On the theory elements and methods of parametrical regulation of the national economy
evolution based on discrete stochastic dynamic models
Abdykappar A. Ashimov1, Bahyt T. Sultanov2, Yuriy V. Borovskiy3, Dmitry A. Novikov4,
Simon Ya. Serovajsky5, Askar A. Ashimov6
1)
The head of state scientific-technical program, Kazakh National Technical
University named after K. Satpayev. Address: KazNTU, 22 Satpayev str., 050013, Almaty
city, Kazakhstan. E-mail: Ashimov37@mail.ru
2)
The consultant of state scientific-technical program, Kazakh National Technical
University named after K. Satpayev. Address: KazNTU, 22 Satpayev str., 050013, Almaty
city, Kazakhstan. E-mail: sultanov_bt@pochta.ru
3)
The vice-head of state scientific-technical program, Kazakh National Technical
University named after K. Satpayev. Address: KazNTU, 22 Satpayev str., 050013, Almaty
city, Kazakhstan. E-mail: yuborovskiy@gmail.com
4)
The Deputy Director on scientific research, Institute of Control Sciences RAS.
Address: 65 Profsoyuznaya st., 117997, Moscow, Russian Federation. Email: novikov@ipu.ru
5)
The head of a chair, Al-Farabi Kazakh National University. Address: 39/47
Masanchi st., 050012, Almaty city, Kazakhstan. E-mail: serovajskys@mail.ru
6)
The researcher of a state scientific-technical program, Kazakh National Technical
University named after K. Satpayev. Address: KazNTU, 22 Satpaev str., 050013, Almaty city,
Kazakhstan. E-mail: A.Ashimov@kcp.kz
Preferred address for correspondence: Yuriy V. Borovskiy, 313 Furmanov str, apt. 22,
050059, Almaty city, Kazakhstan. E-mail: yuborovskiy@gmail.com
The work presents results on development of the parametrical regulation theory for the
class of discrete stochastic dynamic systems with additive noise. Within the framework of
development of the parametrical regulation theory for discrete dynamic systems we
formulated and proved the theorem about sufficient conditions for existence of solutions to
the problems of variational calculus on synthesis of optimal laws of parametrical regulation
and the theorem about sufficient conditions for continuous dependence of criterions' optimal
values of the considered problem on values of unregulated parameters. The discrete stochastic
model, taken as example, was obtained from the one deterministic computable model of
general equilibrium of economic sectors by adding additive noise to the right parts of dynamic
equations of the model. The problem of parametrical identification of the deterministic variant
of the model was preliminary solved based on the statistical data for the Republic of
Kazakhstan. For the stochastic and deterministic variants of the investigated model problems
of optimal economic growth of national economy of the Republic of Kazakhstan were
formulated and solved (applying the methods of parametrical regulation theory).
Key words: dynamic system with additive noise, parametrical identification,
parametrical regulation.
1
Introduction
As it is known [1] presented in the literature models of the national economy reflects in
mathematical form the most important properties of the economic system and do not account
for a number of emerging shocks, such as violations of the supply side (production and supply
of labor), violations on the demand side (preferences, specific investment, government
spending), the effect of increasing costs or margins (margin, supplements to wages and
salaries, the risk premium), violation of money circulation (interest rates, etc.).
In this paper we adopted the assumption that these and other possible irregularities in
the mathematical model of the national economy can be approximated by adding additive
noise to the right sides of the dynamical equations of the corresponding mathematical model
of economic system. Below the theory of parametrical regulation of the national economy (the
effectiveness of which is shown in the class of models in the form of continuous or discrete
dynamical systems [2], [3]), is developing to the class of discrete dynamic systems with
additive noise, an important subclass of which are computable general equilibrium models
(the so-called CGE-models [4]) with additive noise. In particular, in the framework of the
parametrical regulation theory development in the present work for discrete dynamic systems
with additive noise we formulated the following theorems:
- about sufficient conditions of existence of solutions to problems of variational
calculus on the choice of parameters’ optimal values in a given set of their values (on
synthesis of optimal laws of regulation). In particular if these conditions are met, this ensures
finiteness of mathematical expectation of phase trajectories on a finite time interval.
