## The Time-optimal Problems for the Fuzzy R-solution of the Control Linear Fuzzy Integro-Differential Inclusions

**Andrej V Plotnikov**^{1,}, **Tatyana A. Komleva**^{2}

^{1}Department of Applied Mathematics, Odessa State Academy Civil Engineering and Architecture, Odessa, Ukraine

^{2}Department of Mathematics, Odessa State Academy Civil Engineering and Architecture, Odessa, Ukraine

### Abstract

In this paper, we show some properties of the fuzzy R-solution of the control linear fuzzy integro-differential inclusions and consider the time-optimal problems for it. For such problems we receive necessary conditions of optimality.

**Keywords:** fuzzy integro-differential inclusions, control problems, time-optimal problems, fuzzy R-solution

*American Journal of Modeling and Optimization*, 2013 1 (2),
pp 6-11.

DOI: 10.12691/ajmo-1-2-1

Received December 29, 2012; Revised May 17, 2013; Accepted May 18, 2013

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Plotnikov, Andrej V, and Tatyana A. Komleva. "The Time-optimal Problems for the Fuzzy R-solution of the Control Linear Fuzzy Integro-Differential Inclusions."
*American Journal of Modeling and Optimization*1.2 (2013): 6-11.

- Plotnikov, A. V. , & Komleva, T. A. (2013). The Time-optimal Problems for the Fuzzy R-solution of the Control Linear Fuzzy Integro-Differential Inclusions.
*American Journal of Modeling and Optimization*,*1*(2), 6-11.

- Plotnikov, Andrej V, and Tatyana A. Komleva. "The Time-optimal Problems for the Fuzzy R-solution of the Control Linear Fuzzy Integro-Differential Inclusions."
*American Journal of Modeling and Optimization*1, no. 2 (2013): 6-11.

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### 1. Introduction

In the last decades, a number of works devoted to problems of optimal control of set-valued trajectories (fuzzy trajectories, trajectory bundles or an ensemble of trajectories) appeared. These works fall into a subdivision of the optimal control theory, namely, the theory of process control under uncertainty and fuzzy conditions. This is caused by the fact that, in actual problems arising in economy and engineering in the course of construction of a mathematical model, it is practically impossible to exactly describe the behavior of an object. This is explained by the following fact. First, for some parameters of the object, it impossible to specify exact values and laws of their change, but it is possible to determine the domain of these changes. Second, for the sake of simplicity of the mathematical model being constructed, the equations that describe the behavior of the object are simplified and one should estimate the consequences of such a simplification. Therefore, if is possible to divide the articles devoted to this direction into two types characterized by the following distinctive features:

1) there exists an incomplete or fuzzy information on the initial data [1-12]^{[1]};

2) the equations describing the behavior of the object to be controlled are assumed to be inexact, for example, they can contain some parameters whose exact values and laws of variation are unknown but the domain of their values is fuzzy [12-26]^{[12]}.

In the second case, fuzzy differential inclusions are frequently used to describe behavior of objects. The reason is that, first this approach is most obvious and, second, theory of fuzzy and ordinary differential inclusions is well found and is rapidly developed at the present time [27-34]^{[27]}.

In this article we consider some properties of the fuzzy R-solution of the control linear fuzzy integro-differential inclusions and research the time-optimal problems for it.

### 2. The Fundamental Definitions and Designations

Let be a family of all nonempty (convex) compact subsets from the space with the Hausdorff metric

where , is -neighborhood of set .

Let be the set of all such that satisfies the following conditions:

1) is normal, i.e. there exists an such that ;

2) is fuzzy convex, i.e.

for any and ;

3) is upper semicontinuous, i.e. for any and exists such that whenever ;

4) the closure of the set is compact.

If , then is called a fuzzy number, and is said to be a fuzzy number space.

The set is called the -level of a fuzzy number for . The closure of the set is called the -level of a fuzzy number .

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

Then is a metric in . Further we know that ^{[35]}:

i) is a complete metric space,

ii) for all ,

iii) for all and .

