International Journal of Control Science and Engineering

p-ISSN: 2168-4952    e-ISSN: 2168-4960

2013;  3(2): 68-72

doi:10.5923/j.control.20130302.05

Full Averaging of Control Fuzzy Integrodifferential Inclusions with Terminal Criterion of Quality

Andrej V. Plotnikov1, 2, Tatyana A. Komleva3

1Department of Applied Mathematics, OdessaStateAcademy of Civil Engineering and Architecture, Odessa, 65029, Ukraine

2Department of Optimal Control and Economic Cybernetics, Odessa National University, Odessa, 65026, Ukraine

3Department of Mathematics, OdessaStateAcademy of Civil Engineering and Architecture,Odessa, 65029, Ukraine

Correspondence to: Andrej V. Plotnikov, Department of Applied Mathematics, OdessaStateAcademy of Civil Engineering and Architecture, Odessa, 65029, Ukraine.

Email:

Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

In this paper we consider the fuzzy control system with fuzzy terminalqualitycriterion (Mayer's fuzzy problem) when the behaviour of system is described by thefuzzy controlled integrodifferential inclusion with a small parameter. We use an averaging method of Krylov-Bogolyubov and set in correspondence to the given problem the Mayer's full-averaged fuzzy problem that is more simple for solving. In paper we receive the conditions when the optimal fuzzy solutions of these problems are close.

Keywords: Fuzzy Control Integrodifferential Inclusion, Averaging Method, Mayer Fuzzy Problem

Cite this paper: Andrej V. Plotnikov, Tatyana A. Komleva, Full Averaging of Control Fuzzy Integrodifferential Inclusions with Terminal Criterion of Quality, International Journal of Control Science and Engineering, Vol. 3 No. 2, 2013, pp. 68-72. doi: 10.5923/j.control.20130302.05.

1. Introduction

Many important problems of analytical dynamics are described by the nonlinear mathematical models that as a rule are presented by the nonlinear differential or the integrodifferential equations. The absence of exact universal research methods for nonlinear systems has caused the development of numerous approximate analytic and numerically-analytic methods that can be realized in effective computer algorithms.
The averaging methods combined with the asymptotic representations (in Poincare sense) began to be applied as the basic constructive tool for solving the complicated problems of analytical dynamics described by the differential equations. Averaging theory for ordinary differential equations has a rich history, dating to back to the work of N.M. Krylov and N.N. Bogoliubov[1], and has been used extensively in engineering applications[2-6]. Books that cover averaging theory for differential equations and inclusions include[7-11].
In recent years, the fuzzy set theory introduced by Zadeh[12] has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of fuzzy set theory can be found in many branches of science as physical, mathematical, differential equations and engineering sciences. Recently there have been new advances in the theory control fuzzy integrodifferential equations[13-17] and control fuzzy integrodifferential inclusions[18].
In work[19] full schemes of an average for the fuzzy integrodifferential inclusions have been considered. In this article we prove the substantiation of the method of full averaging for the control fuzzy integrodifferential inclusions with small parameter and terminal criterion of quality (Mayer fuzzy problem). Thereby we expand a circle of systems to which it is possible to apply Krylov-Bogolyubov method of averaging is a template.

2. Preliminaries

Let be a set of all nonempty (convex) compact subsets from the space ,
beHausdorff distance between sets and , is -neighborhood of set .
Let be the set of all such that u satisfies the following conditions:
1) is normal, that is, there exists an such that ;
2) is fuzzy convex, that is,
for any and ;
3) is upper semicontinuous;
4) is compact.
If , then is called a fuzzy number, and is said to be a fuzzy number space. For , denote .
Then from 1)-4), it follows that the -level set for all .
Let be the fuzzy mapping defined by if and.
Define by the relation
,
where is the Hausdorff metric defined in . Then is a metric in .
Further we know that[20]:
1) is a complete metric space,
2) for all ,
3) for all and .
Definition 1.[21] A fuzzy mapping is measurable if for all the set-valued map defined by is Lebesgue measurable.
Definition 2.[21] A fuzzy mapping is said to be integrably bounded if there is an integrable function such that for every .
Definition 3.[21] The integralofafuzzy mapping
isdefinedlevelwiseby
.The setof allsuch that
is a measurable selection for for all .
Definition 4.[21] A measurable and integrably bounded fuzzy mapping is said to be integrable over
if .
Note that if is measurable and integrably bounded, then is integrable. Further if is continuous, then it is integrable.
Now we consider following fuzzy integrodifferential inclusion
(1)
wheremeans; is the state; ;
is a fuzzy mapping; is a fuzzy mapping; .
We interpret[22-25] the fuzzy integrodifferential inclusion (1) as a family of integrodifferential inclusions
(2)
where the subscript indicates that the -level set of a fuzzy set is involved (the system (2) can only have any significance as a replacement for (1) if the solutions generate fuzzy sets (fuzzy R-solution)[25]).
Let denotes the fuzzy R-solution of the fuzzy integrodifferential inclusion (1).
Now we consider following control integrodifferential equations with the fuzzy parameter
(3)
where is the control; are fuzzy parameters; , , .
Definition 5. The set of all measurable single-valued branches of the set is the set of the admissible controls.
Further we consider following control fuzzy integrodifferential inclusions
(4)
where, , are fuzzy maps such that , , .
Obviously, the control fuzzy integrodifferential inclusion (4) turns into the ordinary fuzzy integrodifferential inclusion
(5)
if the control is fixed and .
Let denotes the fuzzy R-solution of the fuzzy integrodifferential inclusion (5), then denotes the fuzzy R-solution of the control fuzzy integrodifferential inclusion (4) for the fixed .
Definition 6.The set be called the attainable set of the fuzzy system (4).

