6. Discussion of results

The case n <l þ m appears, when X∩Y 6¼ ∅.

lems (development of control recommendations).

effective functioning of the stochastic system.

sustainable development will take the form:

<sup>G</sup>ð Þ¼ <sup>Σ</sup> <sup>P</sup><sup>n</sup>

H Yð Þ<sup>V</sup> ¼ A, H Yð Þ<sup>R</sup> ¼ B, r Xð Þ≤ r,

i¼1 Pn j¼1

Σ ∈ GΣ, a∈ Ha,

σ2

characteristics of a particular system S.

>>>>>>>>>>>>>>>>>>>:

Σ>0,

<sup>V</sup>; <sup>h</sup><sup>0</sup> R

Sustainability Assessment at the 21st Century

entropy from some initial point h<sup>0</sup>

ij n o

8

>>>>>>>>>>>>>>>>>>><

state h Z<sup>0</sup> � � <sup>¼</sup> <sup>h</sup><sup>0</sup>

matrixΣ<sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup>

Xi, i = 1, 2, ..., l.

Figure 9.

34

Within the framework of the proposed concept, along with the tasks of monitoring complex systems discussed above, it is possible to solve management prob-

The idea of vector entropy control is the transfer of the vector h(Z) from the

For a Gaussian system, the vector entropy control consists in directing the

matrix Σ<sup>0</sup> to the final point ð Þ hV\*; hR\* with a minimal change of the covariance

The problem of the vector entropy control of the Gaussian system to ensure

<sup>V</sup>; <sup>h</sup><sup>0</sup> R � � <sup>¼</sup> H Z<sup>0</sup> � �

<sup>σ</sup>ij � <sup>σ</sup><sup>0</sup> ij � �<sup>2</sup>

ij < σiiσjj, σij ¼ σji, σii>0 ∀ 1≤i, j≤n,

where A ¼ hV\*, B ¼ hR\*, a—the expectations vector of the components

An illustration of vector entropy management to ensure sustainable development of the system.

The last constraint in Eq. (12) means positive definiteness of the matrix Σ. Note that the performance criterion in Eq. (12) may be different, depending upon the

� � in the state h Zð Þ¼ \* ð Þ hV\*; hR\* , which corresponds to the

and the expectations vector a<sup>0</sup> and acceptable risk (Figure 9).

ai � <sup>a</sup><sup>0</sup> i � �<sup>2</sup> ! min

<sup>þ</sup> <sup>P</sup> j

i¼1

<sup>V</sup>; H Z<sup>0</sup> � �

R � � with the covariance

<sup>a</sup>, <sup>Σ</sup> ,

(12)

Thus, on the basis of the use of two original models—a vector entropy and multidimensional risk—it was succeeded to formalize the new concept of sustainable development of complex systems. Both models are successfully approved on real data.

This concept can be implemented by means of monitoring of the studied system. As observed parameters efficiency factors of the functioning of a system and its risk factors are used. The direction of development is given by an entropy vector, and stability is provided due to an acceptable risk level.

Management recommendations are formed in the form of the solution to an extreme problem Eq. (12). This problem is solved by methods of penalty functions. Currently, the work is at a stage of practical approbation of monitoring of sustainable development of Sverdlovsk region.
