Advances in Systems Science and Applications (2014) Vol.14 No.2 101-128

The Theory of Parametric Control of Macroeconomic
Systems and Its Applications(II)

A. Ashimov, Zh. Adilov, R. Alshanov, Yu. Borovskiy and B. Sultanov
Kazakh National Technical University named after K. Satpayev, 22 Satpaev

Street, Almaty, 050013, Kazakhstan

Abstract

This work consists of three parts and presents the recent results of development
of the theory of parametric control of macroeconomic systems and some its ap-
plications for solving a number of concrete problems.
Keywords Mathematical Model, Structural Stability, Parametrical Identification,
Parametric Control

Part 2. Mathematical Foundations of the Parametric Control Theory

2.1 Sufficient Conditions for the Existence of Solutions for the Problems on Synthesis
and Choice of Optimal Parametric Control Laws

2.1.1 Conditions for the Existence of Solution for the Variational Calculus Problem on
Synthesis of Optimal Parametric Control Law of Continuous Dynamical System

Consider continuous controllable system*

ẋ(t) = f(x(t), µ(t), a(t)), t ∈ [0, T ] (1)

x(0) = x0 (2)

where t is time; x = x(t) = (x1(t), ..., xm(t)) is a vector function of system’s
state; µ = µ(t) = (µ1(t), ..., µq(t)) is a vector function of control; a = a(t) =
(a1(t), ..., as(t)) is a known vector function; s0 = (x10, ..., x

m
0 ) is an initial condition

of the system, a known vector; f-a known vector function of its arguments.
The problem of synthesis of optimal economic tools values consists in finding

the extremum of the following criterion

K =

∫ T

0
F (t, x(t))dt (3)

where F is a known function, at the phase constraints

x(t) ∈ X(t), t ∈ [0, T ] (4)

where X(t) is a given set and at explicit control constraints

u(t) ∈ U(t), t ∈ [0, T ] (5)

* All of the formulated in this part results for non-autonomous dynamical systems remain true
for autonomous dynamical systems as well, when an exogenous vector function a(·) is taken as
constant.



102 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

where U(t) is a given set.
Define the variational calculus problem on synthesis of optimal parametric con-

trol laws for the continuous dynamical system.
Problem 2.1. At given function a(·) to find the control u(·), satisfying the

condition (5), so that dynamical system (1), (2)solution meets the condition (4)
and gives the maximum (the minimum) to functional (3).

To prove the solubility of the Problem 2.1 we first need the single-valued sol-
ubility of Cauchy problem (9), (10). To obtain this result we use known result
from the theory of ordinary differential equations. Let be given: number T > 0,
metrizable compact U and continuous function φ : [0, T ] × Rm × U → Rm such
that for any ρ ≥ 0 exists such σ ≥ 0, that the following inequality is true

|φ(t, y, µ)− φ(t, y′, µ)| ≤ σ|y − y′|∀t ∈ [0, T ], y, y′ ∈ ρB, µ ∈ U (6)

where is a unit ball in Rm and there is such a constant η ≥ 0 that the following
inequality is true

|yφ(t, y, µ)| ≤ η(1 + |y|2)∀t ∈ [0, T ], y ∈ Rm, µ ∈ U (7)

Consider Cauchy problem

ẏ(t) = φ(t, y(t), µ(t))∀t ∈ [0, T ] (8)

y(0) = y0 (9)

where y0 ∈ Rm. The following statement is true ([1], Lemma 4.1):
Lemma 2.1. Given above mentioned constraints (6), (7) for any measurable

mapping µ : [0, T ] → U the problem (8), (9) has the only solution y : [0, T ] → Rm,
satisfying the estimate

|y(t)| ≤
(
|y0|2 + 2ηT

)1/2
eηT∀t ∈ [0, T ] (10)

The following statement is true.
Theorem 2.2. Let function a(·) be continuous on segment [0, T ], U is a

compact in Rq, function f is continuous, for any ρ ≥ 0 there is such σ ≥ 0, that
the following inequality is true

|f(x, µ, a(t))− f(x′, µ, a(t))| ≤ σ|x− x′|∀t ∈ [0, T ], x, x′ ∈ ρB, µ ∈ U (11)

and exists such a constant η ≥ 0 that the following inequality is true

|xf(x, µ, a(t))| ≤ η(1 + |x|2)∀t ∈ [0, T ], x ∈ Rm, µ ∈ U (12)



Advances in Systems Science and Applications (2014) Vol.14 No.2 103

Then for any measurable mapping µ : [0, T ] → U , Cauchy problem (2.1), (2.2)
has the only solution X : [0, T ] → Rm satisfying the estimate

|(t)| ≤
(
|x0|2 + 2ηT

)1/2
eηT∀t ∈ [0, T ] (13)

Proof. Let us introduce notation: φ(t, x, µ) = f (x, µ, a(t)) and f functions
continuity results in continuity of function φ. (6), (7) inequalities truth follows
from relations (11), (12). Thus the theorem statements follow from Lemma 2.1.
The theorem is proved.

For proof of solubility of the Problem 2.1 we use known result from the optimal
control theory for systems, described by differential equations. Let us specify a
closed subset E ⊂ [0, T ] × Rm × U . Consider the system, described by Cauchy
problem (8), (9) when Lemma 2.1 conditions are satisfied, herewith by control we
mean a measurable mapping µ : [0, T ] → U . Acceptable pair for the system in
question is such a pair “state-control”, which satisfies the relations (8), (9) and
inclusion

(t, x(t), u(t)) ∈ E (14)

Let us specify Carathéodory function Φ with non-negative values on the set

[0, t]× (Rm × U). Let us specify a functional I =
T∫
0

Φ (t, x(t), u(t)) dt.

Problem 2.2. To find the minimum for the functional I on the acceptable
pairs set of the system (8), (9).

For any point (t, x) ∈ [0, t]×Rm define a section Et,x = {u ∈ U : (t, x, u) ∈ E}.
Let us specify a set Γt,x = {φ(t, x, u) : (t, x, u) ∈ Et,x} and a function g : [0, T ]×
Rm×Rm → R using an equality g(t, x, y) = min{Φ(t, x, u)| (t, x, u) ∈ E,φ(t, x, u)
= y}.

It is known the following statement about solubility of optimization problem
([1], Statement 4.2), where R̄ implies the set of real numbers, supplemented with
the values −∞ and +∞:

Lemma 2.3. Let, when Lemma 2.1 condition is satisfied, the set Γt,x be con-
vex for all t ∈ [0, T ].x ∈ Rm, and the function g(t, x, ·) : Rm → R̄ be convex for
all (t, x) ∈ [0, T ]×Rm. Then the Problem 2.2has a solution.

Let be a closure of the sum
∪

t∈[0,T ]

X(t). The following statement is true.

Lemma 2.4. Let be a compact, function F be continuouson [0, T ]×X. Then
function Φ, specified by equality Φ(t, x, u) = F0 − F (t, x) where F0 is the maxi-
mum of function F on the set [0, T ]×X satisfies Lemma 2.3 conditions.

Proof. Existence of the maximum for F0 follows from the Weierstrass theorem.
Non-negativity of the determined above function Φ is obvious. It is Carathéodory
function because of the continuity of F. At last, the convexity of function g(t, x, ·)



104 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

for all (t, x) ∈ [0, T ]×Rm is implemented, as function F does not depend on con-
trol u. The lemma is proved.

Let U be a closure of the sum
∪

t∈[0,T ]. For reduction the constraints (4), (5)
to the form (14) we specify the set E ⊂ [0, T ] × Rm × U the way to hold the
relation: Et0 = {(t, x, u) ∈ E| t = t0} = X(t0)× U(t0), t0 ∈ [0, T ].

