The model of optimal economic growth is considered, in which natural capital serves as the primary factor of production. The evolution of natural capital is described using the logistic Verhulst equation, while economic output is modeled using a Cobb-Douglas type production function. Taking into account the negative economic impact of natural capital degradation, the model assumes that only a portion of total output, reduced according to a damage function, is available for consumption. Key challenges in analyzing this problem arise from several factors: the infinite time horizon itself, the unbounded nature of the geometric control constraints, the singularity in the objective functional, and the non-concavity of the Hamiltonian with respect to the phase variable. The main contribution of the paper is the theorem that proves the existence and local boundedness of an optimal admissible control.
Keywords: optimal control, economic growth, natural capital, infinite horizon, existence of optimal control
Received January 12, 2026
Revised January 21, 2026
Accepted January 26, 2026
Sergey Mironovich Aseev, Dr. Phys.-Math. Sci., Principal research scholar, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia, e-mail: aseev@mi-ras.ru
REFERENCES
1. Barbier B.E. Capitalizing on nature: ecosystems as natural assets. Cambridge, NY, Cambridge Univer. Press, 2011, 321 p. ISBN: 978-0-521-18927-9 .
2. Schumacher E.F. Small is beautiful: economics as if people mattered. London, Blond & Briggs Publ., 1973, 223 p.
3. Nordhaus W.D. Managing the global commons: the economics of climate change. Cambridge, Mass., MIT Press, 1994, 213 p. ISBN-10: 0262140551 .
4. Aseev S., Manzoor T. Optimal exploitation of renewable resources: lessons in sustainability from an optimal growth model of natural resource consumption. In: G. Feichtinger, R. Kovacevic, G. Tragler (eds.) Control systems and mathematical methods in economics. Cham, Springer, 2018, Ser. Lecture notes in economics and mathematical systems, vol. 687, pp. 221–245. https://doi.org/10.1007/978-3-319-75169-6_11
5. Verhulst P.F. Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys., 1838, vol. 10, pp. 113–121.
6. Cobb C.W., Douglas P.H. A theory of production. The Amer. Econ. Rev., 1928, vol. 18, no. 1, Suppl., Papers and Proc. Fortieth Annual Meeting of the Amer. Economic Association (Mar., 1928), pp. 139–165. https://www.jstor.org/stable/1811556
7. Aseev S.M., Besov K.O., Kryazhimskiy A.V. Infinite-horizon optimal control problems in economics. Russ. Math. Surv., 2012, vol. 67, iss. 2, pp. 195–253. https://doi.org/10.1070/RM2012v067n02ABEH004785
8. Carlson D.A., Haurie A.B., Leizarowitz A. Infinite horizon optimal control. Deterministic and stochastic systems. Berlin, Springer-Verlag, 1991, 332 p. https://doi.org/10.1007/978-3-642-76755-5
9. Hartman P. Ordinary differential equations, NY, London, Sydney, John Wiley & Sons, Inc., 1964, 640 p. Translated to Russian under the title Obyknovennyye differentsial’nye uravneniya, Moscow, Mir Publ., 1970, 720 p.
10. Seierstad A., Sydsaeter K. Sufficient conditions in optimal control theory. Inter. Econom. Review., 1977, vol. 18, no. 2, pp. 367–391.
11. Balder E.J. An existence result for optimal economic growth problems. J. Math. Anal. Appl., 1983, vol. 95, iss. 1, pp. 195–213.
12. Lavrentiev M. Sur quelques probl`emes du calcul des variations. Ann. Mat. Pura Appl., 1927, vol. 4, pp. 7–28. https://doi.org/10.1007/BF02409983
13. Aseev S.M. Existence of an optimal control in infinite-horizon problems with unbounded set of control constraints. Proc. Steklov Inst. Math., 2017, vol. 297, iss. 1, pp. S1–S10. https://doi.org/10.1134/S0081543817050017
14. Aseev S.M. On the boundedness of optimal controls in infinite-horizon problems. Proc. Steklov Inst. Math., 2015, vol. 291, pp. 38–48. https://doi.org/10.1134/S0081543815080040
15. Aseev S.M. Conditional cost function and necessary optimality conditions for infinite horizon optimal control problems. Dokl. RAN. Matem., inform., prots. upr., 2023, vol. 514, no. 1, pp. 5–11 (in Russian). https://doi.org/10.31857/S2686954323700315
16. Aseev S.M., Veliov V.M. Another view of the maximum principle for infinite-horizon optimal control problems in economics. Russ. Math. Surv., 2019, vol. 74, iss. 6, pp. 963–1011. https://doi.org/10.1070/RM9915
17. Aseev S.M., Veliov V.M. Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions. Proc. Steklov Inst. Math., 2015, vol. 291, iss. 1, pp. S22–S39. https://doi.org/10.1134/S0081543815090023
Cite this article as: S.M. Aseev. On the existence of optimal control in a model of economic growth based on natural capital.Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 29–43.
