phone +7 (3412) 91 60 92

Archive of Issues

Russia Yekaterinburg
Section Mathematics
Title On necessary boundary conditions for strongly optimal control in infinite horizon control problems
Author(-s) Khlopin D.V.a
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa
Abstract In the paper we consider the infinite horizon control problems in the free end case. We obtain the necessary conditions of strong optimality. The method of the proof actually follows the classic paper by Halkin, and the boundary condition for infinity that we construct in our paper is a stronger variety of the Seierstad condition. The complete system of relations of the maximum principle that was obtained in the paper allows us to write the expression for the adjoint variable in the form of improper integral that depends only on the developing trajectory. S.M. Aseev, A.V. Kryazhimskii, and V.M. Veliov obtained the similar condition as a necessary condition for certain classes of control problems. As we note in our paper, the obtained conditions of strong optimality lead us to a redefined system of relations for sufficiently broad class of control problems. An example is considered.
Keywords control problem, strong optimal control, infinite horizon problem, necessary conditions of optimality, transversality condition for infinity, Pontryagin maximum principle
UDC 517.977.5
MSC 49K15, 49J45, 37N40
DOI 10.20537/vm130106
Received 11 February 2013
Language Russian
Citation Khlopin D.V. On necessary boundary conditions for strongly optimal control in infinite horizon control problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 49-58.
  1. Aseev S.M., Kryazhimskii A.V. The Pontryagin maximum principle and optimal economic growth problems, Proc. Steklov Inst. Math., 2007, vol. 257, pp. 1–255.
  2. Aseev S.M., Besov K.O., Kryazhimskii A.V. Infinite-horizon optimal control problems in economics, Russ. Math. Surv., 2012, vol. 67, no. 2, pp. 195–253.
  3. Clarke F.H. Optimization and nonsmooth analysis, New York: J. Wiley, 1983.
  4. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Mathematical theory of optimal processes, New York: Interscience Publishers, John Wiley and Sons, 1962.
  5. Khlopin D.V. On transversality condition in control problems for infinity horizon, Differential Equations and Optimal Control: Abstracts of Int. Conf. Dedicated to the 90th Anniversary of E.F. Mishchenko, Steklov Inst. Math., Moscow, 2008, pp. 144–146.
  6. Khlopin D.V. On $\tau$-vanishing adjoint variable for infinity-horizon control problems, Abstracts of Int. Conf. on Differential Equations and Dynamical Systems, Lomonosov Moscow State University, Moscow, 2012, pp. 173–174.
  7. Aseev S.M., Kryazhimskii A.V. The Pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons, SIAM J. Control Optim., 2004, vol. 43, pp. 1094-1119.
  8. Aseev S.M., Kryazhimskii A.V. Shadow prices in infinite-horizon optimal control problems with dominating discounts, Applied Mathematics and Computation, 2008, vol. 204, no. 2, pp. 519–531.
  9. Aseev S.M., Kryazhimskii A.V., Tarasyev A.M. The Pontryagin maximum principle and transversality conditions for an optimal control problem with infinite time interval, Proc. Steklov Inst. Math., 2001, no. 233, pp. 64–80.
  10. Aseev S.M., Veliov V.M. Needle variations in infinite-horizon optimal control, IIASA Interim Rept. IASA. Laxenburg, Research Report, 2012–04, September 2012, 22 pp.
  11. Aseev S.M., Veliov V.M. Maximum principle for infinite-horizon optimal control problems with dominating discount, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 2012, vol. 19, no. 1–2, pp. 43–63.
  12. Aubin J.P., Clarke F.H. Shadow prices and duality for a class of optimal control problems, SIAM J. Control Optim., 1979, vol. 17, pp. 567–586.
  13. Bogucz D. On the existence of a classical optimal solution and of an almost strongly optimal solution for an infinite-horizon control problem, J. Optim. Theory Appl., 2013, vol. 156, no. 2, pp. 650–682.
  14. Carlson D.A. Uniformly overtaking and weakly overtaking optimal solutions in infinite-horizon optimal control: when optimal solutions are agreeable, J. Optim. Theory Appl., 1990, vol. 64, no. 1, pp. 55–69.
  15. Carlson D.A., Haurie A.B., Leizarowitz A. Infinite horizon optimal control. Deterministic and stochastic systems. Berlin: Springer, 1991.
  16. Chakravarty S. The existence of an optimum savings program, Econometrica, 1962, vol. 30, pp. 178–187.
  17. Halkin H. Necessary conditions for optimal control problems with infinite horizons, Econometrica, 1974, vol. 42, pp. 267–272.
  18. Khlopin D.V. Necessity of vanishing shadow price in infinite horizon control problems,
  19. Pickenhain S. On adequate transversality conditions for infinite horizon optimal control problems - a famous example of Halkin. In: Crespo Cuaresma J.; Palokangas T.; Tarasyev A. (Eds.): Dynamic Systems, Economic Growth, and the Environment, Springer, 2010, pp. 3–22.
  20. Seierstad A. Necessary conditions for nonsmooth, infinite-horizon optimal control problems, J. Optim. Theory Appl., 1999, vol. 103, no. 1, pp. 201–230.
  21. Stern L.E. Criteria of optimality in the infinite-time optimal control problem, J. Optim. Theory Appl., 1984, vol. 44, no. 3, pp. 497–508.
  22. Wachs A.O., Schochetman I.E., Smith R.L. Average optimality in nonhomogeneous infinite horizon Markov decision processes, Math. Oper. Res., 2011, vol. 36, no. 1, pp. 147-164.
  23. Ye J.J. Nonsmooth maximum principle for infinite-horizon problems, J. Optim. Theory Appl., 1993, vol. 76, no. 3, pp. 485–500.
Full text
<< Previous article
Next article >>