1. Introduction

The term "sustainable development" originally was used in 1972 at the United Nations Conference on the Human Environment in Stockholm. In 1987 in a report entitled as "Our common future" of the World Commission on Environment and Development (WCED) where Norwegian Prime Minister Gro Harlem Brundtland was a chair, the term definition of "sustainable development" has been formulated: "development that meets the needs of the present without compromising the ability of future generations to meet their own needs" [1]. The "sustainable development" interpretation is very common and does not show a particular way to move into practice [2]. One can point out the main conceptual complexity, which is that the concept of "sustainable development" includes two terms "sustainability" and "development." Moreover, each of these terms is interpreted in different points.

These terms are multidirectional. Really, maximizing efficiency usually increases risks, reducing the stability of the functioning of a system. Rather, excessive stability leads to an increase in the costs of its maintenance, reducing the efficiency of functioning of a system.

Thirdly, formalization of sustainable development is a complicated complexity of the studied systems and the phenomena. Currently, there is no unambiguous, accurate interpretation of a concept of the complex system. However, there are characteristic signs, such as multidimensionality, multiconnectivity, a multiloop, multileveled (hierarchy), the composite and multipurpose nature of construction, and also indeterminacy and stochasticity of behavior. We will give below the definition which, in our opinion, most adequately characterizes the concept of the complex system.

Definition 1. A complex system is called a system in the model of which there is not enough information to effectively manage this system [3].

This fact leads to different understandings of "sustainable development" in relation to particular systems [4–11]. For example, in [12], there are more than 50 different interpretations of the "sustainable development" concept.

The implementation of sustainable development implies that certain monitoring of the studied system or phenomenon should be carried out. Monitoring is understood as a system of constant overseeing by the current of any phenomenon for the establishment of its compliance to the initial assumptions or desirable result. This phenomenon can occur in any sphere—in social relations, in nature, in the financial and economic sphere, etc. Within monitoring, there is assessment, control of the system, and the formation of management recommendations (management of its state) depending on the impact of particular factors.

Therefore, the formulation of the formalized concept of monitoring sustainable development, which could be concretized for particular cases, seems to be an urgent problem. One of the possible ways is to use a systems approach [13]. Let us make an attempt to formulate monitoring of the sustainable development concept of complex systems with the example of territories.

3. Vector entropy model for the effective functioning of systems

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

effective functioning of the system.

A model of the system as a structure model.

DOI: http://dx.doi.org/10.5772/intechopen.89287

Figure 1.

vector of Y with a probability density:

þ ð∞

�∞

… þ ð∞

�∞

Hð Þ¼� Y

quently differential entropy [22].

multidimensional random value of Y.

23

Consider the problem of developing an integral indicator that characterizes the

Multicriteriality of complex systems functioning, including territorial, and the diversity of their elements functioning, makes the development of universal formal indicators difficult which characterizes the effectiveness of systems as a whole. It is known that entropy is a fundamental property in any systems with probabilistic behavior [16]. The concept of entropy is flexible and allows interpretation in terms of the branch of science, where it is applied. Therefore, entropy modeling is one of the promising lines of research of complex stochastic systems [17–20].

However, the frequent use of entropy for modeling of open systems, in contrast to thermodynamics, is insufficiently formalized and has generally qualitative and private character; there are no rather simple and adequate mathematical models that allow associating entropy with the actual characteristics of conditions of multidimensional systems. Common in these works is the use of Shannon's information entropy [21]. But, as it is noted in [15], the information entropy allows

However, in the same work [21], Shannon heuristically offered a formal analog of a concept of information entropy for the m-dimensional continuous random

p<sup>Y</sup> y1, y2, … , ym

This value Kolmogorov together with Gelfand and Yaglom was called subse-

The differential entropy, being the functional given on the set of the probability density of a random vector of Y, represents a number. Therefore it cannot be an adequate mathematical model of a multidimensional system. However, the practical use of entropy (1) is complicated by the need to know the distribution law of a

� � ln <sup>p</sup><sup>Y</sup> <sup>y</sup>

� �dy1dy2…dym: (1)

developing adequate entropy models only for particular problems.
