THREE LAWS OF OPTIMAL DEVELOPMENT
Girlin S.K.1), Bugerko N.V.2)
1)
Professor, PHD, D. of Science, h. c.,
2)
Student, 6 course,
Institute of economic and management, RHES “Crimean University for the Humanities”
(Yalta, RC, the Russian Federation)
e-mail:sk.girlin@gmail.com
e-mail: buherko@gmail.com
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 largeterm 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:
t
mi (t )   i (t, ) yi ( )mi ( )d  xi (t ) zi (t ) f (t ),
(1)
ai (t )
t
ci (t )   [ i (t, )(1  yi ( ))mi ( )d  kc (1  xi (t )) zi (t ) f (t )] ,
(2)
ai (t )
t
Pi (t )   mi ( )d ,
ai ( t )
0  ai (t )  t , ai (t0 )  0, 0  xi , yi , zi  1, z1  z2  1, t  [t0 , T ], 0  t0  T  ,
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 t ; m(t ) is the rate of arrive of the first new generalized product (internal resource)
quantity at the instant t , which provides the fulfillment of the internal functions of DS, that is,
restoration of itself and creation of the second kind product ( mi (t ) is measured in the units of
f(t)) ; y ( )m( ) is a share of m( ) for fulfillment of internal functions in the subsystem A of
restoration and perfection of the system as a whole,   t ;  (t, ) is the efficiency index for
functioning of the subsystem A along the channel  (t, ) y( )m( )  m(t ), i.e. the number of
units of m(t ) created in the unit of time starting from the instant t per one unit of y ( )m( ) ;
a(t ) is a special temporal bound: the new product creating before a(t ) is never used at the
instant t , but created after a(t ) is used entirely; c(t ) is the rate of creation of the second kind
new generalized product quantity at the instant t , which provides the realization of the external
functions of DS; kc is coefficient of measure accordance among f, m, and c; (1  y( ))m( ) and
 (t, ) are similar to y ( )m( ) and  (t, ) respectively but for the subsystem B of creation of the
second kind product; G(t ) is the total quantity of the obsolete product at the instant t ; t0 is the
starting point for modeling; it is the prehistory on [0, t0 ] 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. m( )  m0 ( ),
y( )  y0 ( ) ,  [0, t0 ]).
Let ai (t )  ai (t0 )  0, αi (t,τ)  αi1(t)αi2 ( ), i  1,2.
Let us put ai (t )  ai (t0 )  0, αi (t,τ)  αi1(t)αi2 ( ), i  1,2.
The equations (1), (2) can be rewritten as
t
mi (t )  i1(t )  i 2 ( ) yi ( )mi ( )d  mi(t )  xi (t ) zi (t ) f (t ),
t
(3)
0
t
ci (t )   i (t , )(1  yi ( ))mi ( )d  ci(t )  kc (1  xi (t )) zi (t ) f (t ),
(4)
t0
where
t0
mi(t )  i1(t )  i 2 ( ) yi 0 ( )mi 0 ( )d ,
0
ci (t ) 
t0
 i (t, )(1  yi 0 ( ))mi 0 ( )d , i  1,2.
0
Let ai (t )  ai (t0 )  0, αi (t,τ)  αi1(t)αi2 ( ), i  1,2.
Let us put ai (t )  ai (t0 )  0, αi (t,τ)  αi1(t)αi2 ( ), i  1,2.
Theorem 1. Let non-negative piecewise continuous functions yi0 (τ) , mi0 (τ) be given
on [0, t0 ] (i.e., the initial prehistory be given), also continuous functions
0  t0  T  , positive function
f (t )
i
on [t0 , T ] ,
, piecewise nonnegative continuous functions
xi (t ), yi (t ), zi (t ), 0  xi , yi , zi  1, z1 (t )  z2 (t )  1, i  1, 2, be given. Then the system (3),(4) has
the unique piecewise continuous solution
t
mi (t )   Ai (t , )ri (t , ) xi ( ) zi ( ) f ( )d  mi(t ) Ai (t, t0 )  xi (t ) zi (t ) f (t ),
t
(5)
0
t
ci (t )   Bi (t , ) xi ( ) zi ( ) f ( )d  Bi (t )  ci(t )  kc (1  xi (t )) zi (t ) f (t ),
(6)
t0
where
t
t

s
Ai (t, )  1   ki ( s) exp(  ki (u)du)ds, ri (t, )  i1(t )i 2 ( ) yi ( ), ki (t )  i1(t )i 2 (t ) yi (t ),
t
t
t0

Bi (t , )   xi ( ) zi ( ) f ( )[  i (t, u)(1  yi (u)) Ai (u, )ri (u, )du  i (t, )(1  yi ( ))]d ,
t
Bi (t )   i (t , u )(1  yi (u ))mi(u ) Ai (u, t0 )du, t  [t0 , T ],
t
0
and on [t0 , T ] the functions mi (t ) and ci (t ) are piecewise continuous , i  1,2 .
Proof. It is similar to the proof [12].
If we put z  z1, z2  1  z , we can consider the following task: to determine the control
function z*(t) with the restrictions (3), (4), for which the functional
T
I ( z)  max I ( z )  max  ( c1(t )  c2 (t ))dt
0 z1
0 z1
t0
Theorem 2.Under conditions of theorem 1, the functional I(z) has the form:
T
I ( z )  I 0    (T , t ) f (t )z (t )dt ,
(7)
t0
T
(T , t )   [ x1(t ) E1( , t )  x2 (t ) E2 ( , t )  1( , t )(1  y1(t )) x1(t )  2 ( , t )(1  y2 (t )) x2 (t )]d 
t
 kc ( x2 (t )  x1(t )) ,