- about sufficient conditions of continuous dependence (of variational calculus
problems’ of synthesis laws of parametrical regulation) optimal values of criterion on values
of uncontrolled parameters.
In this paper the discrete stochastic model, obtained from deterministic computable
model of the general equilibrium of economic sectors of Makarov [4] by adding additive
noise to the right sides of the model’s dynamical equations, was considered as an example of
theoretical results applications. The problem of parametrical identification of the researched
model was solved based on statistical data on economy evolution of the Republic of
Kazakhstan. The problems of optimal (in sense of some criterion) economic growth of
national economy of the Republic of Kazakhstan were formulated and solved applying
methods of the parametrical regulation theory.
1. Elements of the parametrical regulation theory based on the discrete dynamic
system with additive noise
We consider discrete stochastic regulated system
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝛼) + 𝜉(𝑡), 𝑡 = 0, … , 𝑛 − 1,
x(0)  x0 ,
(1)
(2)
where t – time taking nonnegative integer values;
x  x(t )  x1 (t ),..., x m (t ) – function of the system (1) state, random vector-function of


discrete argument (a vector random process);
u  u (t )  u1 (t ),..., u q (t ) – regulation, vector-function of discrete argument;


2
a   a1 ,..., a s  – deterministic vector of uncontrollable parameters, 𝑎 ∈ 𝐴, 𝐴 – given
set, 𝐴 ⊂ 𝑅 𝑠 ;
𝜉 = 𝜉(𝑡) = (𝜉1 (𝑡), … , 𝜉 𝑚 (𝑡)) – known vector random process, which expresses the
noise (as such may be, for example, an additive Gaussian white noise);
f – a known vector-function of its arguments;
x0  x01 ,..., x0m – initial state of the system, a deterministic vector.


We define the optimality criterion to be the maximization for 𝑎 fixed:
𝐾𝛼 = 𝐄{∑𝑛𝑡=1 𝐹𝑡 [𝑥(𝑡)]}.
(3)
Here Ft - are known functions, Е – mathematical expectation, 𝑥(𝑡) - a solution to the system
(1), (2) for a given 𝛼.
We introduce the phase constraints for the system (1), (2):
E[ x(t )]  X (t ), t  1,..., n,
(4)
where X (t ) – given set. Hereinafter the mathematical expectation of a vector chance quantity
means the vector of mathematical expectations of the coordinates of this value.
In the below considered tasks, explicit constraints on regulation are assumed:
u (t )  U (t ), t  0,..., n  1,
(5)
where 𝑈(𝑡) – given set, 𝑈(𝑡) ⊂ 𝑅 𝑞 . Sets X (t ) , U (t ) for all determined above t values are the
closures of bounded open sets.
The method of parametrical regulation is used in the formulation and solution of the
following variational problem, so called the problem of variational calculus of the synthesis of
the optimum law of parametrical regulation.
Problem 1. Given the vector of unregulated parameters 𝑎 ∈ 𝐴 find such regulation u,
that satisfies the condition (5), so that corresponding to it the solution of dynamical system
(1), (2) satisfies the condition (4) and provides the maximum of the functional (3).
For the fixed 𝑎 ∈ 𝐴 let us define the set of allowable regulations for the system (1) – (2)
in the following manner:
𝑈𝛼 = {𝑢|𝑢(𝑡) ∈ 𝑈(𝑡), 𝑡 = 0, … , 𝑛 − 1; 𝐄[𝑥(𝑡)] ∈ 𝑋(𝑡), 𝑡 = 1, … , 𝑛} ⊂ 𝑅 𝑛𝑞 ,
where x(t ) is the solution to the system (1), (2) corresponding to the regulation u and set 𝑎
parameter value. Then the problem 1 is reduced to maximizing the functional 𝐾 = 𝐾𝛼 (𝑢)
which is defined by the formula (3), on the set of allowable regulations of the system 𝑈𝛼 . We
call the problem 1 non-trivial, if the set 𝑈𝛼 is non-empty and contains some open set.
The theorem on sufficient conditions for existence of solution to the problem 1 (set in
accordance with the method of parametrical regulation) and the theorem of continuous
dependence of the optimal value of 𝐾𝛼 criterion of the problem 1 on the parameter a are
presented below.