Lemma 2.5. Let the mappings t → X(t), t → U(t) be continuous at each
point t ∈ [0, T ] in the following sense: if the inclusions xk ∈ X(tk), uk ∈ U(tk)
are true, where tk ∈ [0, T ], k = 1, 2, ... and convergence of the sequences is met
tk → t, xk → x, uk → u, then the inclusions x ∈ X(t), u ∈ U(t) are true.Then the
set is closed.

Proof. Consider such an sequence {(tk, uk, xk)} of elements of the set , that
there is a convergence (tk, uk, xk) → (t, u, x). From the inclusion (tk, uk, k) ∈ E,
because of specifying the set , follow the inclusions tk ∈ [0, T ], xk ∈ X(tk), uk ∈
U(tk), From closure of segment [0, T ] follows the inclusion t ∈ [0, T ]. Using lemma
conditions, ascertain that x ∈ X(t), u ∈ U(t) Therefore, the inclusion (t, u, x) ∈ E
is true, that is the set is closed. The lemma is proved.

Let us make sure that lemma conditions are not excessively restrictive. Con-
sider, for instance, a typical situation for scalar case, when the set X(t) =
{x| a(t) ≤ x ≤ b(t)} , t ∈ [0, T ] is given, where functions and b are continuous.
Let the inclusion xk ∈ X(tk) be met, where tk ∈ [0, T ], k = 1, 2, ... and there is the
convergence tk → t, xk → x, Ipso facto, the inequalities a(tk) ≤ xk ≤ b(tk), k =
1, 2, ... are true. By proceeding here to the limit allowing for continuity of func-
tions and b, we get the result a(t) ≤ x ≤ b(t) what results in that x ∈ X(t)
Thereby, the mapping t ∈ X(t) is continuous on the segment [0, T ].

Analogously ascertains the truth of Lemma 2.5 conditions and in more gener-
al cases, when boundaries of sets of acceptable control and condition values are
continuous functions of time.

For reduction the equation (1) to the form (8) it is enough to specify φ(t, x, u) =
f(x, u, a(t)) Then the set Γt,x, appearing in Lemma 2.2 conditions, defines in the
following way

Γt,x = {f (x,w, a(t))|w ∈ U(t)} (15)

Denote by V the set of acceptable pairs “state-control” of the system (1), (2) in
question at given known function , that is such vector function pairs (x, u) which
satisfy the relations (1), (2), (4), (5). From Lemma 2.3 directly follows the next
statement.

Theorem 2.6. Let function (·) be continuous on the segment [0, t], U be a
compact in Rq, function f be continuous in X × U × A and for any ρ ≥ 0 there
is such σ ≥ 0, that the following inequality is true∣∣f(x, u, a(t))− f(x′, u, a(t))

∣∣ ≤ σ
∣∣x− x′

∣∣ , x, x′ ∈ ρB, u ∈ U (16)



Advances in Systems Science and Applications (2014) Vol.14 No.2 105

and there is such a constant η ≥ 0, that the following inequality is true

|xf(x, u, a(t))| ≤ η
(
1 + |x|2

)
∀t ∈ [0, T ], x ∈ Rm, u ∈ U (17)

Let X be a compact, function F be continuous on [0, T ]×X. Let, moreover, the
mappings t → X(t), t → U(t) be continuous for t ∈ [0, T ] in the following sense:
if the inclusion xk ∈ X(tk), uk ∈ U(tk) are true, where tk ∈ [0, T ], k = 1, 2, ... and
there is convergence of sequences tk → t, xk → x, uk → u, then the inclusions
x ∈ X(t), u ∈ U(t) are true. Then, in the case of non-emptiness of the set Va
and convexity of the set Γt,x for all t ∈ [0, T ], x ∈ X(t) the Problem 2.1 has a
solution in the class of measurable functions.

2.1.2 Conditions for the Existence of Solution for the Variational Calculus Problem on
Choice (Among Given Finite Algorithms Set) of Optimal Parametric Control Law of
Continuous Dynamical System

The continuous controllable system in question (1), (2). Control u is chosen here
among the set of given control laws:

uj(t) = Gj (v, x(t)) , t ∈ (0, T ), j = 1, ..., r (18)

Here Gj is known vector function of its arguments; υ = (υ1, ..., υl) is vector of
control parameters. On control parameters lay the constraints like

v ∈ V (19)

where V is some subset of the space Rl. Moreover, it is assumed that control
parameters should be such that corresponding control law (18) satisfies the con-
dition (5), that is the following inclusion is met

Gj (v, x(t)) ∈ U(t), t ∈ (0, T ) (20)

where U(t) is given set. There are phase constraints on the system:

x(t) ∈ X(t), t ∈ (0, T ) (21)

where X(t) is given set.
Consider optimality criteria

Kj = Kj(a, v) =

∫ T

0
F [t, xj(t)]dt (22)

where xj = xj(t) =
(
x1j (t), ..., x

m
j (t)

)
is solution of Cauchy problem (1), (2) at

given function (·) and control u = uj(t) =
(
u1j (t), ..., u

q
j(t)

)
, that is for chosen



106 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

j-th control law (26).
Consider the next subsidiary extremal problem:
Problem 2.3*. At given function a(·) for each of r control laws to find such

a vector of control parameters v, that corresponding it solution x = xj of the
problem (1), (2) with control law u = uj , determining by formula (18), satisfies
the conditions (19)-(21) and gives the maximum for the functional (22).

Define the following variational calculus problem on choice (among given finite
algorithms set) of optimal parametric control law for non-autonomous continuous
system.

Problem 2.3. Given known function a(·) among all optimal control laws in
sense of the Problem 2.3* to choose the one, which corresponds to the maximal
optimality criterion value (22).

Substituting into the equation (1) control value from formula (19), we get

ẋ(t) = f
(
x(t), Gj (v, x(t)) , a(t)

)
, t ∈ (0, T ) (23)

where for short we omit the index j in denoting condition function of the system,
corresponding to given control law. Denote by W j

a a set of acceptable pairs
“state-control parameter” of the system in question, that is such pairs (x, v),
which satisfy both the equalities (23), (2), and the inclusions (19)-(21). Thus,

the Problem 2.3* comes to the maximization of functional K =
∫ T
0 F [t, x(t)]dt

on the set W j
a .

Ascertain first a solubility of Cauchy problem (23), (2). The following theorem
directly results from Lemma 2.1.

Theorem 2.7. Let the function (·) is continuous on the segment [0, T ], sets U
and V are compact, functions f and Gj are continuous, for any ρ ≥ 0 there exist
such σ ≥ 0, χ ≥ 0, that the following inequalities are true

|f(x, u, a(t))−f(x′, u′, a(t))| ≤ σ(|x−x′|+|u−u′|)∀t ∈ [0, T ], x, x′ ∈ ρB, u, u′ ∈ U
(24)∣∣Gj (v, x)−Gj

(
v, x′

)∣∣ ≤ χ
∣∣x− x′

∣∣∀x, x′ ∈ ρB, v ∈ V (25)

and exists such a constant η ≥ 0 that the following inequalitiy is true

|xf(x,Gj(v, x), a(t))| ≤ η(1 + |y|2)∀t ∈ [0, T ], x ∈ Rm, v ∈ V (26)

Then for any v ∈ V the problem (2), (2) has the only solution x : [0, T ] →
Rm,satisfying the estimate

|(t)| ≤
(
|0|2 + 2ηT

)1/2
eηT∀t ∈ [0, T ] (27)

Ascertain solubility of the Problem 2.3*. The following theorem is true.
Theorem 2.8. Assume that when the conditions of the theorem 2.7 are satis-

fied for given function (·) and given value of j ∈ {1, ..., r} the setW j
a is not empty,



Advances in Systems Science and Applications (2014) Vol.14 No.2 107

the sets V, U, X are compact, the sets X(t), U(t) are compact for all t ∈ [0, T ],
and function F is continuous. Then the Problem 2.3* has a solution.

