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Asgarov S. (Azerbaijan), Alakbarov M. (Azerbaijan), Aliev Z. (Azerbaijan), Babayev N. (Uzbekistan), Chiladze G. (Georgia), Datskovsky I. (Israel), Garbuz I. (Moldova), Gleizer S. (Germany), Ershina A. (Kazakhstan), Kobzev D. (Switzerland), Kohl O. (Germany), Ktshanyan M. (Armenia), Lande D. (Ukraine), Ledvanov M. (Russia), Makats V. (Ukraine), Miletic L. (Serbia), Moskovkin V. (Ukraine), Murzagaliyeva A. (Kazakhstan), Novikov A. (Ukraine), Rahimov R. (Uzbekistan), Romanchuk A. (Ukraine), Shamshiev B. (Kyrgyzstan), Usheva M. (Bulgaria), Vasileva M. (Bulgar).

The article is a further
development of the ideas first established by academicians L.V. Kantorovich [9,10],
V.M. Glushkov and developed in future professors V.V. Ivanov, Yu. P. Yatsenko
[7,8,11, 13-17], S.K. Girlin [5,12]. There are two basic classes of developing systems (DS) (or
evolutionary systems): 1) DS that are already have been created and have
initial prehistory, 2) DS that are not have been created and have not initial
prehistory. The second class is named originating DS [2]. Each of these classes
divides, in turn, on three classes: Artificial DS (ADS) that are have been
created by human beings and are functioning with their participation; Natural
DS (NDS), in particular, the cell and cell associated objects [14]; and Joined
DS (JDS): ADS and NDS as a whole [12]. So we have 6 classes of DS in all. Some
examples of ADS are: industry, science, any educational center, including
school, college, university, and education as a whole, art, health services,
etc. The examples of NDS are the cell and cell associated objects, a separate
plant, a separate organism, a population of animals, the biosphere, etc. We can
consider the neosphere (in the sense by academician V.I. Vernadsky) as JDS
that is the combination of two DS, one of which is human activity as ADS and
another of which is the other part of our planet as NDS [13, p. 10]. The main
elements of ADS are work places (WP). A work place is usually localized in the
time and space aggregate of labor functions of the respective ware: material,
energy, and information, which should be fulfilled by a respective specialist.
The result of WP functioning in industry are various goods and services or
products. The main characteristics or indices of the WP functioning are
efficiency (the quantities of products produced per unit expenditure and per
unit time). There are three important classes of WP: one that enters the DS
from external environment or from other DS, the second reproduces or creates
new more effective WP for the DS itself, and the third reproduces external
goods with respect to DS. There are two branches of industry: one is called a
subsystem A of DS in which DS creates new WP, and the second is called a subsystem
B of DS in which DS creates or produces goods and services that are external
with respect to DS. The distribution of WP by some control function *y*(*t*)
between the subsystems A and B is very important. The problem of this optimal
distribution was formulated by V.M. Glushkov. This problem was investigated
in [7]. The main result is for small-term period of the time the desired *y*(*t*)
is minimally possible, but for large-term period of the time the desired* y*(*t*)
may differ from the minimally possible on the larger initial part of the time
segment*. *

The consideration of distribution between subsystems A and B not only internal, but external resources too, allowed to generalize mathematical model of DS [2-6] (in particular, to investigate originating developing systems without given prehistory of DS [2,3]).

A set of the obtained results suggests that it has been created a new science - mathematical theory of development [14].

We will consider the simplest case of cooperative interaction of two-product developing systems. It is the case of passive systems interaction, at which the systems distribute between itself external resources and not exchanged between itself the products of its functioning. Equations and inequalities of the passively interactive DS look like:

_{} (1)

_{}, (2)

_{}

_{}_{}_{}_{}_{}_{}

where i is the number of DS, i=1,2; *f(t)
*is the rate of the resource inflow from the outside at the instant _{}_{ }; _{}is
the rate of arrive of the first new generalized product (internal resource)
quantity at the instant _{},
which provides the fulfillment of the internal functions of DS, that is,
restoration of itself and creation of the second kind product (_{}is
measured in the units of _{};
_{}is
a share of _{} for
fulfillment of internal functions in the subsystem A of restoration and
perfection of the system as a whole, _{}; _{}is
the efficiency index for functioning of the subsystem A along the channel _{}i.e.
the number of units of _{} created
in the unit of time starting from the instant _{} per one unit of _{}; _{}is
a special temporal bound: the new product creating before _{} is
never used at the instant _{},
but created after _{} is
used entirely; _{} is
the rate of creation of the second kind new generalized product quantity at the
instant _{},
which provides the realization of the external functions of DS; _{} is
coefficient of measure accordance among *f*,
*m*, and *c*;_{}
and _{}are
similar to _{}and
_{} respectively
but for the subsystem B of creation of the second kind product; _{}is
the total quantity of the obsolete product at the instant _{};_{}
is the starting point for modeling; it is the prehistory on _{}_{ }of DS, for which all the functions are given (their values will be
noted by the same symbols but with the sign “0”, e.g. _{}_{}, _{}

Let _{}_{}_{}

Let us put_{}_{}_{}

_{The equations (1), (2) can
be rewritten as}

_{} (3)

_{} (4)

_{where}

_{}, _{}_{}

Let _{}_{}_{}

Let us put _{}_{}_{}

_{ }**Theorem 1. **Let non-negative piecewise
continuous functions _{}, _{}_{ }be given

on _{} (i.e.,
the initial prehistory be given), also continuous functions _{} on
_{}, _{}_{ }positive function _{} ,
piecewise nonnegative continuous functions _{}_{}_{} be
given. Then the system (3),(4) has the unique piecewise continuous solution

_{} (5)

(6)

where

_{}_{}_{}

_{}

_{}_{}

and on _{} the
functions _{}and_{}are piecewise continuous ,_{}.