Theorem 1. Suppose given fixed 𝑎 ∈ 𝐴 in non-trivial problem 1 for any 𝑡 = 1 ÷ 𝑛
chance quantity 𝜉(𝑡) are absolutely continuous and have zero mathematical expectations;
functions f and Ft satisfy the Lipschitz condition. Functions f (for 𝑢 ∈ 𝑈𝛼 ) and Ft by module
3
are constrained by some linear functions of |x|. That is, for some positive number b the
inequalities
|𝑓(𝑥, 𝑢, 𝛼)| ≤ 𝑏(1 + |𝑥|), |𝐹𝑡 (𝑥)| ≤ 𝑏(1 + |𝑥|)
are carried out. Then the problem 1 has a solution.
Theorem 2. Suppose when 𝑎 ∈ 𝐴 and 𝑡 = 1 ÷ 𝑛 for non-trivial problem 1 chance
quantity 𝜉(𝑡) are absolutely continuous and have zero mathematical expectations, functions f,
Ft satisfy the Lipschitz condition. Functions f and Ft do not exceed a linear function with
respect to |x| in some neighborhood of the point a. Then the mapping 𝑎 → max 𝐾𝑎 (𝑢) is
𝑢∈𝑈𝑎
continuous in A.
Proofs of Theorem 1 and 2 are presented in Appendix.
2. Representation of computable general equilibrium models in the form of
discrete stochastic dynamic systems and the problem of parametrical regulation of the
national economy evolution on the basis of computable models
Deterministic computable general equilibrium model in general form is represented by
the following system of relations [4, Ch. 3].
1) The subsystem of difference equations, binding the values of endogenous variables
for two consecutive years:
x(t  1)  f ( x(t ), y (t ), z (t ), u (t ),  ) , x(0)  x0 ,
(6)
x (t )  ( x(t ), y(t ), z (t ))  R m – vector
Here t – year number, discrete time, t  0, 1, 2, ..., n  1 ; ~
of the system’s endogenous variables;
x(t )  ( x 1 (t ), x 2 (t ),..., x m1 (t ))  X 1 (t ) , y(t )  ( y 1 (t ), y 2 (t ),..., y m2 (t ))  X 2 (t ) ,
z(t )  ( z 1 (t ), z 2 (t ),..., z m3 (t ))  X 3 (t ) ;
x(t ) variables include values of fixed assets of producing sectors, balances in agents’ bank
accounts, etc.; y (t ) include agents’ demand and supply values in different markets, etc.; z (t )
– different values of market prices and a budget share in the market with state prices for
different economic agents; m1  m2  m3  m ; u and  are vectors of exogenous parameters;
u  (u 1 (t ), u 2 (t ),..., u q (t ))  U (t )  R q – vector of controlled (regulated) parameters; X1(t),
X2(t), X3(t), U(t) – compact sets with nonempty interior;    1 ,  2 ,...,  s   A  R m - vector
of uncontrolled parameters, A – open connected set; f : X 1 (t )  X 2 (t )  X 3 (t )  U (t )  A  R m1
– continuous mapping for t  0,1,2,..., n  1 .
2) The subsystem of algebraic equations, describing behavior and interaction of agents
in different markets during the selected year, these equations allow the expression of variables
y (t ) in terms of exogenous parameters and the remaining endogenous variables:
y (t )  g ( x(t ), z (t ), u (t ),  ) .
Here g : X 1 (t )  X 3 (t )  U (t )  A  R m2 continuous mapping, t  0,1,2,..., n .
4
(7)
3) The subsystem of recurrent relations for iterative calculations of equilibrium values
of market prices in different markets and of budget shares in markets with state prices for the
different economic agents:
z (t )[Q  1]  h( z (t )[Q], y (t )[Q], L, u (t ),  ) .
(8)
Here Q  0, 1, 2, ... – iteration number; L – set of positive numbers (adjustable iteration
constants (When their values decrease the economic system reaches the equilibrium state
faster, but the risk that the price go to the negative domain increases);
h : X 2 (t )  X 3 (t )  (0,) m3  U (t )  A  R m3 – continuous mapping (which is contracting
when x(t )  X 1 (t ), u (t )  U (t ) ,   A are fixed and some fixed L. In this case h mapping
has a single fixed point, to which the iterative process (7), (8) converges), t  0,1,2,..., n .