Proof. From the condition (21) and continuity of function F follows that
this function is bounded. Then, because of non-emptiness of the set W j

a , upper
boundary sup K of function K exists on set W j

a of acceptable pairs of the system
in question. Then there exists such an elements sequence {(xk, vk)} from the set

W j
a , that convergence takes place (here Kk =

∫ T
0 F [t, xk(t)]dt)

Kk → supK (28)

The inclusion (xk, vk) ∈W j
a assumes the following relations to be true

ẋk(t) = f
(
xk(t), Gj (vk, xk(t)) , a(t)

)
, t ∈ (0, T ) (29)

xk(0) = x0 (30)

xk(t) ∈ X(t), t ∈ (0, T ) (31)

vk ∈ V (32)

Gj (vk, xk(t)) ∈ U(t), t ∈ (0, T ) (33)

From the Theorem 2.7 results the estimate

|xk(t)| ≤ (|x0|2 + 2ηT )
1
2 eηT∀t ∈ [0, T ] (34)

Thus, using BolzanoCWeierstrass theorem allowing for boundedness of set V,
after subsequence separation we get the convergences

k(t) → (t)∀t ∈ [0, T ] (35)

vk → v (36)

Limiting value v satisfies the inclusion (19) because of closure of set V and
condition (32). Using the conditions (35), (36) and continuity of function Gj , we
get the convergence

f
(
xk(t), Gj (vk, xk(t)) , a(t)

)
→ f

(
x(t), Gj (v, x(t)) , a(t)

)
, t ∈ (0, T ) (37)

Function xk satisfies the integral relation

xk(t) = x0 +

∫ t

0
f
(
xk(τ), Gj (vk, xk(τ)) , a(τ)

)
dτ, t ∈ (0, T ) (39)

By proceeding here to the limit allowing for the conditions (35), (38), we get
the equality

x(t) = x0 +

∫ t

0
f
(
x(τ), Gj (v, x(τ)) , a(τ)

)
dτ, t ∈ (0, T ) (40)



108 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

By differentiating both parts of the equality (40) with respect to t, we ascertain
that the equation (1) is true. Consequently, after proceeding to the limit in
equality (30) taking into account the condition (35), we ascertain that the initial
condition (2) is true. Thus, the limiting pair (x, v) is acceptable.

Continuity of function F from the condition (35) results in convergence

F [t, xk(t)] → F [t, x(t)] , t ∈ [0, T ] (41)

therefore ∫ T

0
F [t, xk(t)]dt→

∫ T

0
F [t, x(t)]dt (42)

From (42) and (28) follows the equality
T∫
0

F [t, x(t)]dt = supK. Thus, we found

the pair (x, u) ∈ W j
a for which the upper boundary of functional K(on the set

of all acceptable pairs of the system in question) is obtained. The theorem is
proved.

Obviously, the Problem 2.3 comes to the search of maximum of function
Φa(j) = max

(x,v)∈W j
a
Kj on finite set {1, ..., r} at given function (·). The maximal

value of considered functional at given function (·) and j-th control law is placed
on the right side of previous equality. The existence of this value is proved in the
Theorem 2.8. As the maximum of function on finite set is obtained always, we
get the following statement.

Theorem 2.9. When Theorem 2.8 conditions are satisfied, the Problem 2.3
has a solution.

2.1.3 Conditions for the Existence of Solution for the Variational Calculus Problem on
Synthesis of Optimal Parametric Control Law of Discrete Dynamical System

Consider discrete non-autonomous controllable system

x(t+ 1) = f(x(t), u(t), a(t)), t = 0, 1, ..., n− 1 (43)

x(0) = x0 (44)

where t is time. Here x = x(t) =
(
x1(t), ..., xm(t)

)
is vector function (of discrete

argument) of systems state; u = u(t) =
(
u1(t), ..., uq(t)

)
is control, vector function

of discrete argument; a = a(t) =
(
a1(t), ..., as(t)

)
is known vector function of

discrete argument; x0 =
(
x10, ..., x

m
0

)
is initial condition of the system, known

vector; f is known vector function of its arguments.
The problem of choosing optimal economic tools values consists in finding the

extremum of the following criterion

K =

n∑
t=1

F [t, x(t)] → max (min) (45)



Advances in Systems Science and Applications (2014) Vol.14 No.2 109

where F is known function, at phase constraints on system (43)-(44) solution like

x(t) ∈ X(t), t = 1, ..., n (46)

where X(t) is given set, and following constraints on control:

u(t) ∈ U(t), t = 0, 1, ..., n− 1 (47)

where U(t) is given set.
Define the variational calculus problem on synthesis of optimal parametric con-

trol laws for discrete dynamical system.
Problem 2.4. Given known function a(·), to find control u(·), satisfying the

condition (47), so that dynamical system solution (43), (44), corresponding it,
satisfies the condition (46) and gives the maximum (minimum) for functional
(45).

Denote by Va the set of acceptable pairs “state-control” of the system in
question at given known function a(·), that is such pairs of vector functions
(x, u), which satisfy the relations (43), (44), (46), (47). Introduce notations:

X =
n∪

t=1
X(t), U =

n−1∪
t=0

U(t). The following statement is true.

Theorem 2.10. Let for given function a(·) the set Va be not empty; sets X(t)
and U(tC1) are closed and bounded for all t = 1, ..., n; mapping f is continuous
by the first two arguments on set X × U , and function F is continuous by the
second argument on set X. Then the Problem 2.4 has a solution.

Proof of this theorem is based on properties of functions which continuous on
compact.

2.1.4 Conditions for the Existence of Solution for the Variational Calculus Problem on
Choice (Among Given Finite Algorithms Set) of Optimal Parametric Control Law of
Discrete Dynamical System

Consider discrete non-autonomous controllable system (43), (44). The following
phase constraints are imposed on this system:

x(t) ∈ X(t), t = 1, ..., n (48)

where X(t) is given set. In equation (43) control is chosen among the set of given
control laws:

uj(t) = Gj (v, x(t)) , t = 1, ..., n, j = 1, ..., r (49)

where Gj is known vector function of its arguments, v = (v1, ..., vl) is vector of
control parameters. The following constraints are imposed on control parameters

v ∈ V (50)



110 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

where V is some subset of space Rl. Moreover, we will assume that control param-
eters should be such that corresponding control law (49) satisfies the condition

Gj (v, x(t)) ∈ U(t), t = 0, ..., n− 1 (51)

where U(t) is given set.
Consider the following optimality criteria:

Kj = Kj(a, v) =

n∑
t=1

F [t, xj(t)] (52)

where xj = xj(t) =
(
x1j (t), ..., x

m
j (t)

)
is problem (43), (44) solution at given func-

tion (·) and control u = uj(t) =
(
u1j (t), ..., u

q
j(t)

)
, that is for chosen j-th control

law.
Define the next subsidiary extremal problem:
Problem 2.5*. Given known function (·) for each of r control laws to find

such a vector of control parameters v, that corresponding it solution x = xj of
problem (43), (44) with control law u = uj , determining by formula (2.49), sat-
isfies the conditions (48), (50), (51) and gives the maximum for functional (52).

The following problem is called as variational calculus problem on choice a-
mong given finite set of algorithms of optimal parametric control laws for discrete
non-autonomous system.

Problem 2.5. Given known function (·) among all optimal control laws in
sense of the Problem 2.5* to find the one, which corresponds to the maximal
optimality criterion value (52).

Ascertain first the solubility of the Problem 2.5* for fixed control law. Substi-
tuting into equation (43) the control value from formula (49), we get

x(t+ 1) = f
(
x(t), Gj (v, x(t)) , a(t)

)
, t = 0, 1, ..., n− 1 (53)

where for short we omit the index j in denoting condition function of the system,
corresponding to j-th control law. Denote by W j

a the set of acceptable pairs
“state-control parameter” of the system in question, that is such pairs (x, v)
which satisfy both the equalities (44), (53), and the inclusions (48), (49), (50).
Thus, the Problem 2.5* comes to the maximization of functional Kj = on the set

W j
a . Let the sets and U be determined by relations X =

n∪
t=1

X(t), U =
n−1∪
t=0

U(t).

