DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

THEORY OF FUNCTIONAL DIFFERENTIAL EQUATIONS
AND SOME PROBLEMS IN ECONOMIC DYNAMICS
V. P. MAKSIMOV
This paper focuses on boundary value problems and control problems for functional
differential equations in the abstract form. Key questions of developing techniques for
the computer-assisted study of such problems are discussed. Within the framework of
the general approach, the problems of impulse and hybrid control are considered with
regard to applications in economic mathematical modeling.
Copyright © 2006 V. P. Maksimov. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Economic dynamics is one of the possible and very actively developing area in applications
of the theory of functional differential equations (FDEs). The subject to study is an
in time developing process as a sequence of the states of an economy system with possible
structural breaks. An essential feature of any economic process is the presence of a
lag which means a period of time between the moment of an external action and a reply
of the system, for instance, between capital investments moment and a moment of an
actual growth in output. Thus a model governing the dynamics of the economic system
under consideration can be written in the form of FDE. First we give below some preliminaries
from the theory of FDEs in an abstract form. Those are concerned around
boundary value problems (BVPs) and control problems (CPs). Next some corollaries
from the general theorems are formulated in a form that allows one to apply the results to
some problems that arise in economic dynamics, also some questions of the computerassisted
study of BVPs and CPs are discussed. Finally, we present two problems from
economic dynamics that are formulated in the form of impulse (hybrid) control problems.
2. Preliminaries
Let D and B be Banach spaces such that D is isomorphic to the direct product B × R n (in
what follows we write D ≃ B × R n ).
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 757–765