Computable model (6), (7), (8) given fixed values of exogenous variables for each
x (t ) that correspond to the demand
moment of t time defines values of endogenous variables ~
and supply equilibrium (in the markets of agents’ goods and services) in the framework of the
algorithm below.
1) On the first step it is assumed that t=0 and the initial values of x(0) variables are set.
2) On the second step the initial values of z (t )[ 0] variables are set for the current t in
different markets and for different agents; with the help of (7) the values of
y (t )[0]  G ( x(t ), z (t )[0], u (t ), a) are calculated (initial values of demand and supply of agents
in markets of goods and services).
3) On the third step the iteration process (8) is run for current t. Meanwhile current
values of demand and supply for every Q are found from (7): y (t )[Q]  G ( x(t ), z (t )[Q], u, a)
through the refinement of market prices and budget shares of economic agents.
The condition for completion of iteration process is the equality of demand and supply
values in different markets. As a result equilibrium values of market prices are determined in
every market and budget shares in markets with government prices for different economic
agents. Q index is omitted for such equilibrium values of endogenous variables.
4) On the next step values of xt 1 variables for the next moment of time are found in
accordance with the obtained equilibrium solution for t moment with the help of difference
equations (6). The value of t increases by unity. Transition to the step 2. The number of
reiteration of steps 2, 3, 4 is defined according to the problems of calibration, forecasting and
regulation for time intervals selected in advance.
Stochastic computable general equilibrium model (stochastic computable model)
obtained from the deterministic model (6), (7), (8), is a model to the dynamical equation’s (6)
right side of which we added an additive noise 𝜉(𝑡):
𝑥(𝑡 + 1) = 𝑓(𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡), 𝑢(𝑡), 𝛼) + 𝜉(𝑡), 𝑡 = 0, … , 𝑛 − 1; 𝑥(0) = 𝑥0 ,
(9)
i.e., the model of type (9), (7), (8).
We formulate the problem of variational calculus on the synthesis of the parametrical
regulation optimal law for a stochastic computable model.
Problem 2. Given the vector of unregulated parameters 𝑎 ∈ 𝐴 find such a control u(t),
which satisfies the condition (5), so that the corresponding to it the solution of the dynamical
system (9), (7), (8) satisfies the following condition
𝐄[𝑥̃(𝑡)] ∈ 𝑋1 (𝑡) × 𝑋2 (𝑡) × 𝑋3 (𝑡), 𝑡 = 1, … , 𝑛
5
and provides maximum to the functional
𝐾𝛼 = 𝐄{∑𝑛𝑡=1 𝐹𝑡 [𝑥̃(𝑡)]}.
It is not difficult to check, that formulated above theorems 1, 2 hold true for the
computable model with the continuous mappings f, g, h and with convergent iterative process
(7), (8).
3. Example. Application of the parametrical regulation theory by the example of a
stochastic computable model of economic sectors
3.1. Results of the parametrical identification of the deterministic computable
model of economic sectors
The considered model according to the statistical data of the Republic of Kazakhstan is
presented by nineteen economic agents (sectors), among which there are 16 agents-producers,
and there is a sector – an aggregate consumer, a government sector and a banking sector.
The model under consideration is presented in the framework of the general expressions
of relations (6), (7), (8) respectively m1  67 , m2  597 , m3  34 relations, with the help of
which the values of its 698 endogenous variables are calculated. The model also contains
2045 exogenous evaluated parameters.
In this case the problem of identification (calibration) of exogenous parameters ω of the
model comes to finding the global minimum of some objective function, which is set by the
model itself. The constraints on the set of optimization (  ) are also set by using the same
model. To solve the problem of parametric identification of the considered computable model
(based on the apparent assumption that the mismatch (in general case) of points of minimum
two different functions) the two criteria of the following types were proposed:
K IA ( ) 
1
n T
 y i (t )  y i* (t ) 




i
i*


y
(
t
)
t 1 i 1


T
2
nA
K IB ( ) 
1
n T
 y i (t )  y i* (t ) 




i
i*


y
(
t
)
t 1 i 1


2
nB
T
Here y i (t ) , y i* (t ) – calculated and observed values of the model’s output variables
respectively, K IA ( ) – auxiliary criterion, K IB ( ) – basic criterion; n B  n A ;  i  0 и
 i  0 – some weight coefficients, which values are determined by solving the problem of
nA
parametric identification of the dynamic systems;

i 1
i
 n ,
nB

i 1
i
 n .
