Controllability of a nonholonomic macroeconomic system. (English) Zbl 1257.49022

Summary: This paper studies optimal control problems and sub-Riemannian geometry on a nonholonomic macroeconomic system. The main results show that a nonholonomic macroeconomic system is controllable either by trajectories of a single-time driftless control system (single-time bang-bang controls), or by nonholonomic geodesics or by sheets of a two-time driftless control system (two-time bang-bang controls). They are strongly connected to the possibility of describing a nonholonomic macroeconomic system via a Gibbs-Pfaff equation or by four associated vector fields, based on a contact structure of the state space and our isomorphism between thermodynamics and macroeconomics that praises three laws of a nonholonomic macroeconomic system.


49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
53C17 Sub-Riemannian geometry
91B64 Macroeconomic theory (monetary models, models of taxation)
Full Text: DOI


[1] Udrişte, C.: Thermodynamics versus economics. UPB Sci. Bull, Series A 69(3), 89–91 (2007)
[2] Udrişte, C., Ferrara, M., Zugrăvescu, D., Munteanu, F.: Geobiodynamics and Roegen type economy. Far East J. Math. Sci. (FJMS) 28(3), 681–693 (2008) · Zbl 1147.91045
[3] Udrişte, C.: Geometric Dynamics. Kluwer, Amsterdam (2000)
[4] Udrişte, C., Dogaru, O., Ţevy, I.: Extrema with Nonholonomic Constraints. Monographs and Textbooks, vol. 4. Geometry Balkan Press, Bucharest (2002)
[5] Udrişte, C., Ferrara, M., Opriş, D.: Economic Geometric Dynamics. Monographs and Textbooks, vol. 6. Geometry Balkan Press, Bucharest (2004)
[6] Udrişte, C., Ferrara, M.: Black hole models in economics. Tensor NS 70(1), 53–62 (2008) · Zbl 1193.91103
[7] Stamin, C., Udrişte, C.: Nonholonomic geometry of Gibbs contact structure. U.P.B. Sci. Bull., Series A 72(1), 153–170 (2010) · Zbl 1212.53073
[8] Liu, W., Sussmann, H.: Abnormal sub-Riemannian minimizers. IMA Preprint Series # 1059 (1992) · Zbl 0823.49026
[9] Udrişte, C.: Multitime controllability, observability and bang–bang principle. J. Optim. Theory Appl. 139(1), 141–157 (2008) · Zbl 1156.93013
[10] Udrişte, C.: Simplified multitime maximum principle. Balkan J. Geom. Appl. 14(1), 102–119 (2009) · Zbl 1180.49023
[11] Udrişte, C.: Nonholonomic approach of multitime maximum principle. Balkan J. Geom. Appl. 14(2), 111–126 (2009)
[12] Udrişte, C., Ţevy, I.: Multitime linear-quadratic regulator problem based on curvilinear integral. Balkan J. Geom. Appl. 14(2), 127–137 (2009)
[13] Udrişte, C., Ţevy, I.: Multitime dynamic programming for curvilinear integral actions. J. Optim. Theory Appl. 146(1), 189–207 (2010) · Zbl 1202.49027
[14] Udrişte, C.: Equivalence of multitime optimal control problems. Balkan J. Geom. Appl. 15(1), 155–162 (2010)
[15] Udrişte, C.: Multitime maximum principle for curvilinear integral cost. Balkan J. Geom. Appl. 16(1), 128–149 (2011) · Zbl 1220.49002
[16] Udrişte, C., Bejenaru, A.: Multitime optimal control with area integral costs on boundary. Balkan J. Geom. Appl. 16(2), 138–154 (2011) · Zbl 1220.49003
[17] ShankarSastry, S., Montgomery, R.: The structure of optimal controls for steering problem. In: NOLCOS, Conf Proc., Bordeaux, France (1992)
[18] Chernavski, D.S., Starkov, N.I., Shcherbakov, A.V.: On some problems of physical economics. Phys. Usp. 45(9), 977–997 (2002)
[19] Georgescu-Roegen, N.: The Entropy Law and Economic Process. Harvard University Press, Cambridge (1971)
[20] Ruth, M.: Insights from thermodynamics for the analysis of economic processes. In: Kleidon, A., Lorenz, R. (eds.) Non-equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond, pp. 243–254. Springer, Heidelberg (2005)
[21] Smulders, S.: Entropy, environment and endogenous economic growth. Journal of International Tax and Public Finance 2, 317–338 (1995)
[22] Sergeev, V.: The thermodynamical approach to market (translated from Russian and edited by Leites, D.). Max Planck Institute, Preprint no: 76, (2006); arXiv:0803.3432v1 [physics.soc-ph] 24 Mar (2008)
[23] Nardini, F.: Technical Progress and Economic Growth. Springer, Berlin (2001) · Zbl 0964.91031
[24] Vrănceanu, G.: Lectures of Differential Geometry Vol. I (in Romanian). Didactical and Pedagogical Editorial House, Bucharest (1962). Vol. II, (1964)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.