A.S. Aseev, S.P. Samsonov. On the problem of optimal stimulation of demand ... P. 23-38

We study the problem of optimal stimulation of demand based on a controlled version of Kaldor’s business cycle model. Using the approximation method, we prove a version of Pontryagin’s maximum principle in the normal form, containing an additional pointwise condition on the adjoint variable. The results obtained develop and strengthen the previous results in this direction.

Keywords: optimal control, Kaldor’s business cycle model, Pontryagin’s maximum principle

Received April 1, 2024

Revised April 5, 2024

Accepted April 8, 2024

Anton Sergeevich Aseev, Lomonosov Moscow State University, Leninskiye Gory 1, Moscow, 119991 Russia, e-mail: anton.ser.as@gmail.com

Sergey Petrovich Samsonov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Leninskiye Gory 1, Moscow, 119991 Russia, e-mail: samsonov@cs.msu.ru

REFERENCES

1.   Ramsey F.P. A mathematical theory of saving. The Economic Journal, 1928, vol. 38, pp. 543–559. doi: 10.1111/ecoj.12229

2.   Barro R.J., Sala-i-Martin X. Economic growth. 2nd ed. Cambridge, The MIT Press, 2004. ISBN: 978-0-262-02553-9 . Translated to Russian under the title Ekonomicheskii rost, Moscow, BINOM. Laboratoriya znanii Publ., 2010, 824 p.

3.   Acemoglu D. Introduction to modern economic growth. Princeton N.J.: Princeton Univ. Press, 2009, 1008 p. ISBN: 978-0-691-13292-1 .

4.   Carlson D.A., Haurie A.B., Leizarowitz A. Infinite horizon optimal control. Deterministic and stochastic systems. Berlin, Springer-Verlag, 1991, 332 p. doi: 10.1007/978-3-642-76755-5

5.   Aseev A.S. Optimal economic growth problem. J. Math. Sci., 2023, vol. 276, no. 1, pp. 37–47. doi: 10.1007/s10958-023-06723-4

6.   Kaldor N. A model of trade cycle. The Economic J., 1940, vol. 50, no. 197, pp. 78–92. doi: 10.2307/2225740

7.   Aseev A.S. Optimal stationary regimes in Kaldor’s business-cycle controlled model. Math. Models Comput. Simul., 2019, vol. 11, iss. 5, pp. 750–758. doi: 10.1134/S2070048219050028

8.   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. doi: 10.1134/S0081543807020010

9.   Weitzman M.J. Income, wealth, and the maximum principle, Cambridge MA, Harvard Univ. Press, 2003, 358 p. ISBN-13: 978-0-674-01044-4 .

10.   Seierstad A., Sydsæter K. Optimal control theory with economic applications, Amsterdam, North-Holland Publ., 1987, 472 p. ISBN-13: 978-0444879233 .

11.   Aseev S.M., Besov K.O., Kryazhimskiy A.V. Infinite-horizon optimal control problems in economics. Russian Math. Surveys, 2012, vol. 67, no. 2, pp. 195–253. doi: 10.1070/rm2012v067n02abeh004785

12.   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, suppl. 1, pp. S22–S39. doi: 10.1134/S0081543815090023

13.   Pickenhain S. Infinite horizon optimal control problems in the light of convex analysis in Hilbert spaces. J. Set-Valued Var. Anal., 2015, vol. 23, no. 1, pp. 169–189. doi: 10.1007/s11228-014-0304-5

14.   Tauchnitz N. The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon. J. Optim. Theory Appl., 2015, vol. 167, no. 1, pp. 27–48. doi: 10.1007/s10957-015-0723-y

15.   Cannarsa P., Frankowska H. Value function, relaxation, and transversality conditions in infinite horizon optimal control. J. Math. Anal. Appl. , 2018, vol. 457, no. 2, pp. 1188–1217. doi: 10.1016/j.jmaa.2017.02.009

16.   Ye J.J. Nonsmooth maximum principle for infinite-horizon problems. J. Optim. Theory and Appl., 1993, vol. 76, no. 3, pp. 485–500. doi: 10.1007/BF00939379

17.   Cesari L. Optimization — theory and applications. Problems with ordinary differential equations, NY, Springer-Verlag, 1983. doi: 10.1007/978-1-4613-8165-5

18.   Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. (Two volumes in one, translated from the first Russian edition 1957–1961). Martino Fine Books, United States, 2012, 280 p. ISBN: 1614273049 . The 4th edition of Russian text published in Elementy teorii funktsii i funktsional’nogo analiza. Moscow, Nauka Publ., 1976, 544 p.

19.   Clarke F.H. Optimization and nonsmooth analysis, NY, Wiley Interscience, 1983, 308 p. ISBN: 9780471875048 .

20.   Filippov A.F. On some issues in the theory of optimal regulation. Vestnik Moskovskogo Universiteta, 1959, vol. 2, pp. 25–32 (in Russian).

21.   Hartman P. Ordinary differential equations. NY, London, Sydney, The Johns Hopkins university John Wiley & Sons, Inc., 1964, 640 p. Translated to Russian under the title Obyknovennyye differentsial’nye uravneniya, Moscow, Mir Publ., 1970, 720 p.

Cite this article as: A.S. Asev, S.P. Samsonov. On the problem of optimal stimulation of demand. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 2, pp. 23–38.