The following theorem is true.
Theorem 2.11. Let given known function (·) and j-th control law the set

W j
a is not empty, the sets V, X(t) and U(t − 1) are closed and bounded for all

t = 1, ..., n, function f is continuous by the first two arguments on the set X × U



Advances in Systems Science and Applications (2014) Vol.14 No.2 111

function Gj is continuous on the set V ×X and function F is continuous by the
second argument on the set . Then the Problem 2.5* has a solution.

Proof of this theorem is based on property of continuous on compact functions
to reach their maximal and minimal values. As on finite set the maximum of
function is obtained always, we get the following statement.

Theorem 2.12. When the Theorem 2.11 condition is satisfied, the Problem
2.5 has a solution.

2.1.5 Conditions for the Existence of Solution for the Variational Calculus Problem on
Synthesis of Optimal Parametric Control Law of Discrete Stochastic Dynamical System

Consider discrete stochastic controllable system like

x(t+ 1) = f(x(t), u(t), a) + ξ(t), t = 0, 1, ..., n− 1 (54)

x(0) = x0 (55)

Here x = x(t) ∈ Rm is function of the (54), (55) systems state, random vector
function of the discrete argument (vector random process); x0 is initial condition
of the system, deterministic vector; u = u(t) ∈ Rq is vector of controllable param-
eters, vector function of the discrete argument; a ∈ Rs is vector of uncontrollable
parameters, a ∈ A,A ⊂ Rs is given set. ξ = ξ(t) = (ξ1(t), ..., ξm(t)) is known
vector random process, expressing noises (as such a noise can be, for instance,
Gaussian noise); f is known vector function of its arguments.

Specify optimality criterion, which is subject to maximization for the present
problem is of the form

K = E

{
n∑

t=1

Ft(x(t))

}
(56)

Here Ft are known functions, E is mathematical expectation.
There are phase constraints on the system:

E[x(t)] ∈ X(t), t = 1, ..., n (57)

where X(t) is given set.
Determined earlier constraints on control hold in the problems considered fur-

ther:
u(t) ∈ U(t), t = 0, 1, ..., n− 1 (58)

where U(t) ∈ Rq is given set.
Define the following variational problem, called as the variational calculus

problem on synthesis of optimal parametric control law for discrete stochastic
dynamical system.

Problem 2.6. Given known vector of uncontrollable parameters a ∈ A to find
the parametric control law u, satisfying the condition (58), so that dynamical



112 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

system (54), (55) solution corresponding it satisfies the condition (57) and gives
the maximum for functional (56).

Determine the set of acceptable controls Uad for system under study in terms
of the set of such control laws u(t), satisfying the constraint (58), for which the
mathematical expectation E[x(t)] of corresponding solution of stochastic system
satisfies the inclusion (57).

The following theorem is true.
Theorem 2.13. Let in the Problem 2.6 given a ∈ A for any t = 1, ..., n random

variables ξ(t) are absolutely continuous and has zero mathematical expectations,
the sets X(t), U(t) are closed and bounded for all t, function f satisfies the Lip-
schitz condition, and functions Ft are continuous by Lipschitz. Functions f (for
u ∈ U and a ∈ A) and Ft by absolute value does not exceed some linear rela-
tively |x| functions. Then, if the set of acceptable controls Uad is not empty, the
Problem 2.6 will be soluble.

Proof. According to the Weierstrass theorem, continuous function on non-
empty bounded set reaches its maximum. Thus, it is enough to show that the
multivariable function K = K(u), determined by (56), is continuous, and the
set Uad is closed and bounded. Its non-emptiness is one of the conditions of the
theorem.

We will show that mathematical expectations of the magnitudes, entering into
phase constraint (57) are exist. Indeed, according to equation (54), we have

E[x(t+ 1)] = E[f(x(t), u(t), a)] + E[ξ(t)]

The second summand of the right-hand-side of this equality has sense under
the theorem conditions, and the first is calculated by formula

E[f(x(t), u(t), λ)] =

∫
Rn

f(ω, u(t), a)px(t)(ω)dω

if the last integral absolutely converges (here by px(t) is denoted probability den-
sity function of random variable x(t)). The latter fact is really takes place under
constraints on increase of function f and existence of mathematical expectation
of the magnitude x(t) for any t = 1, ..., n (this fact is tested using the mathemat-
ical induction method).

Existence of mathematical expectation on the right-hand-side of equality (56)
results from constraints on increase of function Ft and existence of mathematical
expectation of the variable x(t). Let convergence of vectors uk → u, uk ∈ Uad

takes place. From Equation (54) follows the equality

|xk(t+ 1)− x(t+ 1)| = |f(xk(t), uk(t), a)− f(x(t), u(t), a)|



Advances in Systems Science and Applications (2014) Vol.14 No.2 113

where uk and x are solutions of the problem (2.54), (2.55) at controls uk and u,
correspondingly. Then the following relation is true

|xk(t+ 1)− x(t+ 1)| ≤ Lf [|xk(t)− x(t)|+ |uk(t)− u(t)|]

where Lf is the Lipschitz constant of function f . By repeating analogous reason-
ing and taking into account that under the condition (55), xk(0) = x(0) we will
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)|

≤
t∑

s=0

(Lf )
s+1|uk(t− s)− u(t− s)| ≤ εk

where εk ≤ 0 under k → ∞.
By denoting the maximal of the Lipschitz constants of functions Ft by LF for

t = 1, ..., n we get the estimate

|Ft[xk(t)]− Ft[x(t)]| ≤ LF εk

After calculating mathematical expectations of both of parts of this inequality,
we get inequality

E{|Ft[xk(t)]− Ft[x(t)]|} ≤ LF εk

This implies that E{|Ft[xk(t)]−Ft[x(t)]|} → 0 and convergence of the sequence
in question E{|Ft[xk(t)] − Ft[x(t)]|}. Hence, function K = K(u) is continuous
because of (56).

Boundedness of the set Uad results from boundedness of the set U(t). Closure
of the set Uad results from continuity of the mapping Uad → X, determined by
defining of the set Uad and compactness of the set X(theorem about closure of
complete preimageof compact at continuous mapping). Now, existence of solution
for the problem under study follows from the Weierstrass theorem.

The theorem is proved.

2.1.6 Conditions for the Existence of Solution for the Variational Calculus Problem on
Choice (Among Given Finite Algorithms Set) of Optimal Parametric Control Law of
Discrete Stochastic Dynamical System

Controllable discrete dynamical system with given additive noise, described by
equations (54), (55) with phase constraints (57) is considered again in the follow-
ing parametric control problem for discrete dynamical system. In this problem
control is chosen among the set of given control laws:

uj(t) = Gj(v, x(t)), t = 0, ..., n− 1, j = 1, ..., r (59)



114 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

where Gj is known vector function of its arguments, v = (v1, ..., vl) is parameters
vector of control law Gj .

On adjustable coefficients v impose the constraints

v ∈ V (60)

where V is compact in space Rl. Moreover, it is assumed that parameters
of control law should be such that corresponding control law (59) satisfies the
condition (58), that is the following inclusion would be met

E
[
Gj(v, x

v
j (t))

]
∈ U(t) (61)

Here xvj is solution of problem (54), (55) at chosen v coefficient values, uncon-
trollable parameter a and j-th parametric control law.

Consider optimality criteria

Kj = Kj(v, a) = E

{
n∑

t=1

Ft(x
v
j (t))

}
(62)

Define the following variational problem, called as the variational calculus prob-
lem on choice of optimal parametric control law for discrete stochastic dynamical
system.

