Skip to main content
SpringerLink
Log in

The Pontryagin maximum principle and optimal economic growth problems

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

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

References

  1. V. V. Aleksandrov, V. G. Boltyanskii, S. S. Lemak, N. A. Parusnikov, and V. M. Tikhomirov, Optimization of the Dynamics of Control Systems (Mosk. Gos. Univ., Moscow, 2000) [in Russian].

    Google Scholar 

  2. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1979; Springer, New York, 1997).

    Google Scholar 

  3. V. I. Arnold, “Optimization in Mean and Phase Transitions in Controlled Dynamical Systems,” Funkts. Anal. Prilozh. 36(2), 1–11 (2002) [Funct. Anal. Appl. 36, 83–92 (2002)].

    Article  Google Scholar 

  4. A. V. Arutyunov, “Perturbations of Extremal Problems with Constraints and Necessary Optimality Conditions,” in Itogi Nauki Tekh., Ser. Mat. Anal. (VINITI, Moscow, 1989), Vol. 27, pp. 147–235 [J. Sov. Math. 54 (6), 1342–1400 (1991)].

    Google Scholar 

  5. A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems (Faktorial, Moscow, 1997; Kluwer, Dordrecht, 2000).

    Google Scholar 

  6. S. M. Aseev, “A Method of Smooth Approximation in the Theory of Necessary Optimality Conditions for Differential Inclusions,” Izv. Ross. Akad. Nauk, Ser. Mat. 61(2), 3–26 (1997) [Izv. Math. 61, 235–258 (1997)].

    MathSciNet  Google Scholar 

  7. S. M. Aseev, “Extremal Problems for Differential Inclusions with State Constraints,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 233, 5–70 (2001) [Proc. Steklov Inst. Math. 233, 1–63 (2001)].

    MathSciNet  Google Scholar 

  8. 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. Akad. Nauk 394(5), 583–585 (2004) [Dokl. Math. 69 (1), 89–91 (2004)].

    MathSciNet  Google Scholar 

  9. S. M. Aseev, A. V. Kryazhimskii, and A. M. Tarasyev, “The Pontryagin Maximum Principle and Transversality Conditions for an Optimal Control Problem with Infinite Time Interval,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 233, 71–88 (2001) [Proc. Steklov Inst. Math. 233, 64–80 (2001)].

    Google Scholar 

  10. S. M. Aseev and A. I. Smirnov, “First-Order Necessary Optimality Conditions for the Problem of Optimal Passage through a Given Domain,” in Nonlinear Dynamics and Control, Ed. by S.V. Emel’yanov and S.K. Korovin (Fizmatlit, Moscow, 2004), Issue 4, pp. 179–204 [in Russian].

    Google Scholar 

  11. N. A. Bobylev, S. V. Emel’yanov, and S. K. Korovin, Geometrical Methods in Variational Problems (Magistr, Moscow, 1998; Kluwer, Dordrecht, 1999).

    Google Scholar 

  12. J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).

    MATH  Google Scholar 

  13. B. V. Gnedenko, Theory of Probability (URSS, Moscow, 2001; Gordon and Breach, Newark, NJ, 1997).

    Google Scholar 

  14. B. P. Demidovich, Lectures on the Mathematical Theory of Stability (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  15. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 1979).

    Google Scholar 

  16. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  17. B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  18. L. S. Pontryagin, Ordinary Differential Equations (Nauka, Moscow, 1974; Addison-Wesley, Reading, 1962).

    Google Scholar 

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

    Google Scholar 

  20. A. F. Filippov, “On Some Problems in Optimal Control Theory,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., Astron., Fiz., Khim., No. 2, 25–32 (1959).

  21. A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer, Dordrecht, 1988).

    Google Scholar 

  22. P. Aghion and P. Howitt, Endogenous Growth Theory (MIT Press, Cambridge, MA, 1998).

    Google Scholar 

  23. K. J. Arrow, “Applications of Control Theory to Economic Growth,” in Mathematics of the Decision Sciences (Am. Math. Soc., Providence, RI, 1968), Part 2, pp. 85–119.

    Google Scholar 

  24. K. J. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy (J. Hopkins Univ. Press, Baltimore, MD, 1970).

    Google Scholar 

  25. A. V. Arutyunov and S. M. Aseev, “Investigation of the Degeneracy Phenomenon of the Maximum Principle for Optimal Control Problems with State Constraints,” SIAM J. Control Optim. 35, 930–952 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  26. S. M. Aseev, “Methods of Regularization in Nonsmooth Problems of Dynamic Optimization,” J. Math. Sci. 94(3), 1366–1393 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Aseev, G. Hutschenreiter, and A. Kryazhimskii, “A Dynamic Model of Optimal Allocation of Resources to R&D,” IIASA Interim Rept. IR-02-016 (Laxenburg, Austria, 2002).

