The Pontryagin maximum principle and optimal economic growth problems

  • S. M. Aseev
  • A. V. Kryazhimskii


Maximum Principle Hamiltonian System Optimal Control Problem STEKLOV Institute Curve Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 2.
    V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1979; Springer, New York, 1997).Google Scholar
  3. 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)].CrossRefGoogle Scholar
  4. 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. 5.
    A. V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems (Faktorial, Moscow, 1997; Kluwer, Dordrecht, 2000).Google Scholar
  6. 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)].MathSciNetGoogle Scholar
  7. 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)].MathSciNetGoogle Scholar
  8. 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)].MathSciNetGoogle Scholar
  9. 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. 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. 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. 12.
    J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972; Nauka, Moscow, 1977).MATHGoogle Scholar
  13. 13.
    B. V. Gnedenko, Theory of Probability (URSS, Moscow, 2001; Gordon and Breach, Newark, NJ, 1997).Google Scholar
  14. 14.
    B. P. Demidovich, Lectures on the Mathematical Theory of Stability (Nauka, Moscow, 1967) [in Russian].Google Scholar
  15. 15.
    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 1979).Google Scholar
  16. 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. 17.
    B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control (Nauka, Moscow, 1988) [in Russian].MATHGoogle Scholar
  18. 18.
    L. S. Pontryagin, Ordinary Differential Equations (Nauka, Moscow, 1974; Addison-Wesley, Reading, 1962).Google Scholar
  19. 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. 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).Google Scholar
  21. 21.
    A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer, Dordrecht, 1988).Google Scholar
  22. 22.
    P. Aghion and P. Howitt, Endogenous Growth Theory (MIT Press, Cambridge, MA, 1998).Google Scholar
  23. 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. 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. 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).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    S. M. Aseev, “Methods of Regularization in Nonsmooth Problems of Dynamic Optimization,” J. Math. Sci. 94(3), 1366–1393 (1999).MATHCrossRefMathSciNetGoogle Scholar
  27. 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. 28.
    S. Aseev, G. Hutschenreiter, and A. Kryazhimskii, “Optimal Investment in R&D with International Knowledge Spillovers,” WIFO Working Paper No. 175 (2002).Google Scholar
  29. 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).MATHCrossRefMathSciNetGoogle Scholar
  30. 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).MATHCrossRefMathSciNetGoogle Scholar
  31. 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. 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).MATHCrossRefGoogle Scholar
  33. 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. 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. 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).MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    F.-C. Bagliano and G. Bertola, Models for Dynamic Macroeconomics (Oxford Univ. Press, New York, 2004).MATHGoogle Scholar
  37. 37.
    E. J. Balder, “An Existence Result for Optimal Economic Growth Problems,” J. Math. Anal. Appl. 95, 195–213 (1983).MATHCrossRefMathSciNetGoogle Scholar
  38. 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).MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    R. J. Barro and X. Sala-i-Martin, Economic Growth (McGraw Hill, New York, 1995).Google Scholar
  40. 40.
    R. Bellman, Dynamic Programming (Princeton Univ. Press, Princeton, NJ, 1957).Google Scholar
  41. 41.
    J.-P. Benassy, “Is There Always Too Little Research in Endogenous Growth with Expanding Product Variety?,” Eur. Econ. Rev. 42, 61–69 (1998).CrossRefGoogle Scholar
  42. 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).MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    J. Blot and P. Michel, “First-Order Necessary Conditions for Infinite-Horizon Variational Problems,” J. Optim. Theory Appl. 88(2), 339–364 (1996).MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    V. F. Borisov, G. Hutschenreiter, and A. V. Kryazhimskii, “Asymptotic Growth Rates in Knowledge-Exchanging Economies,” Ann. Oper. Res. 89, 61–73 (1999).MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control. Deterministic and Stochastic Systems (Springer, Berlin, 1991).MATHGoogle Scholar
  46. 46.
    D. Cass, “Optimum Growth in an Aggregative Model of Capital Accumulation,” Rev. Econ. Stud. 32, 233–240 (1965).CrossRefGoogle Scholar
  47. 47.
    L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, Berlin, 1959).MATHGoogle Scholar
  48. 48.