Problem 2.7. Given known vector of uncontrollable parameter a ∈ A to find
each of r control laws to find such a vector of adjustable coefficients v, so that
corresponding it solution x = xj of problem (54), (55) with control law u = uj ,
determined by formula (59), satisfies the conditions (60), (61) and gives the max-
imum for functional (62) with consequent choice of the best of found optimal
control laws, i.e. the one, to which corresponds the maximal value of optimality
criterion.

Now we get sufficient conditions for the existence of solution of the problem
2.7.

Denote by xvj solution of the system (54), (55) for chosen j-th parametric con-
trol law (59), its adjustable coefficient v and parameter α:

xvj (t+ 1) = f
(
xvj (t), Gj

(
v, xvj (t)

)
, a
)
+ ξ(t), t = 0, 1, ..., n− 1 (63)

xvj (0) = x(0) (64)

For considered problem denote the set of acceptable values of adjustable coeffi-
cients as the set V j

ad, consisting of such values v ∈ V satisfying the condition (60),
for which corresponding solution of the problem (63), (64) satisfies the inclusions

E
[
Gj(v, x

v
j (t))

]
∈ U(t), t = 0, 1, ..., n− 1 (65)



Advances in Systems Science and Applications (2014) Vol.14 No.2 115

E
[
xvj (t)

]
∈ X(t), t = 1, ..., n− 1 (66)

We will call the Problem 2.7 as nontrivial, if the set V j
ad, corresponding it, is

not empty and contains some open set for each j = 1, .., r.
Theorem2.14. Let in the Problem 2.7 a ∈ A, the sets U(t), X(t), V are

compact, functions f , Gj , Ft satisfy the Lipschitz condition. These functions
satisfy constraints on increase as well: functions |f(x,Gj(v, x), a)|, |Ft(x)| do not
exceed linear relatively |x| functions uniformly by v ∈ V . Random variable ξ(t) is
absolutely continuous and has zero mathematical expectation. Then in the case
of non-emptiness of the sets V j

ad the Problem 2.7 has a solution.
Proof. It is enough to ascertain that all of the functions (62) Kj are contin-

uous, and all of the sets V j
ad are closed and bounded, where j = 1, ..., r. The

existence of all used below mathematical expectations is proved in the same way
as when proving the Theorem 2.13.

Allowing for mathematical expectation additivity, we find the values of

Kj = Kj(v) =

n∑
t=1

E
{
Ft[x

v
j (t)]

}
whence it follows inequality for v, w ∈ V j

ad

|Kj(v)−Kj(w)| = |
n∑

t=1

E
{
Ft[x

v
j (t)]

}
−

n∑
t=1

E
{
Ft[x

w
j (t)]

}
|

≤
n∑

t=1

|E
{
Ft[x

v
j (t)]

}
− E

{
Ft[x

w
j (t)]

}
|

From relations (63), (64) follows that

|xvj (t+ 1)− xwj (t+ 1)| =
∣∣f (xvj (t), Gj

(
v, xvj (t)

)
, a
)
− f

(
xwj (t), Gj

(
w, xwj (t)

)
, a
)∣∣

≤ Lf

[
|xvj (t)− xwj (t)|+ |Gj

(
v, xvj (t)

)
−Gj

(
w, xwj (t)

)
|
]

where Lf is the Lipschitz constant of function f . After denoting by La the
maximal one of the Lipschitz constants of function Gj , we will get inequality

|xvj (t+1)−xwj (t+1)| ≤ Lf (1+La)|xvj (t)−xwj (t)|+LfLG|v−w|, t = 0, 1, ..., n−1

Taking into account equality |xvj (0)− xwj (0)| = 0, we get the estimate

|xvj (t+ 1)− xwj (t+ 1)| ≤ LfLG

t∑
l=0

[Lf (1 + LG)]
l |v − w| ≤ β|v − w|∀v, w ∈ V j

ad



116 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

where

β = LfLG

t∑
l=0

[Lf (1 + LG)]
l

After denoting by LF the maximal one of the Lipschitz constants of function
Ft, we will have∣∣F [xvj (t)]− F [xwj (t)]

∣∣ ≤ Lf

∣∣xvj (t)− xwj (t)
∣∣ ≤ LFβ|v − w|∀v, w ∈ V j

ad

Thus, in the case of sufficient smallness of difference of adjustable coefficients v
and w the values of xvj (t) and x

w
j (t) (and the same F [xvj (t)] and F [x

w
j (t)]) will be

arbitrary close to each other. Determine converged sequence w = vk → v. Then,
after finding mathematical expectations of LHS and RHS of the last inequality,
we will get the following inequality:

E
[∣∣F [xvj (t)]− F [xwj (t)]

∣∣] ≤ LFβ|v − w|

Hence results the following convergence

E
{
Ft

[
xvkj (t)

]}
→ E

{
Ft

[
xwk
j (t)

]}
from which follows continuity of function Kv

j .

As V j
ad ⊂ V , then all of the sets V j

ad are bounded. Closure of the sets V
j
ad results

from closure of the sets U(t), X(t + 1), of proved above continuity of mappings
v → E[xvj (t + 1)], v → E[Gj(v, x

v
j (t))] and determining the set V j

ad as complete
preimageof pointed sets at continuous mappings. The theorem statement follows
from the Weierstrass theorem about reaching continuous on compact function of
its upper boundary.

2.2 Sufficient Conditions for Continuous Dependence of Optimal Criteria Values of the
Parametric Control Problems on Uncontrollable Functions

In this section, within the framework of developing the 4th component of the
parametric control theory, there are derived sufficient conditions of continuous
dependence on uncontrollable functions a(·) of optimal criteria values for all of
considered above parametric control problems for non-autonomous deterministic
dynamical systems and sufficient conditions of continuous dependence on uncon-
trollable parameters of optimal criteria values for considered above parametric
control problems for autonomous stochastic dynamical systems. All of the results
defined for non-autonomous systems remain true for autonomous dynamical sys-
tems as well, where exogenous vector function a(·) is taken as constant.

The following theorem determines sufficient conditions of continuous depen-
dence of optimal criterion values for the Problem 2.4.

Theorem 2.15. Assume that when conditions of the Theorem 2.10 are met,



Advances in Systems Science and Applications (2014) Vol.14 No.2 117

in neighborhood (in Euclidean topology) of function a(·) function f is continuous
by the third argument and satisfies the Lipschitz condition by the first argument
on uniformly by the second and third arguments.Then optimal criterion value
for the Problem 2.4 continuously depends on uncontrollable function at the point
a(·).

Proof. Let the following convergence occurs

ak → aBRsn (67)

According to the Theorem 2.10 the Problem 2.4 given the values of uncontrollable
functions and ak has a solutions, which we denote, correspondingly, by u and
uk. Denote by x[b, w] solution of condition equation (43), (44), corresponding to
uncontrollable function b and control w. Ipso facto the function x[b, w] satisfies
relations

x [b, w] (t+ 1) = f (x [b, w] (t), w(t), b(t)) , t = 0, 1, ..., n− 1 (68)

x [b, w] (0) = x0 (69)

Then the following inequalities are true

0 ≥ K (x [a, uk])−K (x [a, u]) ,K (x [ak, uk])−K (x [ak, u]) ≥ 0

where by k(Y ) is denoted the value of criterion K at given value of system state
. Consequently, we get the following relations:

0 ≤ K (x [a, u])−K (x [a, uk]) ≤ K {(x [a, u])−K (x [ak, u])}+
{(x [ak, u])− (x [k, uk])}+ {K(x [ak, uk])−K (x [a, uk])}
≤ 2 sup

w∈U
|K(x [a,w])−K (x [ak, w])|

(70)

