Professor, Shuakayev M. K.,  Ph.D. Nazarbekova S.T., Shuakayeva A.

JSC "MMC Kazakhaltyn", Republic of Kazakhstan,
Kazakh National University by named al Farabi, Republic of Kazakhstan,                   JSC "BTA Bank", Republic of Kazakhstan

Stochastic Volterra`s Models of Control Systems for Describing    of Biological Systems


In this paper stochastic model of growing fungus is presented, which is a 3-dimensional bilinear stochastic system and is managed by 3 characteristics: temperature, moisture components and nutrition with minerals. After that managing problems and managing initial system problems are investigated, which helps to manage the process of fungus growth and by that allows us to describe the model of a rather complicated technological process in more details. New approach for control system development is also presented in this paper.

Introduction

Investigators quite often analyze differential equation system for the description of mathematical models of biological systems.  This differential equation system does not let manage the process, occurring in the biological system and gives accurate information only at specific moments and for fixed characteristics. It is quite difficult to trace the whole difficult technological process of growing fungus, which requires the processing of input parameters of the system that are to be effectively arranged by any means.

Process control system of growing fungus can be described with bilinear stochastic system as follows:

                                                                        (1)

                                                                                              (2)

where À and  – matrix with 3õ3 dimension, and   w (t) is three dimensional vector– column, with the following  data  w1(t) – keeping a specific temperature, moisture  w2(t) –and w3(t) – feeding with mineral resources,

Ñ (.) – dimension vector  n x 1, y (.) – is called system’s output.

In case u (t) = 0 , then the system (1) determines an initial condition, i.å. matrix À determines an initial condition in which initial data is input, which included biological data of the object, environment temperature etc.

In this case the system’s parameter (1) has to be chosen so that matrix À is stable, i.e.  its eigenvalues lies on the left half-plane.

Example1. If   matrix A represented

À= .

Then  its eigenvalues is defines by following equation

det ( A-E ) = 0, 

solving it we obtain eigenvalues matrix A equally  -2 and -1. Consequently matrix A is stable.

The equations (1) - (2)  generalizes systems of cultivation of microbes from [4]. In (1)-(2), matrix A defines specific growth rate  , matrix B – concentration of flowing into growth, input y(t) is the same in [4].

For determination cross-correlation functions we formulate by following theorem.

Theorem1. For the system (1) – (2) there is a Volterra`s model, for which  the sequence of relative correlation functions is determined by the kernel of bilinear determined system  [8], i.e. by the following:

        .                                                              (3)

        

Then Hankel`s matrix H for system  (1) – (2) is chosen as follows:

                                                                                    (4)

        

In this case for the successful appliance of algorithm of realization it is obligatory to require its full range, i.e

(H) ≠ Î.                                                                                           (5)

System (1) – (2) can be used for the description of mathematical models for higher fungus and lower fungus as well.

Proof of  this theorem as  follows by using Ito formulas of Stochastic derivation we can exactly fined of sequence mutually correlation functions and
after we obtain  (3 – (5).

System (1) - (2)  can  be  used  for  describing  the  mathematical  models for both lower and higher types of cultivation of microbes.
According to Theorem 1, the process  of  constructing  the Hankel`s matrix H,  we can   conclude  that  the  deterministic  controllability matrix is ​​defined as:

   ,                                                                                     (6)

       where parameters

,

and the matrix of determined observation N is a three-dimensional  matrix with parameters as follows:

,                                                                             (7)

 

Then object (1) – (2) deterministically controlled, if conditions are as follows:

 

de t (Ñ) ≠ Î ,                                                                                               (8)

and we obtained condition observeability, if

de t (N) ≠ Î ,                                                                                               (9)

In this case  Hankel`s  matrix Í is determined as follows:

 Í = Ñ ∙ N,                                                                                             (10)

Condition (4) labels that the process of growing fungus is both deterministically controlled and can be observed.

In our opinion complicated biological process of growing fungus which was described shortly above, is to be investigated in effective system of data handing. That’s why we developed the following subsystems: «cosmopolitan», «commonly occurring», «rare in occurring», «new for the area of research » and «dominant».   

However, data base is developed only for the major part of «Cosmopolitan», subsystem and other subsystems: «commonly occurring», «rare in occurring», «new for the area of research » and «dominant» are contained in it, based on their similar qualities and for the data handing we also develop only one system of data handing which is «Cosmopolitan».

            In this paper working outed Packet of Appling Program, including Control Systems Bases of  Dates, Constraction Hankel matrixe, Woking Out of Volterras model (series) and other subsystems.

Referents:

1. Assaubayev K.Sh., Shuakayev M.K. / Volterra series and control theory. - Vol.1, Alma-Ata,1993,p.167.
2. Assaubayev K.Sh., Shuakayev M.K. /Algebras and liable groups. Volterra series and control theory. Vol. 2, Almaty, 1993, p.187.
3. Assaubayev K.Sh., Shuakayev M.K. /Volterra series and control theory. - Vol. 3, Almaty, 1993, p.187.
4. Williamson D. / Observation of Bilinear Systems with Application to biological Control// Automatica, vol.13, 1977, p.p. 243-254                                                                            5. Nazarbekova S.T, Ashurov H., Mirzakhmedov Sh.,  Dalimov D.N., Azizov S.T. The study of the role of some micromycetes in enzymatic hydrolysis of the walls of seed crops little oil-bearing  / Materials XI Congress of Russian Botanical Society / / Botanical research in Asian Russia. Barnaul. 2003, Vol.1, p.p. 45-46
6. On the current state of lichen herbarium of Botany Department KazNU / Development of plant science in Central Asia and its integration into the production / / Proceedings of the International Conference. Tashkent. 2004. S.280-282
7. Lebedev O.V., Ruban I.N., Voropayeva N.L., Nazarbekova S.T .The use of information and nano-biotechnology for analysis, forecasting, and improving the environmental situation in the area of ​​oil production in the Aral Sea and the Aral Sea / International scientific-practical conference, .Schuchinsk. 2007. P.265-269
8. Bruni C., Di Pillo G. and Koch G. Bilinear systems: IEFF TR. Aut. Coutr., 1974,p.p.334–348                                                                                                        9. Barab, A. -L.; Oltvai, Z. (2004). "Network biology: understanding the cell's functional organization". Nature reviews. Genetics 5 (2): 101–113.                                   10.  Covert; Schilling, C.; Palsson, B. (2001). "Regulation of gene expression in flux balance models of metabolism". Journal of Theoretical Biology 213 (1): 73–88.                                                                                                                                                             11. Covert, M. W.; Palsson, B. . (2002). "Transcriptional regulation in constraints-based metabolic models of Escherichia coli". The Journal of Biological Chemistry 277 (31): 28058–28064.