**Definition 1.** ^{[36]} *A mapping ** is measurable if for all ** the set-valued map ** defined by ** is measurable. *

**Definition 2.** ^{[36]} *A mapping ** is said to be integrably bounded if there is an integrable function ** such that ** for every **.*

**Definition 3.** ^{[36]} *The integral of a fuzzy mapping ** is defined levelwise by ** : ** is a measurable selection of ** for all **.*

**Definition 4.** ^{[36]} *A measurable and integrably bounded 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 control differential equations with the fuzzy parameter

(1) |

where means ; ; ; is the control; is the fuzzy parameter; .

Let be the measurable set-valued map.

**Definition 5.** *The set ** of all measurable single-valued branches of the set-valued map ** is the set of the admissible controls.*

Further we consider following control fuzzy differential inclusions

(2) |

where is the fuzzy map such that .

Obviously, the control fuzzy differential inclusion (2) turns into the ordinary fuzzy differential inclusion [29-32]^{[29]}

(3) |

if the control is fixed and .

If right-hand side of the fuzzy differential inclusion (3) satisfies some conditions (for example look ^{[32]}) then the fuzzy differential inclusions (3) has the fuzzy R-solution ^{[32]}.

Let denote the fuzzy R-solution of the differential inclusion (3), then denotes the fuzzy R-solution of the control differential inclusion (2) for the fixed .

**Definition 6.** *The set ** is called the attainable set of the fuzzy system (2).*

### 3. The Some Properties of The Fuzzy R-Solution

Further consider the following control linear fuzzy integro-differential inclusions

(4) |

where are -dimensional matrix-valued functions; is - dimensional matrix-valued function; is - dimensional matrix-valued function; is fuzzy set.

In this section, we consider some properties of the fuzzy R-solution of the control fuzzy integro-differential inclusion (4).

Let the following condition be true.

**Condition A:**

A1. are measurable on ;

A2. There exist such that

almost everywhere on ;

A3. is measurable on ;

A4. There exists such that almost everywhere on ;

A5. The set-valued map is measurable on ;

A6. There exists such that almost everywhere on .

**Theorem 1. **^{[37]} *Let condition A be true.*

*Then for every ** there exists the fuzzy R-solution ** such that *

*1)** **the fuzzy map ** is equal to*

where ; is Cauchy matrix of the differential equation ; is solution of the system

2) for every ;

3) the fuzzy map is the absolutely continuous fuzzy map on .

**Theorem 2.** ^{[37]} Let condition A be true.

Then the attainable set is compact and convex.

We obtained the basic properties of the fuzzy R-solution of systems (4). Now, we consider some fuzzy control problems.

### 4. Time-Optimal Problems

Consider the following time-optimal problem ^{[10, 26]}: it is necessary to find the minimal time and the control such that the fuzzy R-solution of system (4) satisfies one of the conditions:

(5) |

(6) |

(7) |

where is the fuzzy terminal set.

It is obvious that optimum time and optimum controls for these problems will be different.

**Theorem 3. (necessary optimal condition for time-optimal problem (4),(5)).** *Let condition A be true and the pair ** be optimal in control problem (4),(5).*

*Then there exists the vector-function **, that is the solution of the system*

such that

1)

almost everywhere on ;

2) ,

where .

**Proof.** Let be the optimal control and be the optimal fuzzy R-solution of problem (4),(5), i.e.

1)

2)

From 1) and 2) we have

(8) |

for all .

Consequently

From we have

for all .

From Theorem 1 we have that the function is continuous on .

If for all then we have . Hence there exists such that . Consequently we have

for all , i.e. .

It contradicts that is optimal time.

If ,

and , than we have a contradiction. Hence there exists such that

Consequently

Then we have

for almost every . If

than theorem 3 is proved.

Following theorems are similarly proved.

