Elements of Parametric Control Theory of Market Economic Development
The application of mathematical models of a national economy is an important subject area for the analysis of an effective public policy in the area of the economic growth [73].
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Elements of Parametric Control Theory of Market Economic Development
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Components of Parametric Control Theory of Market Economic Development
The application of mathematical models of a national economy is an important subject area for the analysis of an effective public policy in the area of the economic growth [73]. Many dynamical systems, including the national economic system [33, 30], after some transformations, can be described by the following systems of nonlinear ordinary differential equations:
x ðtÞ ¼ f ðxðtÞ; uðtÞ; aÞ;
(1.1)
xðt0 Þ ¼ x0 :
(1.2)
with the initial condition
Here t is the time, t 2 ½t0 ; t0 þ T ; T>0, is a fixed number; x ¼ xðtÞ 2 Rm is the state of system (1.1), (1.2); x0 2 Rm is the initial state of the system (deterministic vector); u ¼ uðtÞ 2 Rq is the vector of controlled (regulated) parameters; the functions uðtÞ and their derivatives are to be uniformly bounded; a 2 A Rs is the vector of uncontrolled parameters; and A is an open connected set. For a solution to system (1.1), (1.2) to exist, let’s assume that the vector function f satisfies the Lipschitz condition and the following linear constraints on its growth rate: jf ðx; u; aÞj cð1 þ jxjÞ;
A.A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a National Economy, DOI 10.1007/978-1-4614-4460-2_1, # Springer Science+Business Media New York 2013
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1 Elements of Parametric Control Theory of Market Economic Development
where c is a positive constant. As is well known, the solution (evolution) to the considered system of ordinary differential equations depends on both the vector of initial values x0 and the values of vectors of controlled (u) and uncontrolled (a) parameters. Therefore, the result of evolution (development) of the nonlinear dynamical system, with a given vector of the initial values x0 , is defined by the values of vectors of both controllable and uncontrollable parameters. It is also known [3] that the process described by (1.1) may be judged by the solutions of this system only if the qualitative image of the trajectories of this system is invariable under small—in some sense—disturbances of the right-hand side part of (1.1). In other words, system (1.1) must possess the property of robustness or structural stability. For the reason just mentioned, the theory of parametric control of the market economic development is proposed in [7, 8, 53–55]. This theory consists of the following components: 1. The methods for forming the set (library) of macroeconomic mathematical models. These methods are oriented toward the description of various specific socioeconomic situations, taking environmental safety conditions into consideration. 2. The methods for estimating the conditions for robustness (structural stability) of the models of national economic systems from the library without parametric control. Here, the conditions of belonging to the considered mathematical models of the Morse–Smale class of systems, the class of O-robust systems, the class of uniformly robust systems, the cla
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