The algorithm of solution to the problem of parametrical identification of the model was
selected in the form of the following steps.
1. The problems A and B (problems of finding minimums of the functions K IA () and
K IB ( ) respectively) are solved simultaneously for some vector of initial values of
parameters 1   , as a result points  K0 IA и  K0 IB are found.
2. If K IB ( K0 IB )   , then the problem of parametrical identification of the model (6, 7,
8) is considered to be solved.
3. Otherwise, the problem A is solved taking the point  K0 IB as a starting point  1 and
the problem B is solved taking the point  K0 IA as a starting point  1 . Proceed to Step 2.
6
Sufficiently large number of iterations of steps 1, 2, 3 gives an opportunity for searched
values of parameters to come out from neighborhood of points of nonglobal minimums of one
criterion with the help of another criterion and thus to solve the problem of parametrical
identification.
As a result of simultaneous solving of the problems A and B applying the stated
algorithm with the help of the Nelder-Mead algorithm [5] the values K IA  0.015 and
K IB  0.0129 were obtained. Meanwhile the relative magnitude of the deviations of
calculated values of the variables used in the basic criteria from the corresponding observed
values was less than 1.29%.
3.2. Finding optimal values of regulated parameters based on the stochastic
computable model of economic sectors
The stochastic computable model of economic sectors was obtained from the
corresponding deterministic model (with found by solving the problem of parametrical
identification estimate values of exogenous parameters) by adding discrete white noise to the
right side of the dynamic equations of the model. Values to the order of magnitude smaller
compared to the values of deterministic parts of the corresponding dynamic equations were
taken as the standard deviations of Gaussian random variables defining the white noise.
In computational experiments with the stochastic computable model as the optimization
criterion we used the criterion
 1 2015

K 1  E   Y (t )  max
 6 t  2010 
(10)
– average value of gross output of the country in prices of 2000 for 2010 - 2015 and according
considered realizations of a random process.
The value of K1 criterion for the basic computational variant (applying the values of
exogenous parameters, obtained as a result of the model’s parametrical identification) equals
to K1 = 0.9891∙1013.
During the experiments with the optimization criterion (10) the constraints on growth of
consumer prices of the following type was used:
E( P(t ))  1.09E( P (t )), t  2010  2015 .
Here P (t ) – calculated consumer price level of the model without parametrical regulation,
P (t ) – consumer price level with parametrical regulation.
In the computational experiments the regulation of 1536 exogenous parameters – j-th
agent-producer’s budget shares, assigned to purchase of goods and services that are produced
by i-th agent-producer for the years of 2010-2015 was carried out:
Oi j (t ); t  2010  2015; i, j  1  16 . Here
16
O
i
i 1
j
(t )  1 for the selected values of t. Basic
values of the selected shares, obtained as a result of solution of the parametrical identification
problem according to the data for 2000-2008 are expressed in terms of Oi j ; i, j  1  16 .
The following problem of finding the values of regulated vectors parameters was
considered. Based on the stochastic computable model of economic sectors find values of
7
agent-producers budget shares ( Oi j (t ); t  2010  2015; i, j  1  16 ), that would provide the
upper boundary of K1 criterion under the following additional constraints on these shares:
0.5  Oi j (t ) / Oi j  2; i, j  1  16; t  2010  2015 .
The solutions of such optimization problems were obtained with the help of NelderMead algorithm [5]. After application of the parametrical regulation of the stochastic model’s
budget shares, the values of the criterion turned out to be K1 = 1.2453∙1013 its value increased
by 25.89 % as compared to the basic variant.
The analogous problem of the parametrical regulation with corresponding constraints
1 2015
and with the criterion (10) analogue K 2   Y (t ) was solved on the basis of the
6 t  2010
.
deterministic CGE model of economic sectors.