**Theorem 4. (necessary optimal condition for time-optimal problem (4),(6)).** *Let condition A be true and the pair ** be optimal in control problem (4),(6).*

*Then there exists the vector-function **, that is the solution of the system*

*such that*

*1)** *

*almost everywhere on **;*

*2)** **for all *

*and there exists *

This theorem can be proved similar to theorem 5 with little changes in condition (8):

for all

for all and there exist and such that

**Theorem 5. (necessary optimal condition for time-optimal problem (4),(7)).** *Let condition A be true and the pair ** be optimal in control problem (4),(7).*

*Then there exists the vector-function **, that is the solution of the system*

*such that *

*1)** *

*almost everywhere on **;*

*2)** **for all *

*and there exists *

Also this theorem can be proved similar to theorem 5 with little changes in condition (8):

for all

for all and there exist and such that

### 5. Conclusions

In this paper we have considered time-optimal problems for systems when the behavior of object is described by linear control fuzzy integro-differential inclusion. For these problems we have received necessary conditions of optimality. These results generalize results of papers ^{[7, 10, 14, 16, 21, 24]}.

### References

[1] | Ovsyannikov, D.A., Mathematical methods for the control of beams, . Univ., , 1980. | ||

In article | |||

[2] | Zubov, V.I., Dynamics of controlled systems, ``Vyssh. Shkola'', Moscow, 1982. | ||

In article | |||

[3] | Zubov, V.I., Stability of motion. Lyapunov methods and their application, ''Vyssh. Shkola'', Moscow, 1984. | ||

In article | |||

[4] | Arsirii, A.V. and Plotnikov, A.V., “Systems of control over set-valued trajectories with terminal quality criterion,” Ukr. Math. J., 61 (8), 1349-1356, 2009. | ||

In article | CrossRef | ||

[5] | Phu, N.D. and Tung, T.T., “Some results on sheaf-solutions of sheaf set control problems,” Nonlinear Anal., 67 (5), 1309-1315, 2007. | ||

In article | CrossRef | ||

[6] | Plotnikov, A.V., “Controlled quasidifferential equations and some of their properties,” Differ. Equations, 34 (10), 1332-1336, 1998. | ||

In article | |||

[7] | Plotnikov, V.A. and Plotnikov, A.V., “Multivalued differential equations and optimal control,” Applications of mathematics in engineering and economics (Sozopol, 2000), 60-67, Heron Press, Sofia, 2001. | ||

In article | |||

[8] | Plotnikov, A.V., Komleva, T.A. and Arsiry, A.V., “Piecewise Constant Controlled Fuzzy Systems”, Int. J. Sensing, Computing & Control, 2 (1), 50-57, 2012. | ||

In article | |||

[9] | Dabbous, T.E., “Adaptive control of nonlinear systems using fuzzy systems,” J. Ind. Manag. Optim., 6 (4), 861-880, 2010. | ||

In article | CrossRef | ||

[10] | Plotnikov, A.V. and Komleva, T.A., “Linear problems of optimal control of fuzzy maps,” Intelligent Information Management, 1 (3), 139-144, 2009. | ||

In article | CrossRef | ||

[11] | Plotnikov, A.V., Komleva, T.A. and Arsiry, A.V., “Necessary and sufficient optimality conditions for a control fuzzy linear problem,” Int. J. Industrial Mathematics, 1 (3), 197-207, 2009. | ||

In article | |||

[12] | Plotnikov, A.V. and Komleva, T.A., “Fuzzy quasidifferential equations in connection with the control problems,” Int. J. Open Problems Compt. Math., 3 (4), 439-454, 2010. | ||

In article | |||

[13] | Konstantinov, G.N., “Sufficient conditions for optimality of a minimax control problem of an ensemble of trajectories,” Soviet Math. Dokl., 36 (3), 460-463, 1988. | ||

In article | |||

[14] | Otakulov, S., “On the approximation of the time-optimality problem for controlled differential inclusions,” Cybernet. Systems Anal., 30 (3), 458-462, 1994. | ||

In article | CrossRef | ||

[15] | Otakulov, S., “On a difference approximation of a control system with delay,” Autom. Remote Control, 69 (4), 690-699, 2008. | ||

In article | CrossRef | ||

[16] | Plotnikov, A.V., “Linear control systems with multivalued trajectories,” Kibernetika (Kiev), (4), 130-131, 1987. | ||

In article | |||

[17] | Plotnikov, A.V., “Compactness of the attainability set of a nonlinear differential inclusion that contains a control,” Kibernetika (Kiev), (6), 116-118, 1990. | ||

In article | |||

[18] | Plotnikov, A.V., “A problem on the control of pencils of trajectories,” Siberian Math. J., 33 (2), 351-354, 1992. | ||