    Google Scholar 

  28. S. Aseev, G. Hutschenreiter, and A. Kryazhimskii, “Optimal Investment in R&D with International Knowledge Spillovers,” WIFO Working Paper No. 175 (2002).

  29. S. M. Aseev, G. Hutschenreiter, and A. V. Kryazhimskii, “A Dynamical Model of Optimal Investment in R&D,” J. Math. Sci. 126(6), 1495–1535 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Aseev, G. Hutschenreiter, A. Kryazhimskiy, and A. Lysenko, “A Dynamical Model of Optimal Investment in Research and Development,” Math. Comput. Modell. Dyn. Syst. 11(2), 125–123 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Aseev and A. Kryazhimskii, “The Pontryagin Maximum Principle for Infinite-Horizon Optimal Controls,” IIASA Interim Rept. IR-03-013 (Laxenburg, Austria, 2003).

    Google Scholar 

  32. 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, 1094–1119 (2004).

    Article  MATH  Google Scholar 

  33. S. Aseev, A. Kryazhimskii, and A. Tarasyev, “First Order Necessary Optimality Conditions for a Class of Infinite Horizon Optimal-Control Problems,” IIASA Interim Rept. IR-01-007 (Laxenburg, Austria, 2001).

    Google Scholar 

  34. S. M. Aseev, A. V. Kryazhimskii, and A. M. Tarasyev, “The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Economic Growth Problems,” in Nonlinear Control Systems 2001: Proc. 5th IFAC Symp., St. Petersburg (Russia), July 4–6, 2001 (Elsevier, New York, 2001), Vol. 1, pp. 71–76.

    Google Scholar 

  35. 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 

  36. F.-C. Bagliano and G. Bertola, Models for Dynamic Macroeconomics (Oxford Univ. Press, New York, 2004).

    MATH  Google Scholar 

  37. E. J. Balder, “An Existence Result for Optimal Economic Growth Problems,” J. Math. Anal. Appl. 95, 195–213 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  38. L. Baratchart, M. Chyba, and J.-B. Pomet, “A Grobman-Hartman Theorem for Control Systems,” J. Dyn. Diff. Eqns. 19(1), 75–107 (2007).

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  41. J.-P. Benassy, “Is There Always Too Little Research in Endogenous Growth with Expanding Product Variety?,” Eur. Econ. Rev. 42, 61–69 (1998).

    Article  Google Scholar 

  42. 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 

  43. J. Blot and P. Michel, “First-Order Necessary Conditions for Infinite-Horizon Variational Problems,” J. Optim. Theory Appl. 88(2), 339–364 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  44. V. F. Borisov, G. Hutschenreiter, and A. V. Kryazhimskii, “Asymptotic Growth Rates in Knowledge-Exchanging Economies,” Ann. Oper. Res. 89, 61–73 (1999).

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  46. D. Cass, “Optimum Growth in an Aggregative Model of Capital Accumulation,” Rev. Econ. Stud. 32, 233–240 (1965).

    Article  Google Scholar 

  47. L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, Berlin, 1959).

    MATH  Google Scholar 

  48. L. Cesari, Optimization—Theory and Applications. Problems with Ordinary Differential Equations (Springer, New York, 1983).

    MATH  Google Scholar 

  49. A. C. Chiang, Elements of Dynamic Optimization (McGraw Hill, Singapore, 1992).

    Google Scholar 

  50. F. H. Clarke, Optimization and Nonsmooth Analysis (J. Wiley, New York, 1983).

    MATH  Google Scholar 

  51. A. K. Dixit and R. S. Pyndyck, Investment under Uncertainty (Princeton Univ. Press, Princeton, NJ, 1994).

    Google Scholar 

  52. A. K. Dixit and J. Stiglitz, “Monopolistic Competition and Optimum Product Diversity,” Am. Econ. Rev. 67, 297–308 (1977).

    Google Scholar 

  53. R. Dorfman, “An Economic Interpretation of Optimal Control Theory,” Am. Econ. Rev. 59, 817–831 (1969).

    Google Scholar 

  54. I. Ekeland, “Some Variational Problems Arising from Mathematical Economics,” in Mathematical Economics (Springer, Berlin, 1988), Lect. Notes Math. 1330, p. 1–18.

    Chapter  Google Scholar 

  55. W. J. Ethier, “National and International Returns to Scale in the Modern Theory of International Trade,” Am. Econ. Rev. 72, 389–405 (1982).

    Google Scholar 

  56. R. V. Gamkrelidze, Principles of Optimal Control Theory (Plenum Press, New York, 1978).

    MATH  Google Scholar 

  57. G. M. Grossman and E. Helpman, Innovation and Growth in the Global Economy (MIT Press, Cambridge, MA, 1991).

    Google Scholar 

  58. H. Halkin, “Necessary Conditions for Optimal Control Problems with Infinite Horizons,” Econometrica 42, 267–272 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  59. P. Hartman, Ordinary Differential Equations (J. Wiley & Sons, New York, 1964).

