Skip to main content
SpringerLink
Log in

On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

For a class of infinite-horizon optimal control problems that appear in studies on economic growth processes, the properties of the adjoint variable in the relations of the Pontryagin maximum principle, defined by a formula similar to the Cauchy formula for the solutions to linear differential systems, are studied. It is shown that under a dominating discount type condition the adjoint variable defined in this way satisfies both the core relations of the maximum principle (the adjoint system and the maximum condition) in the normal form and the complementary stationarity condition for the Hamiltonian. Moreover, a new economic interpretation of the adjoint variable based on this formula is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979; Plenum, New York, 1987).

    Google Scholar 

  2. S. M. Aseev, K. O. Besov, and A. V. Kryazhimskii, “Infinite-horizon optimal control problems in economics,” Russ. Math. Surv. 67(2), 195–253 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  3. S. M. Aseev and A. V. Kryazhimskii, “The Pontryagin maximum principle for an optimal control problem with a functional specified by an improper integral,” Dokl. Math. 69(1), 89–91 (2004).

    MATH  Google Scholar 

  4. S. M. Aseev and A. V. Kryazhimskii, “The Pontryagin maximum principle and optimal economic growth problems,” Proc. Steklov Inst. Math. 257, 1–255 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. M. Aseev and A. V. Kryazhimskii, “On a class of optimal control problems arising in mathematical economics,” Proc. Steklov Inst. Math. 262, 1–16 (2008).

    Article  MathSciNet  Google Scholar 

  6. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).

    Google Scholar 

  7. P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964).

    MATH  Google Scholar 

  8. D. Acemoglu, Introduction to Modern Economic Growth (Princeton Univ. Press, Princeton, NJ, 2008).

    Google Scholar 

  9. S. M. Aseev and A. V. Kryazhimskiy, “The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons,” SIAM J. Control Optim. 43(3), 1094–1119 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  10. S. M. Aseev and V. M. Veliov, “Maximum principle for infinite-horizon optimal control problems with dominating discount,” Dynamics of Continuous, Discrete and Impulsive Systems, Ser. B: Applications and Algorithms 19(1–2), 43–63 (2012).

    MATH  MathSciNet  Google Scholar 

  11. S. M. Aseev and V. M. Veliov, “Needle variations in infinite-horizon optimal control,” in Variational and Optimal Control Problems on Unbounded Domains, Ed. by G. Wolansky and A. J. Zaslavski (Amer. Math. Soc., Providence, RI, 2014), Ser. Contemporary Mathematics, Vol. 619, pp. 1–17.

    Chapter  Google Scholar 

  12. J. P. Aubin and F. H. Clarke, “Shadow prices and duality for a class of optimal control problems,” SIAM J. Control Optim. 17, 567–586 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  13. R. J. Barro and X. Sala-i-Martin, Economic Growth (McGraw Hill, New York, 1995).

    Google Scholar 

  14. R. Bellman, Dynamic Programming (Princeton Univ. Press, Princeton, NJ, 1957).

    MATH  Google Scholar 

  15. L. M. Benveniste and J. A. Scheinkman, “Duality theory for dynamic optimization models of economics: The continuous time case,” J. Econ. Theory 27, 1–19 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  16. D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems (Springer, Berlin, 1991).

    Google Scholar 

  17. R. Dorfman, “An economic interpretation of optimal control theory,” Am. Econ. Rev. 59, 817–831 (1969).

    Google Scholar 

  18. H. Halkin, “Necessary conditions for optimal control problems with infinite horizons,” Econometrica 42, 267–272 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Michel, “On the transversality conditions in infinite horizon optimal problems,” Econometrica 50, 975–985 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  20. F. P. Ramsey, “A mathematical theory of saving,” Econ. J. 38, 543–559 (1928).

    Article  Google Scholar 

  21. A. Seierstad, “Necessary conditions for nonsmooth, infinite-horizon optimal control problems,” J. Optim. Theory Appl. 103, 201–229 (1999).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of the Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. M. Aseev

  2. International Institute for Applied Systems Analysis, Schlossplatz 1, A-2361, Laxenburg, Austria

    S. M. Aseev

Authors
  1. S. M. Aseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. M. Aseev.

Additional information

Original Russian Text © S.M. Aseev, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 4.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aseev, S.M. On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems. Proc. Steklov Inst. Math. 287 (Suppl 1), 11–21 (2014). https://doi.org/10.1134/S0081543814090028

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543814090028

Keywords

Search

Navigation