Ei ( , u)   i ( , t )(1  yi (t )) Ai (t, u)ri (t, u)dt , i  1,2 ,
u
t


I 0  [ 1(t , )(1  y1( ))m1 ( ) A1( , t0 )d   2 (t, )(1  y2 ( ))( A2 ( , u) r2 ( , u) x2 (u) f (u)du 
t0 t0
t0
t0
T t



m2 ( ) A2 ( , t0 )  x2 ( ) f ( ))d  c1(t )  c2(t )  kc (1  x2 (t ) f (t )]dt.
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
1
2
z1(t )  z(t )  ( sign (T , t )  sign (T , t ) ), z2 (t )  1  z(t ), t [t0 , T ].
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 x1  x2 , y1  y2 ,  (T , t )  0 , and the
functional I ( z )  I 0  const does not depend on z .
Theorem 5. If T  t0 is sufficiently small, the function x2 (t )  x1(t ) is continuous on the
interval (t0 , T ] , 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 xi(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 Bi of implementation of external function, accordingly
is greater).
Corollary. If the function x2 (t )  x1(t ) is continuous on some (T−,T], >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  (T , T )  0, the function  (T , t ) be differentiable by variable t on the
segment
 L1 
[t0,T]
and
constants
L1
and
L2,
L1L2>0,
be
known
and
such,
that
 (T , t )
  L2 , t  [t0 , T ] .
t
It is obvious, that for any t  [t0 , T ]
 (T , T )  L2 (T  t )   (T , t )   (T , T )  L1(T  t )
Theorem
[t0 , T 
(T 
6. If the condition A is fulfilled and T  t0  
 (T , T )
,
L1
then on interval
(T , T )
) the maximum of the functional I(z) is achieved at z(t) = 1, on interval
L2
(T , T )
,T ]
L1
at z(t) = 0, and on the segment
[T 
(T , T )
(T , T )
,T 
]
L2
L1
at
1
z (t )  ( sign (T , t )  sign (T , t )) .
2
The proof easily follows from the obvious fact, that a maximum of functional I ( z ) is
1,   0,
 (T , T )
1

),
achieved at z (t )  ( sign  (T , t )  sign (T , t )) , where sign   0,   0, t  [t0 , T 
2
L
1,   0,
2

and  (T , t )  0, t  (T 
 (T , T )
L1
, T ].
Let us show that condition A fulfilled, if, for example, the next condition B is fulfilled.
Condition B. Let i (t , )  i  const  0, i (t , )  i  const  0,

   2 ,
xi (t )  xi  const  0, yi (t )  yi  const  (0,1),  1
,

 x2  2 y2 (1  y2 )  x11 y1 (1  y1 )
Under condition B
x  y (1  y2 ) (2 1 )(T t )
(T , t )
 (T t )
 x11 y1(1  y1)e 1
( 1  2 2 2
e
)0
t
x11 y1(1  y1)
i  1, 2.
and function
function
 (T , t )
is continuous if t  [t0 , T ]. Consequently, by the theorem of Weierstrass
t
 (T , t )
on [t0 , T ] has the most negative value (which can be designated through
t
 L2 ) and the least negative value (which can be designated through − L1 ).
Condition C. Let  (T , T )  0, and
 ( , t )  x1(t ) E1( , t )  x2 (t ) E2 ( , t )  1( , t )(1  y1(t )) x1(t )   2 ( , t )(1  y2 (t )) x2 (t ),
t0  t    T ,
 ( , t )    const  0.
inf
t0 t  
k
Theorem 7. If the condition C is fulfilled and T  t0  c , then the maximum of the

k
k
functional I ( z ) is achieved at z (t )  1 on [t0 , T  c ] and at z (t )  0 on (T  c , T ] .


T
As (T , t )    ( , t )d  kc ( x2 (t )  x1(t ))  kc   (T  t ), t0  t    T , then the proof
t
easily follows from the estimation of sign of function  (T , t ).
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: i1(t )  1, i 2 ( )  i  const  0, i  1, 2 , in [1].
Conclusion.The got results (theorems 4-7) for the case of passive co-operative twoproduct 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 A1 (“science”) has
priority ( A1 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 A2
(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 [t0 , T ] 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".
Abstrac. The task of passive co-operation of the developing systems is investigational
with the purpose of receipt of most exit on the set temporal interval of their general external
product. The problem of the best allocation of external resources is set and decided between two
co-operatively interactive developing systems (at the set allocation of internal resources between
the subsystems of each system).
Keywords: laws of optimal development, developing systems interaction, integral model.
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