After application of the parametrical identification of agent producer’s budget shares the
criterion value of the deterministic model turned out to be equal to K2 = 1.6283∙1013, the value
of the criterion K2 increased by 33.14 % as compared to the basic variant.
If to compare the results of the problem of the variational calculus on the basis of
stochastic and deterministic computable models of general equilibrium, one can say that there
is a reduction in the estimated value of the functional of the variational problem, taking into
account the disturbing violations in the deterministic computable general equilibrium model
in the form of additive noise.
Conclusion
1. Some results on the development of the theory of parametrical regulation for a class
of discrete stochastic dynamic models are presented. We prove theorems on sufficient
conditions for the existence of solution of the stated optimization problem and on the
continuous dependence of optimal values of the criterion of this problem on uncontrollable
parameters.
2. We show the application efficiency of the parametrical regulation theory by the
example of a stochastic computable model of economic sectors. A method for estimating the
optimal values of regulated parameters of economic policy based on the considered
mathematical model is offered and the optimal values of regulated parameters are found.
3. The obtained results can be used in the development and implementation of effective
state economic policy.
The authors acknowledge N.Yu Borovskiy and D.B. Nurseitov for helping to conduct
computational experiments.
References
[1] Samarskiy A.A., Mikhailov A.P. Mathematical modeling: Ideas. Methods.
Examples, Physmatlit, Moscow, 2002 (in Russian).
[2] Ashimov A.A., Sultanov B.T., Adilov Zh.M., Borovskiy Yu.V., Novikov D.A.,
Nizhegorodcev R.V., & Ashimov As.A., Macroeconomic analysis and economic policy based
on parametrical regulation, Physmatlit, Moscow, 2010 (in Russian).
[3] Ashimov A.A., Sagadiyev K.A., Borovskiy Yu.V., Iskakov N.A. & Ashimov As.A.,
On the market economy development parametrical regulation theory. Kybernetes, The
8
international journal of cybernetics, systems and management sciences. Vol. 37, №5, 2008,
623-636.
[4] Makarov V.L., Bakhtizin A.R., & Sulakshin S.S. The use of computable models in
public administration, Scientific Expert, Moscow: 2007.
[5] Nelder J.A. & Mead R. A simplex method for function minimization. The Computer
Journal. №. 7, 1965, 308-313.
Appendix
Proof of theorem 1. According to the Weierstrass theorem, a continuous function on a
nonempty closed bounded set reaches its maximum. Thus, we need to show that a defined
with the help of (3) the function of many variables 𝐾 = 𝐾𝛼 (𝑢) is continuous and the set 𝑈𝛼 is
closed and bounded. Its non-emptiness is a part of the theorem condition.
We show that there are mathematical expectations of values in the phase constraint (4).
Indeed, according to equation (1), we have
𝐄[𝑥(𝑡 + 1)] = 𝐄[𝑓(𝑥(𝑡), 𝑢(𝑡), 𝛼)] + 𝐄[𝜉(𝑡)]
The second term on the right side of this equation makes sense under the conditions of
the theorem, and the first is calculated by the formula
𝐄[𝑓(𝑥(𝑡), 𝑢(𝑡), 𝛼)] = ∫ 𝑓(𝜔, 𝑢(𝑡), 𝛼)𝑝𝑥(𝑡) (𝜔)𝑑𝜔,
𝑅𝑚
if the last integral is absolutely convergent (here the probability density function of a random
variable 𝑥(𝑡) is expressed in terms of 𝑝𝑥(𝑡) ). The latter fact is indeed the case due to
constraints on the growth of f function and due to the existence of the mathematical
expectation of x(t) value for any t=1,…,n (this fact is verified by induction).
The existence of mathematical expectation on the right side of this equation (3)
follows from the constraints on growth of Ft function and the existence of the mathematical
expectation of x (t ). value. We prove K continuous dependence on u in topology of 𝑅 𝑛𝑞 .