3. Main Result

In this section we consider the fuzzy control problem with small parameter
(6)
where is a small parameter.
In this section we associate with the equation (6) the following averaged integrodifferential equation
(7)
where
(8)
(9)
(10)
Remark. In this paper we will consider a case when the limits (8), (9) and (10) exist.
We will set correspondence between the control of the initial inclusion (6) and by the control of averaged inclusion (7).
We will put the control in correspondence a control as it follows:
1) We calculate
,
where is constant.
2) Now we build the control as it follows:
,
where
.
We will put the control in correspondence a control as it follows:
1) We calculate
,
where is constant.
2) Now we build the control as it follows:
,where
Now we consider Mayer fuzzy problem.As is generally known, the Mayer problem, is to maximize, over all solutions to control system defined on fixed time interval, a functional depending on the final position[26,27].
Mayer fuzzy problem: let be a continuous fuzzy map and . For example such that forall,where
, is constant,
.
Definition 6.An admissible control is said to be an optimal control for problem (6) if
(11)
for all admissible control , where , .
Now we obtain the main result of this article.
Theorem. Let in domain
the following hold:
1) is continuous in ;
2) is continuous in ;
3) there exist constants such that
for all , and any ;
4) there exist continuous functions , , and constant such that
,
for any , and any ;
5) there exist constants , and such that
,
,
for any ;
6) there exist continuous functions , , and constants , , such that
, ,
, ,
,
for any and ;
7) the limits (8),(9) exist uniformly in ;
8) is continuous in ;
9) there exists constant such that
for all ;
10) the limit (10) exists;
11) for any and the fuzzy R-solution of the system (7) together with a -neighbourhood belong to the domain , i.e. ;
12) there exists a constant such that for all ;
Then for any and there exists such that for all and the following statements hold:
a) for an optimal control of Mayer fuzzy problem (6) there exists an admissible control of the fuzzy system (7) such that
(12)
b) for an optimal control of Mayer fuzzy problem (7) there exists an admissible control of the fuzzy system (6) such that
(13)
c)
(14)
where are an optimal control of Mayer fuzzy problems (6) and (7);
Proof. Let and are optimal controls of Mayer fuzzy problems (6) and (7).
Let and are fuzzy R-solutions of following fuzzy systems
,
.
Let is an admissible control of the fuzzy system (7) corresponding to the optimal control of , and is an admissible control of the fuzzy system (6) corresponding to the optimal control of .
Let and are fuzzy R-solutions of following fuzzy systems
,
.
By conditions 1)-12) and[44] so that
for all .
Then we get
(15)
(16)
Let then we obtain (15) and (16).
Since are optimal controls of Mayer fuzzy problems (6) and (7), and areadmissible controls of the fuzzy systems (6) and (7), we have
,
.
Also, we obviously have
or
Hence, we obtain (12) - (14). The theorem is proved.
Remark.If we replace (11) on
or
or
,
where , , then a theorem will be just.

4. Conclusions

In this article we prove the substantiation of the method of full averaging for the control fuzzy integrodifferential inclusions with small parameter and terminal criterion of quality (Mayer fuzzy problem). This result generalize the results of A.V. Plotnikov[28, 29] for the control ordinary differential inclusions with small parameter and terminal criterion of quality.

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