    MATH  Google Scholar 

  60. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, London, 1974).

    MATH  Google Scholar 

  61. G. Hutschenreiter, Yu. M. Kaniovski, and A. V. Kryazhimskii, “Endogenous Growth, Absorptive Capacities, and International R&D Spillovers,” IIASA Work. Paper WP-95-092 (Laxenburg, Austria, 1995).

    Google Scholar 

  62. K. Inada, “On a Two-Sector Model of Economic Growth: Comments and a Generalization,” Rev. Econ. Stud. 30(2), 119–127 (1963).

    Article  Google Scholar 

  63. M. D. Intriligator, Mathematical Optimization and Economic Theory (Prentice-Hall, Englewood Cliffs, NJ, 1971).

    Google Scholar 

  64. T. Kamihigashi, “Necessity of Transversality Conditions for Infinite Horizon Problems,” Econometrica 69, 995–1012 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  65. T. C. Koopmans, “Objectives, Constraints, and Outcomes in Optimal Growth Models,” Econometrica 35, 1–15 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  66. A. Liapounoff, Problème général de la stabilité du mouvement (Princeton Univ. Press, Princeton, NJ, 1947), Ann. Math. Stud. 17.

    MATH  Google Scholar 

  67. S. F. Leung, “Transversality Condition and Optimality in a Class of Infinite Horizon Continuous Time Economic Models,” J. Econ. Theory 54, 224–233 (1991).

    Article  MATH  Google Scholar 

  68. H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion (Springer, Berlin, 1989), Lect. Notes Econ. Math. Syst. 334.

    MATH  Google Scholar 

  69. O. L. Mangasarian, “Sufficient Conditions for the Optimal Control of Nonlinear Systems,” SIAM J. Control 4, 139–152 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  70. P. Michel, “On the Transversality Condition in Infinite Horizon Optimal Problems,” Econometrica 50, 975–985 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  71. W. D. Nordhaus, Managing the Global Commons. The Economics of Climate Change (MIT Press, Cambridge, MA, 1994).

    Google Scholar 

  72. F. P. Ramsey, “A Mathematical Theory of Saving,” Econ. J. 38, 543–559 (1928).

    Article  Google Scholar 

  73. P. M. Romer, “Endogenous Technological Change,” J. Polit. Econ. 98(5, Part 2), S71–S102 (1990).

    Google Scholar 

  74. P. A. Samuelson, “Paul Douglas’s Measurement of Production Functions and Marginal Productivities,” J. Polit. Econ. 87(5), 923–939 (1979).

    Article  Google Scholar 

  75. A. Seierstad, “Necessary Conditions for Nonsmooth, Infinite-Horizon, Optimal Control Problems,” J. Optim. Theory Appl. 103(1), 201–229 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  76. A. Seierstad and K. Sydsaeter, Optimal Control Theory with Economic Applications (North-Holland, Amsterdam, 1987).

    MATH  Google Scholar 

  77. S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics (Kluwer, Dordrecht, 2000).

    MATH  Google Scholar 

  78. K. Shell, “Applications of Pontryagin’s Maximum Principle to Economics,” in Mathematical Systems Theory and Economics 1 (Springer, Berlin, 1969), Lect. Notes Oper. Res. Math. Econ. 11, pp. 241–292.

    Google Scholar 

  79. G. V. Smirnov, “Transversality Condition for Infinite-Horizon Problems,” J. Optim. Theory Appl. 88(3), 671–688 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  80. R. M. Solow, Growth Theory: An Exposition (Oxford Univ. Press, New York, 1970).

    Google Scholar 

  81. L. E. Stern, “Criteria of Optimality in the Infinite-Time Optimal Control Problem,” J. Optim. Theory Appl. 44(3), 497–508 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  82. M. L. Weitzman, Income, Wealth, and the Maximum Principle (Harvard Univ. Press, Cambridge, MA, 2003).

    MATH  Google Scholar 

  83. J. J. Ye, “Nonsmooth Maximum Principle for Infinite-Horizon Problems,” J. Optim. Theory Appl. 76(3), 485–500 (1993).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    S. M. Aseev & A. V. Kryazhimskii

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

    S. M. Aseev & A. V. Kryazhimskii

Authors
  1. S. M. Aseev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Kryazhimskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aseev, S.M., Kryazhimskii, A.V. The Pontryagin maximum principle and optimal economic growth problems. Proc. Steklov Inst. Math. 257, 1–255 (2007). https://doi.org/10.1134/S0081543807020010

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543807020010

Keywords

Search

Navigation