Geometric optimal control in economical systems. (Q1608311)

The notion of ``economic'' is similarly undefined as, for example, is ``mechanics''. There are some fields of mechanics; each of them can be characterized by its own sets of models, conjectures, and research methods. Similarly, in economics, one can highlight some fields by distinguishing one from another using fundamental notions and research aims. Let us mention some of them: collective decisions, models of economical balance, and models of economical dynamics. To study these models, an extensive set of mathematical tools are used: the theory of positive matrices, linear programming, the Brouwer fixed point theorem, stability of stationary points of vector fields and mappings, bifurcations, and variational problems. The role of mathematical methods and models is discussed in [O. O. Zamkov, Yu. A. Cheremnykh, and A. V. Tolsto\-pyatenko, Mathematical Methods in Economics [in Russian], Delo i Servis Publ. House (1999)] in detail. In the present paper, we give some directions of simulation that lie between topology, differential geometry, the theory of differential equations, and dynamical systems. A more detailed discussion of these problems can be found in the lectures given by the second author at the Mechanical and Mathematical Department of the Lomonosov Moscow State University. In studying the flow of goods and financial flows in an economical system, one can use various methods of their visualization, for example, methods of computer geometry. In the present section, we give a geometrical approach that can be used for constructing models of ware, financial, and informational flows. We also consider a model of describing financial flows which is closely related to the consideration of special decomposition of the projective space and functions on these decompositions. In section 2 a model of the flow of goods and functions on decompositions of the Möbius band are considered and in section 3 topological models of net marketing are investigated.