From the conditions (68) we derive inequalities

|x [ak, w] (t+ 1)− x [a,w] (t+ 1)|
= |f (x [ak, w] (t), w(t), ak(t))− f (x [a,w] (t), w(t), (t))|
≤ |f (x [ak, w] (t), w(t), ak(t))− f (x [ak, w] (t), w(t), (t))|
+ |f (x [ak, w] (t), w(t), a(t))− f (x [a,w] (t), w(t), a(t))|
≤ sup

∈X,φ∈U
|f (y, φ, ak(t))− f (y, φ, a(t))|+ L |x [ak, w] (t)− x [a,w] (t)| ,

t = 0, 1, ..., n− 1,

where L is the Lipschitz constant of function f by the first argument, which does
not depend on w. Consequently, we get the following relation

|ψk(t+ 1)| ≤ ηk + L |ψk(t)| , t = 0, 1, ..., n− 1



118 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

where
ψk(t) = x [ak, w] (t)− x [a,w] (t), t = 0, 1, ..., n− 1,
ηk = max

t=0,1,...,n−1
sup

∈X,φ∈U
|f (y, φ, ak(t))− f (y, φ, a(t))|

Then we determine relations

|ψk(t+ 1)| ≤ ηk + L|ψk(t)| ≤ ηk + L(ηk + L|ψk(t− 1)|)
= (1 + L)ηk + L2|ψk(t− 1)| ≤ (1 + L)ηk + L2(ηk + L|ψk(t− 2)|)

= (1 + L+ L2)ηk + L3|ψk(t− 2)| ≤ ... ≤
t∑

r=0

Lrηk + Lt+1|ψk(0)|, t = 0, 1, ..., n− 1

From initial condition (69) follows that |ψk(0)| = 0. Then the following estimate
is true

|ψk(t+ 1)| ≤ ηk

t∑
r=0

Lr, t = 0, 1, ..., n− 1

from which follows that

|x[ak, w](t+ 1)− x[a,w](t+ 1)| ≤ ηk

t∑
r=0

Lr, t = 0, 1, ..., n− 1

Note that right-hand-side of this inequality does not depend on w.
From the condition (67) follows the convergence ηk → 0 under continuity of

function f by the third argument, as well as closure and boundedness of the sets
X and U . Then we derive from the last inequality that

x [ak, w] (t) → x [a,w] (t) uniformly by w, t = 1, ..., n− 1.

Hence under continuity of function F follows that

F [t, x [ak, w] (t)] → F [t, x [a,w] (t)] uniformly by w, t = 1, ..., n− 1.a

therefore,

K(x[ak, w]) → K(x[a,w]) uniformly by w, t = 1, ..., n− 1.a

whence the convergence

sup
w∈U

|K(x [a,w])−K (x [ak, w])| → 0 (71)

follows.
Taking the limits in inequality (70) and taking into account the conditions (58),
we will have

K (x [a, uk]) → K (x [a, u]) (72)



Advances in Systems Science and Applications (2014) Vol.14 No.2 119

The following estimate is true.

|K(x [ak, uk])− (Kx [a, u])| ≤ |K(x [ak, uk])− (Kx [a, uk])|
+ |(Kx [a, uk])− (Kx [a, u])|

Taking the limits and taking into account the conditions (71), (72), we will de-
termine the convergence

K (x [ak, uk]) → K (x [a, u])

Hence under defining controls u and uk results that the maximal value of crite-
rion K, corresponding to uncontrollable function ak, converges to its maximum,
corresponding to the extreme value of uncontrollable function . The theorem is
proved.

Determine now sufficient conditions of continuous dependence on uncontrol-
lable functions of optimal criterion values for the variational calculus problem 2.5
on choice (among given finite algorithms set) of optimal parametric control laws
based on discrete non-autonomous dynamical system.

Determine first corresponding result for the Problem 2.5*.
Theorem 2.16. Assume that when conditions of the Theorem 2.11 are met in

neighborhood of point function f is continuous by the third argument and satis-
fies the Lipschitz condition by the first two arguments on X×U uniformly by the
third argument, and function Gj satisfies the Lipschitz condition by the second
argument on uniformly by the first argument.Then optimal criterion value for
the Problem 2.5* continuously depends on uncontrollable function at point .

Proof. Let the convergence occurs

ak → aBRsn (73)

According to the Theorem 2.11 the Problem 2.5* at values of uncontrollable
functions and ak has a solution, which we denote, correspondingly, by v and
vk. Denote by x[b, w] solution of state equations (53), (44), corresponding to
uncontrollable function b(·) and controllable parameter w. Ipso facto function
x[b, w] meets relations

x [b, w] (t+ 1) = f
(
x [b, w] (t), Gj (v, x [b, w] (t)) , b(t)

)
, t = 0, 1, ..., n− 1 (74)

x [b, w] (0) = x0 (75)

Analogously with inequality (70) the following relation is determined

0 ≤ K (x [a, v])−K (x [a, vk]) ≤ 2 sup
w∈V

|K(x [a,w])−K (x [ak, w])| (76)



120 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

After denoting
a = x [a,w] , ak = x [ak, w]

from equalities (74) we derive inequalities

|xk(t+ 1)− x(t+ 1)|
=

∣∣f (xk(t), Gj (w, xk(t)) , ak(t)
)
− f

(
x(t), Gj (w, x(t)) , a(t)

)∣∣
≤

∣∣f (xk(t), Gj (w, xk(t)) , ak(t)
)
− f

(
xk(t), Gj (w, xk(t)) , a(t)

)∣∣
+
∣∣f (xk(t), Gj (w, xk(t)) , a(t)

)
− f

(
x(t), Gj (w, x(t)) , a(t)

)∣∣
≤ sup

y∈X,φ∈U
|f (y, φ, ak(t))− f (y, φ, a(t))|

+ L
[
|xk(t)− x(t)|+

∣∣Gj (w, xk(t))−Gj (w, x(t))
∣∣]

≤ sup
y∈X,φ∈U

|f (y, φ, ak(t))− f (y, φ, a(t))|

+ L(1 +M) |xk(t)− x(t)| , t = 0, 1, ..., n− 1,

where L is the Lipschitz constant of function f by the first two arguments, and
is the Lipschitz constant of function Gj by the second argument. Consequently,
we get inequality

|ψk(t+ 1)| ≤ ηk +N |ψk(t)| , t = 0, 1, ..., n− 1

where
N = L(1 +M)

ψk(t) = xk(t)− x(t), t = 0, 1, ..., n− 1,
ηk = max

t=0,1,...,n−1
sup

y∈X,φ∈U
|f (y, φ, ak(t))− f (y, φ, a(t))|

Using technique, described above in proof of the Theorem 2.15, we determine the
estimate

|ψk(t+ 1)| ≤ ηk

n∑
r=0

N r, t = 0, 1, ..., n− 1

and the convergence ηk → 0. Thereby, |ψk(t)| → 0, t = 1, ..., n therefore,

x [ak, w] (t) → x [a,w] (t) uniformly by w, t = 1, ..., n− 1.

Hence under continuity of function F follows that

F [t, x [ak, w] (t)] → F [t, x [a,w] (t)] uniformly by w, t = 1, ..., n− 1.

therefore,
K (x [ak, w]) → K (x [a,w]) uniformly by w.



Advances in Systems Science and Applications (2014) Vol.14 No.2 121

Then under finite dimensionality of the space of controls and boundedness of the
set U we get

sup
w∈V

|K(x [a,w])−K (x [ak, w])| → 0 (77)

Taking the limits in inequality (76) and taking into account the condition (77),
we will have

K (x [a, vk]) → K (x [a, v]) (78)

The following estimate is true

|K(x [ak, vk])−K (x [a, v])| ≤ |K(x [ak, vk])−K (x [a, vk])|
+ |K(x [a, vk])−K (x [a, v])|

Taking the limits here and taking into account the conditions (77), (78), we
determine the convergence

K (x [ak, vk]) → K (x [a, v])

Hence under definition of control parameters v and vk follows that the maximal
value of criterion K, corresponding to uncontrollable function ak, converges to
its maximum, corresponding to extreme value of uncontrollable function. The
theorem is proved.

