpike properties in optimal control: An overview of discrete-time and continuous-time results. In: Handbook of Numerical Analysis, vol. 23 (eds E. Trélat, E. Zuazua), 2022, 367–400.
  6. B. Geshkovski, E. Zuazua. Turnpike in optimal control of pdes, resnets, and beyond. Acta Numer. 31 (2022), 135–263, doi: 10.1017/S0962492922000046.
  7. L. Grüne, M. Schaller, A. Schiela. Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs. ESAIM Control Optim. Calc. Var. 27, 56 (2021), Paper No. 56, 28 pp, doi: 10.1051/cocv/2021030.
  8. M. Gugat. A turnpike result for convex hyperbolic optimal boundary control problems. Pure Appl. Funct. Anal. 4, 4 (2019), 849–866.
  9. M. Gugat, F. M. Hante. On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems. SIAM J. Control Optim. 57, 1 (2019), 264–289, doi: 10.1137/17M1134470.
  10. M. Gugat, M. Herty. A turnpike result for optimal boundary control of gas pipeline flow. 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2022): 12–16 September 2022, Bayreuth, Germany, https://epub.uni-bayreuth.de/id/eprint/6809/1/MTNS2022_ExtendedAbstracts_2022-12-22.pdf.
  11. M. Gugat, A. Keimer, G. Leugering. Optimal distributed control of the wave equation subject to state constraints. ZAMM, Z. Angew. Math. Mech. 89 6 (2009), 420–444, doi: 10.1002/zamm.200800196.
  12. M. Gugat, M. Schuster, E. Zuazua. The finite-time turnpike phenomenon for optimal control problems: stabilization by non-smooth tracking terms. In: Stabilization of distributed parameter systems: design methods and applications, ICIAM 2019 SEMA SIMAI Springer Ser., vol. 2. Cham, Springer, 2021, 17–41, doi: 10.1007/978-3-030-61742-4_2.
  13. M. Gugat, S. Steffensen. Dynamic boundary control games with networks of strings. ESAIM Control Optim. Calc. Var. 24, 4 (2018), 1789–1813, doi: 10.1051/cocv/2017082.
  14. M. Herty, A. N. Sandjo. On optimal treatment planning in radiotherapy governed by transport equations. Math. Models Methods Appl. Sci. 21, 2 (2011), 345–359, doi: 10.1142/S0218202511005076.
  15. K. Kunisch, P. Trautmann, B. Vexler. Optimal control of the undamped linear wave equation with measure valued controls. SIAM J. Control Optim. 54, 3 (2016), 1212–1244, doi: 10.1137/141001366.
  16. I. Lasiecka, J. L. Lions, R. Triggiani. Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9) 65, 2 (1986), 149–192.
  17. I. Lasiecka, R. Triggiani. Control theory for partial differential equations: continuous and approximation theories. II. Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia Math. Appl., vol. 75. Cambridge, Cambridge University Press, 2000, doi: 10.1017/CBO9780511574801.002.
  18. J.-L. Lions. Optimal control of systems governed by partial differential equations. Die Grundlehren der mathematischen Wissenschaften, Band 170. Translated from the French by S. K. Mitter. New York-Berlin, Springer-Verlag, 1971.
  19. P. A. Samuelson. A catenary turnpike theorem involving consumption and the golden rule. The American Economic Review 55, 3 (1965), 486–496.
  20. Z. Sebestyén, Z. Tarcsay. On the square root of a positive selfadjoint operator. Period. Math. Hungar. 75, 2 (2017), 268–272, doi: 10.1007/s10998-017-0192-1.
  21. E. Trélat, C. Zhang. Integral and measure-turnpike properties for infinite-dimensional optimal control systems. Math. Control Signals Systems 30, 1 (2018), Art, 3, 34 pp, doi: 10.1007/s00498-018-0209-1.
  22. M. Tucsnak, G. Weiss. Observation and control for operator semigroups. Birkhäuser Adv. Texts, Basler Lehrbüch [Birkhäuser Advanced Texts: Basel Textbooks]. Basel: Birkhäuser Verlag, 2009.
  23. A. Wouk. A note on square roots of positive operators. SIAM Rev. 8 (1966), 100–102, doi: 10.1137/1008008.
  24. A. J. Zaslavski. Discrete-Time Optimal Control and Games on Large Intervals. Springer Optim. Appl., vol. 119. Cham, Springer, 2017, doi: 10.1007/978-3-319-52932-5.
  25. A. J. Zaslavski. Turnpike conditions in infinite dimensional optimal control. Springer Optim. Appl., vol. 148. Cham, Springer, 2019, doi: 10.1007/978-3-030-20178-4.