IRJET- Existence of Bounded Solutions of Linear First Order Fuzzy Lyapunov Systems-A New Approac

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Existence of

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Bounded Solutions of Linear First Order Fuzzy Lyapunov Systems- A New Approach P. Sailaja1, Sriram Bhagavathula2, K. N. Murty3*

1Department

of Mathematics, Jawaharlal Nehru Technological University, Hyderabad, TS, India.

2Conns Appliances Inc 3Department of Applied Mathematics, Andhra University, Visakhapatnam, A.P, India ---------------------------------------------------------------------------***--------------------------------------------------------------------------Keywords: Fuzzy sets and systems, Lyapunov systems, Stability Analysis and Fundamental fuzzy matrix solutions.

AMS (MoS) classifications: 34A07, 26A39, 46G05 1. Introduction This paper presents a criteria for the existence of the form ( where ( ), matrix and (

)( )

( )

( )

( )

( )

bounded solution of the linear first order Fuzzy Lyapunov system of

( ) (1.1)

( ) are square matrices of order ( ) and F(t) is a square matrix of order n and , -. The homogeneous system corresponding to (1.1) is

)( )

( )

( )

( )

( ) is an (

) continuous

( ). (1.2)

Lyapunov first order system plays an important role in the theory of differential equations and has significant impact on control engineering problems. Recently, much attention has been paid by many authors to study qualitative properties of linear systems and -bounded solutions of linear first order systems. Among them the results established by Kasi Viswanadh, Yan Wu, K. N. Murty, Rompicherla, Anand, Divya and Pagilla [2-10, 14, 15] need a special mention. Kasi Viswanadh in fact initiated [3] the idea of fuzzy first order systems and his co-authors Narayana S. Ravada, Murty K.N extablished -bounded solutions Linear systems on time scale dynamical system needs a special mention [4], as it unifies both continuous and discrete systems in a single framework. In fact the existence of bounded solutions was first initiated by Kasi Viswanadh V. Kanuri [3] and ( ) –bounded solutions by Rompicherla, Anand [14] on Kronecker product linear system of differential equations. Motivated by these ideas, we investigate ( ) bounded solution of Lyapunov linear system of first order fuzzy differential equation (1.1). This paper is organized as follows: section 2, presents basic results on fuzzy sets and systems and develop variation of parameters formula for (1.1). We study existence of ( ) bounded solutions in section 3. We also establish stability criteria for the Lyapunov homogeneous fuzzy system and deduce existing results as particular case.

Recently much attention has been paid by many authors on the existence of ( , )-bounded solutions of linear first order systems [14,16]. The use of bounded solutions for fuzzy first order systems is due to Kasi Viswanadh [3] and based on this Kronecker product systems were studied by Rompicherla, Anand et.al [14]. In fact the results established in K.V.K. Viswanadh [4] are very useful to understand –level sets and have significant impact on characterization of soils and thereby increasing production on food grains to farmers [15].

2. Preliminaries In this section, we present basic notions on fuzzy sets and systems and define –level sets for understanding the use of fuzzy sets. Most of the results presented in this section form a clear understanding of our main results. For, let X be a non-empty set.

A Fuzzy set A X is characterized by its membership function A: X [0,1] is used to represent intermediate degree of membership. The mapping A is also called the membership function of fuzzy set A. the integrals of fuzzy-number-valued functions is obtained as a natural generalization of set valued functions and have been discussed by Puri and Ralescu [17], Kaleva [1] and other co-authors in [16].

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Let

be a closed interval on

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i.e,

,

-(

)

Let | | denotes the length of . If there exists (where

⋃ *(

)( *(

)

)+ is called a division of

)

(

)+

* ,

-+ where ,

For clarity, we write

Throughout this paper, we For

(

),

)

(

*

(

(

),

‖

.

‖+.

, the support function of A is defined as

),

denotes the unit sphere of

It is clear now that for A, B ( (

(i) (ii)

-.

+

‖

(‖

Defintion 2.1: For A

where

and is an associated point of ,

), the Haussdorff metric between them is defined as

*

)=

- denotes a typical interval of

) to denote the family of all non-empty compact convex subsets of

+,

In addition, for

(

(

and we write

, the addition and scalar multiplication are defined in the usual way as

*

(

)such that

) are non-overlapping subintervals of , then a collection

(

)(

(

)=

(

),

, we have

( )( )= ( ) (

Theorem 2.1: Let

, (.,.) is the inner product of

) (

).

). Then (

)

| (

)

(

)|.

Proof: For the proof of the theorem, we refer P. Diamand and P. Kloeden [18].

Definition 2.2: Let following conditions.