As the maximum of two (therefore, any finite number as well) of continuous
functions is continuous, then from the Theorem 2.16 follows similar result for the
Problem 2.5.

Theorem 2.17. Assume that when conditions of the Theorem 2.11 are met
in neighborhood of point , function f is continuous by the third argument and
satisfies the Lipschitz condition by the first two arguments on X × U uniformly
by the third argument, and function Gj satisfies the Lipschitz condition by the
second argument on uniformly by the first argument.Then optimal criterion val-
ue for the Problem 2.5 continuously depends on uncontrollable function at point
a.

The purpose of the next study is to determine sufficient conditions of contin-
uous dependence on uncontrollable functions of optimal criterion values for the
Problem 2.1 based on continuous non-autonomous dynamical system.

Theorem 2.18. Assume that when conditions of the Theorem 2.6 are met
in neighborhoods of point , function f is continuous by the second argument and
satisfies the Lipschitz condition by the first and third arguments on X × A uni-
formly by the second argument.Then optimal criterion value for the Problem 2.1
continuously depends on uncontrollable function at point .

Proof. Let the convergence occurs

ak → a (C[0, T ])s (79)



122 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

According the Theorem 2.6 the Problem 2.1at values of uncontrollable functions
and ak has solutions, which we denote, correspondingly, by u and uk. Denote
by x[b, w] solution of state equations (1), (2), corresponding to uncontrollable
function b and control w. Ipso facto function x[b, w] meets relations

ẋ [b, w] (t) = f (x[b, w] , w(t), b(t)) , t ∈ (0, T ) (80)

x [b, w] (0) = x0. (81)

Then, by repeating reasoning from proof of the Theorem 2.13, analogously with
relation (70) we determine inequality

0 ≤ K (x [a, u])−K (x [a, uk]) ≤ 2 sup
w∈U

|K(x [a,w])−K (x [ak, w])| (82)

After denoting
x = x [a,w] , xk = x [ak, w]

from the conditions (80) we derive inequalities

ẋk(t)− ẋ(t) = f (xk(t), w(t), ak(t))− f (x(t), w(t), a(t)) , t ∈ (0, T )

By integrating by t and taking into account equalities (81), we get

|xk(t)− x(t)| ≤
∫ t
0 |f (xk(τ), w(τ), ak(τ))− f (x(τ), w(τ), (τ))| dτ

≤ L
∫ t
0 |xk(τ)− x(τ)| dτ+L

∫ t
0 |ak(τ)− (τ)| dτ

≤ L
∫ t
0 |xk(τ)− x(τ)| dτ+LT∥ak−∥Θ, t ∈ (0, T )

where L is the Lipschitz constant of function f , not depending on w. Using the
Gronwall lemma, we will have

|xk(t)− x(t)| ≤ ∥ak − a∥Θ, t ∈ (0, T )

where positive constant depends only on L and on .
Using the condition (79), we get that

xk(t) → x(t), t ∈ (0, T )

and therefore,

x [ak, w] (t) → x [a,w] (t) uniformly by w, t ∈ (0, T )

Hence under continuity of function F follows the convergence

K (x [ak, w]) → K (x [a,w]) uniformly by w



Advances in Systems Science and Applications (2014) Vol.14 No.2 123

Consequently, we get

sup
w∈U

|K(x [a,w])−K (x [ak, w])| → 0 (83)

Taking the limits in inequality (82) and taking into account the conditions (83),
we will have

K (x [a, uk]) → K (x [a, u]) (84)

The following estimate is true

|K(x [ak, uk])−K (x [a, u])|
≤ |K(x [ak, uk])−K (x [a, uk])|+ |K(x [a, uk])−K (x [a, u])|

Taking the limits here and taking into account the conditions (83), (84), we
will get the convergence

K (x [ak, uk]) → K (x [a, u])

Hence under definition of controls u and uk it follows that the maximal value of
criterion K, corresponding to uncontrollable function ak, converges to its maxi-
mum, corresponding to extreme value of uncontrollable function. The theorem is
proved.

Determine sufficient conditions of continuous dependence on uncontrollable
functions of optimal criterion values for the variational calculus problem on choice
(among given finite algorithms set) of optimal parametric control laws based on
continuous non-autonomous dynamical system.

Theorem 2.19. Assume that when conditions of the Theorem 2.8 are met
in neighborhood of point , function f satisfies the Lipschitz condition on the set
X×U ×A, and function satisfies the Lipschitz condition by the second argument
on uniformly by the first argument.Then optimal criterion value for the Problem
2.3* continuously depends on uncontrollable function at point .

Proof. Let the convergence occurs

ak → aB(C[0, T ])s (85)

According the Theorem 2.8 the Problem 2.3* at values of uncontrollable functions
and ak has a solutions, which we denote, correspondingly, by v and vk. Denote
by x[b, w] solution of state equations (23), (2), corresponding to uncontrollable
function b and control w. Ipso facto function x[b, w] meets relations

ẋ [b, w] (t) = f
(
x [b, w] (t), Gj (w, x(t)) , b(t)

)
, t ∈ (0, T ) (86)

x [b, w] (0) = x0 (87)



124 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

Then, by repeating reasoning from proof of the Theorem 2.15, analogously with
relation (70) we determine inequality

0 ≤ K (x [a, u])−K (x [a, uk]) ≤ 2 sup
w∈V

|K(x [a,w])−K (x [ak, w])| (88)

After denoting

x = x [a,w] , xk = x [ak, w]

from the conditions (86) we derive inequalities

ẋk(t)−ẋ(t) = f
(
xk(t), Gj (w, xk(t)) , ak(t)

)
−f

(
x(t), Gj (w, x(t)) , a(t)

)
, t ∈ (0, T )

By integrating by t and taking into account equalities (87), we get

|xk(t)− x(t)| ≤
∫ t
0

∣∣∣f (xk(τ), Gj (w, xk(τ)) , ak(τ)
)
− f

(
x(τ), Gj (w, x(τ)) , a(τ)

)∣∣∣
dτ ≤ L

∫ t
0

[
|xk(τ)− x(τ)|+

∣∣∣Gj (w, xk(τ))−Gj (w, x(τ))
∣∣∣+ |ak(τ)− a(τ)|

]
dτ

≤ L(1+)
∫ t
0 |xk(τ)− x(τ)| dτ+LT∥ak − a∥Θ, t ∈ (0, T ),

where L is the Lipschitz constant of function f , and is the Lipschitz constant of
function by the second argument. Using the Gronwall lemma, we will have

|xk(t)− x(t)| ≤ ∥ak − a∥Θ, t ∈ (0, T )

where positive constant depends only on L and on .
Using the condition (85), we get that

x [ak, w] (t) → x [a,w] (t) uniformly by w, t ∈ (0, T )

Hence under continuity of function F follows that

F [t, x [ak, w] (t)] → F [t, x [a,w] (t)] uniformly by w

therefore,

K (x [ak, w]) → K (x [a,w]) uniformly by w

Consequently, we get

sup
w∈V

|K(x [a,w])−K (x [ak, w])| → 0 (89)

Taking the limits in inequality (88) and taking into account the condition (89),
we will have

K (x [a, uk]) → K (x [a, u]) (90)



Advances in Systems Science and Applications (2014) Vol.14 No.2 125

The following estimate is true

|K(x [ak, uk])−K (x [a, u])|
≤ |K(x [ak, uk])−K (x [a, uk])|+ |K(x [a, uk])−K (x [a, u])|

Taking the limits here and taking into account the conditions (89), (90) we will
get the convergence

K (x [ak, uk]) → K (x [a, u])

Hence under definition of controls u and uk follows that the maximal value of
criterion K, corresponding to uncontrollable function ak, converges to its maxi-
mum, corresponding to extreme value . The theorem is proved.