Suppose we have the convergence of vectors uk  u , 𝑢𝑘 ∈ 𝑈𝛼 . From equation (1) it follows
that
|𝑥𝑘 (𝑡 + 1) − 𝑥(𝑡 + 1)| = |𝑓(𝑥𝑘 (𝑡), 𝑢𝑘 (𝑡), 𝑎) − 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑎)|
where xk and х are problem (1), (2) solutions under regulations uk and и respectively. Then
the following relation
xk (t  1)  x(t  1)  L f  xk (t )  x(t )  uk (t )  u (t )  ,
is true where L f is a Lipschitz constant of the function f. Repeating similar arguments and
taking into account that, in virtue of condition (2) xk (0)  x(0) , we have
|𝑥𝑘 (𝑡 + 1) − 𝑥(𝑡 + 1)| ≤ (𝐿𝑓 )2 |𝑥𝑘 (𝑡 − 1) − 𝑥(𝑡 − 1)| +
+(𝐿𝑓 )2 |𝑢𝑘 (𝑡 − 1) − 𝑢(𝑡 − 1)|+𝐿𝑓 |𝑢𝑘 (𝑡) − 𝑢(𝑡)| ≤
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𝑡
≤ ∑(𝐿𝑓 )𝑠+1 |𝑢𝑘 (𝑡 − 𝑠) − 𝑢(𝑡 − 𝑠)| ≤ 𝜀𝑘
𝑠=0
where  k  0 when k  .
Denoting the maximum of the Lipschitz constant of Ft functions, for t = 1,…,n through
LF we get the estimate
Ft  xk (t )  Ft  x(t )  LF  k .
Having calculated the mathematical expectations of both sides of this inequality, we
obtain the inequality
Е{|𝐹𝑡 [𝑥𝑘 (𝑡)] − 𝐹𝑡 [𝑥(𝑡)]|}≤𝐿𝐹 𝜀𝑘 .
It follows that Е{|𝐹𝑡 [𝑥𝑘 (𝑡)] − 𝐹𝑡 [𝑥(𝑡)]|}→0 and convergence of the considered sequence:
E Ft  xk (t )  E Ft  x(t ) . Hence, this convergence and (3) imply the continuity of the
functions 𝐾𝛼 from u. Boundedness of the set 𝑈𝛼 follows from the boundedness of sets U (t ) .
Closure of the set 𝑈𝛼 follows from the continuous mapping 𝑈𝛼 → 𝑋, set with the help of
definition of the set 𝑈𝛼 and compactness of the set Х (the theorem on the closure of a
complete preimage of a compact under the continuous mapping). Now the existence of
problem 1 solution follows from the Weierstrass theorem. The theorem is proved.
Tentatively we formulate the following definition and auxiliary statement used in the
proof of Theorem 2.
Definition. Let the family of functions 𝐾𝑎 (𝑢), 𝑎 ∈ 𝐴, 𝑢 ∈ 𝑈 be defined for the family of
subsets {𝑈𝛼 } of some set U of Euclidian space with the parameter а (from some subset А of
Euclidian space). The family {𝑈𝛼 } is K-continuous on the set А, if for any 𝜀 > 0 there is a
number 𝛿 > 0, so that when performing the inequality |𝑎 − 𝑏| ≤ 𝛿, 𝑎, 𝑏 ∈ 𝐴 for any 𝑢𝑎 ∈
𝑈𝑎 there is such point 𝑢𝑏 ∈ 𝑈𝑏 , so that |𝐾𝑎 (𝑢𝑎 ) − 𝐾𝑎 (𝑢𝑏 )| < 𝜀 inequality holds.
According to this definition the family of sets {𝑈𝛼 } is K-continuous, if in the case of
enough proximity of the parameter b to а, for any element from the set 𝑈𝑏 there is arbitrarily
close (for a function value) to it element of the set 𝑈𝛼 .
The following lemma is auxiliary to prove the continuity of optimal values of the
problem of variational calculus on the synthesis of optimal laws of parametrical regulation.
Lemma 1. Let A and U – be some subsets of Euclidian spaces; A – open set;
U - compact; the family of closed subsets {𝑈𝛼 } belongs to U. Let the mapping (𝑎, 𝑢) →
𝐾𝑎 (𝑢) be continuous in the product of 𝐴 × 𝑈. Let the family of subsets {𝑈𝛼 } be Kcontinuous in some neighborhood of a point 𝑎0 ∈ А. Then the mapping 𝑎 → max 𝐾𝑎 (𝑢) is
𝑢∈𝑈𝑎
continuous in the point 𝑎0 .