In article | CrossRef | ||

[19] | Plotnikov, A.V., “Two control problems under uncertainty conditions,” Cybernet. Systems Anal., 29 (4), 567-573, 1993. | ||

In article | CrossRef | ||

[20] | Plotnikov, A.V., “Necessary optimality conditions for a nonlinear problems of control of trajectory bundles,” Cybern. Syst. Anal., Vol. 36, No. 5, 2000, pp. 729-733. | ||

In article | CrossRef | ||

[21] | Plotnikov, A.V., “Linear problems of optimal control of multiple-valued trajectories,” Cybern. Syst. Anal., 38 (5), 772-782, 2002. | ||

In article | CrossRef | ||

[22] | Plotnikov, A.V. and Komleva, T.A., “Some properties of trajectory bunches of a controlled bilinear inclusion,” Ukr. Math. J., 56 (4), 586-600, 2004. | ||

In article | CrossRef | ||

[23] | Plotnikov, A.V. and Plotnikova, L.I., “Two problems of encounter under conditions of uncertainty,” J. Appl. Math. Mech., 55 (5), 618-625, 1991. | ||

In article | CrossRef | ||

[24] | Molchanyuk, I.V. and Plotnikov, A.V., “Linear control systems with a fuzzy parameter,” Nonlinear Oscil. (N. Y.), 9 (1), 59-64, 2006. | ||

In article | CrossRef | ||

[25] | Plotnikov, A.V., Komleva, T.A. and Molchanyuk, I.V., “Linear control problems of the fuzzy maps,” J. Software Engineering & Applications, 3 (3), 191-197, 2010. | ||

In article | CrossRef | ||

[26] | Plotnikov, A.V., Komleva, T.A. and Plotnikova, L.I., “The partial averaging of differential inclusions with fuzzy right-hand side,” Journal Advanced Research in Dynamical & Control Systems, 2 (2), 26-34, 2010. | ||

In article | |||

[27] | J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued maps and Viability Theory. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1984. | ||

In article | CrossRef | ||

[28] | G.V. Smirnov, Introduction to the theory of differential inclusions, Graduate Studies in Mathematics, American Mathematical Society. Providence, Rhode Island, vol. 41, 2002. | ||

In article | PubMed | ||

[29] | Aubin, J.-P., “Fuzzy differential inclusions,” Probl. Control Inf. Theory, 19 (1), 55-67, 1990. | ||

In article | |||

[30] | Baidosov, V.A., “Differential inclusions with fuzzy right-hand side,” Soviet Mathematics, 40 (3), 567-569, 1990. | ||

In article | |||

[31] | Baidosov, V.A., “Fuzzy differential inclusions," J. of Appl. Math. and Mechan., 54 (1), 8-13, 1990. | ||

In article | CrossRef | ||

[32] | Hullermeier, E., “An approach to modelling and simulation of uncertain dynamical systems,” Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 5 (2), 117-137, 1997. | ||

In article | CrossRef | ||

[33] | Lakshmikantham, V., Gnana Bhaskar, T. and Devi J. Vasundhara, Theory of set differential equations in metric spaces, Cambridge Scientific Publishers, Cambridge, 2006. | ||

In article | |||

[34] | Lakshmikantham, V., and Mohapatra, R.N., Theory of fuzzy differential equations and inclusions, Series in Mathematical Analysis and Applications, 6. Taylor & Francis, Ltd., London, 2003. | ||

In article | CrossRef | ||

[35] | Puri, M.L. and Ralescu, D.A., “Fuzzy random variables,” J. Math. Anal. Appl., (114), 409-422, 1986. | ||

In article | |||

[36] | Park, J.Y. and Han, H.K., “Existence and uniqueness theorem for a solution of fuzzy differential equations,” Int. J. Math. Math. Sci., 22 (2), 271-279, 1999. | ||

In article | CrossRef | ||

[37] | Vasil'kovskaya, V.S., and Plotnikov, A.V., “Integrodifferential systems with fuzzy noise,” Ukr. Math. J., 59(10), 1482-1492, 2007. | ||

In article | CrossRef | ||