    L. Cesari, Optimization—Theory and Applications. Problems with Ordinary Differential Equations (Springer, New York, 1983).MATHGoogle Scholar
  49. 49.
    A. C. Chiang, Elements of Dynamic Optimization (McGraw Hill, Singapore, 1992).Google Scholar
  50. 50.
    F. H. Clarke, Optimization and Nonsmooth Analysis (J. Wiley, New York, 1983).MATHGoogle Scholar
  51. 51.
    A. K. Dixit and R. S. Pyndyck, Investment under Uncertainty (Princeton Univ. Press, Princeton, NJ, 1994).Google Scholar
  52. 52.
    A. K. Dixit and J. Stiglitz, “Monopolistic Competition and Optimum Product Diversity,” Am. Econ. Rev. 67, 297–308 (1977).Google Scholar
  53. 53.
    R. Dorfman, “An Economic Interpretation of Optimal Control Theory,” Am. Econ. Rev. 59, 817–831 (1969).Google Scholar
  54. 54.
    I. Ekeland, “Some Variational Problems Arising from Mathematical Economics,” in Mathematical Economics (Springer, Berlin, 1988), Lect. Notes Math. 1330, p. 1–18.CrossRefGoogle Scholar
  55. 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. 56.
    R. V. Gamkrelidze, Principles of Optimal Control Theory (Plenum Press, New York, 1978).MATHGoogle Scholar
  57. 57.
    G. M. Grossman and E. Helpman, Innovation and Growth in the Global Economy (MIT Press, Cambridge, MA, 1991).Google Scholar
  58. 58.
    H. Halkin, “Necessary Conditions for Optimal Control Problems with Infinite Horizons,” Econometrica 42, 267–272 (1974).MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    P. Hartman, Ordinary Differential Equations (J. Wiley & Sons, New York, 1964).MATHGoogle Scholar
  60. 60.
    M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, London, 1974).MATHGoogle Scholar
  61. 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. 62.
    K. Inada, “On a Two-Sector Model of Economic Growth: Comments and a Generalization,” Rev. Econ. Stud. 30(2), 119–127 (1963).CrossRefGoogle Scholar
  63. 63.
    M. D. Intriligator, Mathematical Optimization and Economic Theory (Prentice-Hall, Englewood Cliffs, NJ, 1971).Google Scholar
  64. 64.
    T. Kamihigashi, “Necessity of Transversality Conditions for Infinite Horizon Problems,” Econometrica 69, 995–1012 (2001).MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    T. C. Koopmans, “Objectives, Constraints, and Outcomes in Optimal Growth Models,” Econometrica 35, 1–15 (1967).MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    A. Liapounoff, Problème général de la stabilité du mouvement (Princeton Univ. Press, Princeton, NJ, 1947), Ann. Math. Stud. 17.MATHGoogle Scholar
  67. 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).MATHCrossRefGoogle Scholar
  68. 68.
    H.-W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion (Springer, Berlin, 1989), Lect. Notes Econ. Math. Syst. 334.MATHGoogle Scholar
  69. 69.
    O. L. Mangasarian, “Sufficient Conditions for the Optimal Control of Nonlinear Systems,” SIAM J. Control 4, 139–152 (1966).MATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    P. Michel, “On the Transversality Condition in Infinite Horizon Optimal Problems,” Econometrica 50, 975–985 (1982).MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    W. D. Nordhaus, Managing the Global Commons. The Economics of Climate Change (MIT Press, Cambridge, MA, 1994).Google Scholar
  72. 72.
    F. P. Ramsey, “A Mathematical Theory of Saving,” Econ. J. 38, 543–559 (1928).CrossRefGoogle Scholar
  73. 73.
    P. M. Romer, “Endogenous Technological Change,” J. Polit. Econ. 98(5, Part 2), S71–S102 (1990).Google Scholar
  74. 74.
    P. A. Samuelson, “Paul Douglas’s Measurement of Production Functions and Marginal Productivities,” J. Polit. Econ. 87(5), 923–939 (1979).CrossRefGoogle Scholar
  75. 75.
    A. Seierstad, “Necessary Conditions for Nonsmooth, Infinite-Horizon, Optimal Control Problems,” J. Optim. Theory Appl. 103(1), 201–229 (1999).MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    A. Seierstad and K. Sydsaeter, Optimal Control Theory with Economic Applications (North-Holland, Amsterdam, 1987).MATHGoogle Scholar
  77. 77.
    S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics (Kluwer, Dordrecht, 2000).MATHGoogle Scholar
  78. 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. 79.
    G. V. Smirnov, “Transversality Condition for Infinite-Horizon Problems,” J. Optim. Theory Appl. 88(3), 671–688 (1996).MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    R. M. Solow, Growth Theory: An Exposition (Oxford Univ. Press, New York, 1970).Google Scholar
  81. 81.
    L. E. Stern, “Criteria of Optimality in the Infinite-Time Optimal Control Problem,” J. Optim. Theory Appl. 44(3), 497–508 (1984).MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    M. L. Weitzman, Income, Wealth, and the Maximum Principle (Harvard Univ. Press, Cambridge, MA, 2003).MATHGoogle Scholar
  83. 83.
    J. J. Ye, “Nonsmooth Maximum Principle for Infinite-Horizon Problems,” J. Optim. Theory Appl. 76(3), 485–500 (1993).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • S. M. Aseev
    • 1
    • 2
  • A. V. Kryazhimskii
    • 1
    • 2
  1. 1.Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria

Personalised recommendations