Proof of Lemma 1.
Suppose the convergence of some sequence 𝑎𝑘 → 𝑎0 holds, where 𝑎𝑘 ∈ 𝐴 Let 𝑢𝑘
denote the maximum point of the function 𝐾𝑎𝑘 on the set 𝑈𝑎𝑘 , and 𝑢0 denote the maximum
point of the function 𝐾𝑎0 on the set 𝑈𝑎0 .
Taking into account 𝐾-continuity of the family of sets {𝑈𝑎 } and continuity of the
function 𝑓𝑎 (𝑥) on 𝐴 × 𝑈, we conclude that for any value ε > 0 we can find such a number 𝑘0 ,
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that when 𝑘 > 𝑘0 there are points 𝑢′𝑘 ∈ 𝑈𝑎0 for which the inequalities |𝐾𝑎𝑘 (𝑢′𝑘 ) −
𝐾𝑎𝑘 (𝑢𝑘 )| ≤ ε hold and additionally max|𝐾𝑎𝑘 (𝑦) − 𝐾𝑎о (𝑦)| ≤ ε holds.
𝑦∈𝑈
As a result, when 𝑘 > 𝑘0 we get the following inequalities
𝐾𝑎0 (𝑢0 ) ≥ 𝐾𝑎0 (𝑢′ 𝑘 ) ≥ 𝐾𝑎𝑘 (𝑢′ 𝑘 ) − 𝜀 ≥ 𝐾𝑎𝑘 (𝑢𝑘 ) − 2𝜀.
(11)
In the same way the following relation is checked
𝐾𝑎𝑘 (𝑢𝑘 ) ≥ 𝐾𝑎0 (𝑢0 ) − 2𝜀.
(12)
From (11) and (12) it follows that under sufficiently large 𝑘 the inequality |𝐾𝑎𝑘 (𝑢) −
𝐾𝑎0 (𝑢0 )| ≤ 2𝜀 holds, that provides the convergence of the sequence 𝐾𝑎𝑘 (𝑢𝑘 ) → 𝐾𝑎0 (𝑢0 ).
Lemma is proved.
Proof of Theorem 2.
In the proof of theorem 1 we established that sets U a are closed and bounded and K a
function is continuous. There we also established continuity of the mapping u  E  xau (t ) 
from which the K-continuity of the family of sets follows. Here 𝑥𝛼𝑢 (𝑡) denotes the solution of
the system (1), (2) for selected 𝛼 and 𝑢.
We show the uniform on u continuity of the mapping a  Ka (u ). Suppose that there
is convergence ak  a where 𝑎, 𝑎𝑘 ∈ 𝐴. From the condition (1) we get the following estimate


xauk (t  1)  xau (t  1)  f xauk (t ), u (t ), ak  f  xau (t ), u (t ), a  
 L f  xauk (t )  xau (t )  ak  a  ,
where L f is the Lipschitz constant of f function. Given the initial state of the system, we
obtain the inequality
xauk (t )  xau (t )    L f  ak  a .
n
s
s 1
Let LF denote the maximum of Lipschitz constants of Ft functions. Then the following
estimate is true
Ft  xauk (t )   Ft  xau (t )   LF xauk (t )  xau (t )  LF   L f
n
s 1

s
ak  a .
Having proceeded to the mathematical expectations of the right and left sides in the last
inequality, we obtain that Е{|𝐹𝑡 [𝑥𝑎𝑢𝑘 (𝑡)] − 𝐹𝑡 [𝑥𝑎𝑢 (𝑡)]|}→0. From this it follows that, when
𝑢
𝑢
k   for any t = 1,…,n there is convergence Е{𝐹𝑡 [𝑥𝑎𝑘 (𝑡)]}→Е{𝐹𝑡 [𝑥𝑎 (𝑡)]}. uniformly on и
from the sum of all sets U a and hence, uniform on и continuity of the mapping a  Ka (u ).
and continuity on (𝑎, 𝑢) of the function 𝐾𝑎 (𝑢). Having applied Lemma 1 we obtain the
desired result. The theorem is proved.
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