Using the Theorem 2.19 analogously with the Theorem 2.17 (in this case sys-
tem continuity is not a matter of principle), we come to the following statement.

Theorem 2.20. When for all j = 1, ..., r conditions of the Theorem 2.19 are
met, optimal criterion value for the Problem 2.5 continuously depends on uncon-
trollable function at point .

Present statements of the theorems about continuous dependence on uncontrol-
lable parameter of optimal criterion values for the parametric control problems
of stochastic dynamical systems. Proves of these theorems are similar to those
presented above.

Theorem 2.21. Let given any a ∈ A conditions of the Theorem 2.13 are met.
Then optimal criterion value for the Problem 2.6 are continuous function of the
parameter .

Now, study the conditions of continuous dependence of optimal criterion val-
ue for the variational calculus problems on choice of parametric control laws on
uncontrollable parameters.

Theorem 2.22. Let given any a ∈ A conditions of the Theorem 2.14 are met.
Then for chosen number value of the law j Kj optimal criterion values for the
Problem 2.7 are continuous function of the parameter .

Consequence 2.23. When conditions of the Theorem 2.22 are met for all
j = 1, ..., r, optimal criterion value K = maxj=1,...,rKj for the Problem 2.7 are
continuous functions of the parameter a ∈ A.

Obtained results will be used in the next section in proving the existence of
corresponding bifurcation points of extremals of the variational problems.

2.3 Sufficient Conditions for the Existence of Extremals’ Bifurcation Points of the Prob-
lems on Choice of Optimal Parametric Control Laws

Let us introduce a notion of extremals’ bifurcation point of the variational calcu-
lus problems on choice (among given finite algorithms set) of optimal parametric
control laws.Existence of such a bifurcation point for some uncontrollable function
a(·) means that in its neighborhood for the problem in question occurs transfer
from one optimal parametric control law to another.



126 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

Consider abstract variational calculus problem on choice (among given finite
algorithms set) of optimal parametric control laws, generalizing the Problems 2.3,
2.5, and 2.7.

Given: the set of uncontrollable functions (or parameters), set of the acceptable
controls sets (adjustable coefficients values) V j

a , a ∈ A, j = 1, ..., r and set of func-
tionals (optimality criteria) Kj = Kj(a, v) where v ∈ V j

a , a ∈ A, j = 1, ..., r. Give
definition for extremals’ bifurcation point for set of the maximization problems
for given functionals on corresponding sets of acceptable controls, i.e. abstract
variational calculus problem on choice (among given finite algorithms set) of op-
timal parametric control laws.

Definition. The value a ∈ A call asextremals’ bifurcation point for the maxi-
mization problem for mappings u→ Kk(a, v) on the sets V k

a , k = 1, ..., r if there
are exist two different numbers i, j ∈ 1, ..., r such that the following relation is
true

max
v∈V l

a

Ki(a, v) = max
v∈V j

a

Ki(a, v) = max
k=1,...,r

max
v∈V k

a

Ki(a, v)

and in any neighborhood of point there is such a point b ∈ A for which the
value

max
k=1,...,r

max
v∈V j

b

Kj(b, v)

reachs for the only value of k.
Theorem 2.24. Assume that when conditions of the Theorem 2.17(or 2.20,

or 2.23) are met, is a connected set, and there are two different values a0, a1 ∈ A,
such that the maximal values by j = 1, ..., r of function maxima v → Kj(a, v) on
the sets are reached for the different values j0, j1 that is the following inequalities
are true:

max
j=1,...,r,j ̸=j0

max
v∈V j

a0

Kj(a0, v) < max
v∈V j0

a0

Kj0(a0, v)

max
j=1,...,r,j ̸=j1

max
v∈V j

a1

Kj(a1, v) < max
v∈V j1

a1

Kj1(a1, v)

Then there exists a bifurcation point of extremals of the Problem 2.3 (or 2.5, or
2.7) on choice (among given finite algorithms set) of optimal parametric control
law.

Proof. Under connectivity of the set points a0, a1 can be connected by con-
tinuous line a = a(s), s ∈ [0, 1], lying in the set , and the following equalities are
true

a(0) = a0, a(1) = a1

Denote

Kj(s) = max
v∈V j

a(s)

Kj1(a(s), v), s ∈ [0, 1]



Advances in Systems Science and Applications (2014) Vol.14 No.2 127

From the Theorems 2.17 (or 2.20, or2.23) follows continuity of functions s →
Kj(s) in the segment [0, 1], and therefore, continuity in this segment of function
as well, where

Determine the set

∆(j) = {s ∈ [0, 1]|Kj(s) = K∗(s)} , j = 1, ..., r

It is closed, being complete preimageof closed set, consisting of the only point
(zero) for continuous function y = y(s) where y(s) = Kj(s) − K ∗ (s) Thereby,
we present the segment [0, 1] in terms of the following sum

[0, 1] =
∪

j=1,...,r

∆(j)

consisting, according to the theorem conditions, as minimum, of two non-empty
closed sets.

From theorem conditions it follows the following relations as well:

0 ∈ ∆(j0), 1 /∈ ∆(j0)

Then the set of boundary points of the set ∆(j0) which are situated in the
interval (0, 1), is not empty. Consequently, there exists lower boundary s0 of
such boundary points. This value is a boundary point of some another set ∆(j2)
and is a part of it as well. Thereby, at a = a(s0) the maximum by j = 1, ..., r the
values is reached, as minimum, for two different numbers j0 and j2. At the same
time, at 0 ≤ s ≤ s0 this maximum is reached for the only value j0. Thus, a(s0)
actually corresponds to desired bifurcation point. The theorem is proved.

The following statement is direct consequence of the Theorem 2.24.
Consequence 2.25. Assume that when conditions of the Theorem 2.17 (or

2.20, or 2.23) are met, is connected set, and at value control using the law
provides solution of the Problem 2.3 (or 2.5, or 2.7), and at , ( control using this
law does not provide solution of the problem in question, that is the following
inequalities are true

max
j=1,...,r,j ̸=j0

max
v∈V j

a0

Kj(a0, v) < max
v∈V j0

a0

Kj0(a0, v)

max
j=1,...,r,j ̸=j1

max
v∈V j

a1

Kj(a1, v) > max
v∈V j0

a1

Kj0(a1, v)

Then there is at least one bifurcation point of extremals of mentioned problem
2.3 (or 2.5, or 2.7).

At the end we present description of the numerical algorithm for finding bifur-
cational value of function (or parameter) a of one of the problems 2.3 or 2.5 or



128 A. Ashimov: The Theory of Parametric Control of Macroeconomic Systems and ...

2.7 on choice (among given finite algorithms set) of parametric control laws and
when conditions of the theorem 2.24 are met.

Connect the points and by smooth curve. Divide this curve to equal parts with
sufficiently small step. For obtained values (points) are determined the numbers
of parametric control laws- bringing solution of the problem 2.3 or 2.5 or 2.7 given
values. Then we find the first value i, at which corresponding law number differs
from. In this case bifurcational value lies on arc of the curve.

For found part of the curve, the algorithm for determining bifurcation point
with given accuracy consists in using the method of bisections. Consequently
we find the point, on the one hand from which on this arc within the range of
deviation from the value the optimal law is, and on the other hand-within the
range of deviation from the value this law is not optimal. From the Consequence
2.25 follows that extremal’s bifurcation point for the problem under solution lies
on mentioned arc, and as its estimate can be taken any point of the arE.

References

[1] Ekeland I., Temam R. (1976), “Convex analysis and variational problems”,
Amsterdam: Nord-Holland publishing company.

Corresponding author
A. Ashimov can be contacted at: ashimov37@mail.ru