Proof. It is similar to the proof [12].

If we put_{}, we can consider the following task: to determine the control
function z*(*t*) with the restrictions (3), (4), for which the functional

_{}

**Theorem 2.**Under conditions of theorem 1, the functional I(z) has the form:

_{}
(7)

_{}

_{},

_{}, _{},

_{}

_{}

The required type of functional can be found by the direct substitution of the solution (6).

**Theorem 3.**If the conditions of theorem 1 are fulfilled, then

_{}

The proof of this and the following theorems follow directly from the functional (7)-type.

**Theorem 4. **If the systems above are identical, any allocation of the external
resources between the systems is optimal.

The proof is obvious, because in this case _{}, _{},
and the functional _{} does
not depend on _{}_{.}

**Theorem 5. **If _{} is
sufficiently small, the function _{} is continuous on the interval _{}, then for maximization of the functional I(*z*), it is necessary to send all external
resources to the system *i*, *i* = 1,2, for which *x _{i}*(

*T*) is less, i.e., to the system, in which relative part of external resources, sent to the subsystem of self-perfectionist less (thus, the relative part of external resources, sent to the subsystem

*B*of implementation of external function, accordingly is greater).

_{i}
**Corollary**. If the function _{} is
continuous on some (T−e,T], e>0, then at the__ _{ }__end of any cutting-off of time
of planning for maximization of the considered functional, it is necessary to
send all external resources to the system, for which relative part of the
external

_{ resources (sent to the subsystem Bi) is greater.}

**Condition A.** Let _{}_{ }the
function_{}be differentiable
by variable t on_{ }the_{ }

_{segment [t0,T] and constants L1 and L2,
L1}_{³}_{L}_{2>0,
be known and such, that }_{}_{}.

It is obvious, that for any_{}

_{}_{ }

**Theorem 6.** If the condition A is fulfilled and _{} then on interval _{}_{ }the maximum of the functional *I(z)*
is achieved at *z(t) = 1,* on
interval_{}_{ }at *z(t) = 0*, and on
the segment _{}_{ }at_{}*.*

The proof easily follows from the obvious fact, that a
maximum of functional _{} is achieved at _{}*, where _{}_{}* and

_{}

Let us show that condition **A **fulfilled, if, for example, the next condition *B* is fulfilled.

**Condition B.** Let _{}_{}_{}

Under condition **B**

_{}

and function _{}_{ }is continuous if _{}Consequently,
by the theorem of Weierstrass function _{}_{ }on _{} has
the most negative value (which can be designated through _{} )
and the least negative value (which can be designated through −_{}).

**Condition C**. Let _{}_{ }and

_{}

_{}_{}

**Theorem 7. **If the condition **C** is
fulfilled and _{},
then the maximum of the functional _{}_{ }is achieved at _{}* _{ }*on

_{}

_{ }and at

_{}*on*

_{ }_{}.

As _{}_{} then the proof easily follows from the estimation of sign of
function _{}

**Remark.**The optimization task of the best distributing of external resource
examined here between two co-operatively interactive two-product developing
systems was decided for the special case:_{}, in [1].

**Conclusion.**The got results** (t**heorems
4-7) for the case of passive co-operative two-product DS interaction are
analogical the next first two laws
of optimal development of one system [4,14] (we suppose that it is possible to
prove statements analogical and to the third law of optimal development).

These laws can be set forth as follows.

First law of optimal development (“law of altruism”). If the size of planning time is small enough, the sought optimum of functional is arrived at the maximally possible (by virtue of limitations of task) use in the subsystem B of internal and external resources for implementation of basic function of the system.

Second law of optimal development (“law of reasonable egoism” [4,11]). If the size of planning time is great enough, the sought optimum of functional is arrived at the substantial stakes of internal and external resources, using the subsystem of self-perfection on the internal necessities of the system on greater initial part of cutting-off of planning time and maximally possible use in the subsystem B of internal and external resources for implementation of basic function of the system at the end of it. This law was shown out of the theorems at general enough suppositions [5].

Third law of optimal development (“law of hierarchy of
priorities”) [13,14]. If the size of planning time is great enough, the sought
optimum of functional is arrived at the following priorities of allocation of
internal and external resources between the subsystems of developing system:
first of all at the larger initial size of planning time the subsystem _{} (“science”) has priority (_{} is the subsystem , in which new technologies of system products
creation functions of _{} and
_{} kinds),
then at the long time size has priority the subsystem of self-development _{} (the
subsystem, in which new products of the first kind are produced, providing the
fulfillment of the internal function of the system – its existence and
development itself), and at the end of the planning time _{}_{ }the subsystem B has priority, in which products of the second kind
are produced, providing the fulfillment of the main system function to the
system).

We`d like to notice that the law of "reasonable egoism" of the system can be considered as clarification of basic principle of communism: "to each - on necessities, from each - to abilities".

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