We define

,

)

,

-. For any

. U is said to be ab n-dimensional fuzzy number if it satisfies the

u is normal i.e. there exists an such that ( ) ( )u is fuzzy convex i.e., , * ( ) ( )+ for any u is upper semi continuous * + is compact. ( ) , -

(i) (ii) (iii) (iv)

(

*

,

-

and

,

-

) by the equation

,, - , - -

,

then the metric space* + has a linear structure, it can be imbedded isomorphically as a convex cone with vertex Banach space of functions. where (

)

, -

is the unit sphere in

with an embedding function

for all

.

Definition 2.3: A fuzzy number in parametric form is represented by ( , - and

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, - ,

|

( ) defined as

), where

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and has the following properties: (i) (ii) (iii)

is a bounded left continuous monotonic increasing is a left continuous monotonic decreasing function of for , -.

If (

over [0,1].

, is a function then according to Ladeh’s principle, we can extend )( )

(

as

* ( ) ( )+ and

)

Further, , (

(, - , - )

)-

For all

and ,

-

,

and , -

-

,

-, the sum (

And

are defined as

, -

, - .

Definition 2.4: we define (

) and the product

* + by (, - , - ),

)

is the Haussdorff metric defined

(

).

We are now in a position to develop the general solution of the Lyapunov fuzzy homogeneous system (1.2) and then the variation of parameters formula for (1.1).

Theorem 2.2: Any solution of the linear Lyapunov system (1.2) is of the form ( ) for any fixed solution of ( ) , -.

Proof: Let

,

- be fixed and let

( ) be a fundamental matrix solution of be a matrix defined by

(

)

( )

(

)

(

Since ( (

. Then let

)

)( ( )

)

Hence (

)

(

( )

)

) for some constant square matrix . ( )

( )

( ) (Take

).

( ) be any solution of (1.1) and ̅ ( ) be a particular solution of (1.1), then

Proof: It can easily be verified that

( )

̅ ( )

( )

̅ ( )

( )

( )

̅ ( ) is a solution of (1.2) and hence ( )

( )

.

.

( )

and hence

)

. Then

it follows that

or ( (

Theorem 2.3: Let

( )

)

is a fundamental matrix

.

Thus Thus

)

(

where

̅ ( )

( )

( )

( ).

Theorem 2.4: A particular solution ̅ ( ) of (1.1) is given by

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̅ ( )

( )∫ (

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) ( ) ( )(

) ( )

.

Proof: It can easily verified that ( ) ( ) is a solution of (1.2). such a solution cannot be a solution of (1.1) unless ( ) we seek a particular solution of (1.1) in the form ̅ ( ) (

( ) ( )

)

Since (

)

( )

( ) (

)

( ) and (

)

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( )

( ) it follows that

( ) Or ( ) ( )

( ∫(

) ( ) ( )(

) ( )

) ( ) ( )(

) ( )

Or ̅ ( )

( ) ( )

( )

( ) [∫ (

) ( ) ( )(

) ( )

]

( )

( )

( ) ( )

( )

( ) [∫ (

) ( ) ( )(

) ( )

]

( )

Hence

Now, we consider the following systems (

)( )

(

)( )

( )(

( ) (2.1)

and ( ) ( ), (2.2)

where and are square matrices of order n. Let ( ) and ( ) be fundamental matrix solutions of (2.1), (2.2) , -. Then (2.1) and 2.2 can conveniently put in the form respectively for each ,

( )

( )-

, ( )

)

( )),(

(

)( )- (2.3)

The following theorem will be of immense use for our future discussion.

Theorem 2.5: Suppose ( )( ) be a fundamental matrix solution of (2.3). Then it is claimed that fundamental matrix solutions of (2.1) and (2.2) respectively.

( ) and

( ) are

For )( )

( ,

( )

( )

, ( ) ( )-

) ( )

(

( )),( ( )⨂

)( )[

( )

( )]

Multiplying both sides of the above relation by (

) ( ) (

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We get ) ( ),(

(

)( )

( )

( )-

( )[

( )

( )] (2.4)

Equation (2.4) holds if and only if each side of (2.4) is equal to either a null matrix or a unit matrix.

Let us suppose ) ( ),(

( Then (

)( )

( )

)( )

( )

( )

. Then

)( )

( )

(

) ( ),(

(

)( )

( )

(

)( )

, ( )

( )

( )-

.

( ) is a fundamental matrix solution of (2.1) and suppose ( )-

. Then

( ) ( ),

-

which is a contradiction. Similar arguments holds for the R.H.S of (2.4). Conversely, ( ) and ( ) are fundamental matrix solutions of (2.1) and (2.2) respectively. Then it is claimed that ( ) ( ) is a fundamental matrix solution of (2.3). Since , -, we have for each (

)( )

(

( )

( ) ( ))

=( ( )

( )

=( ( )

( ) and ( (

( )

( ))

(

)(

)( )

( )

( )) + ( ( )

)( )

( )

( ). Then

( ) ( ))

( ))(

(

( ))

)( )

Hence the claim. Similar result hold for Lyapunov system (1.2) 3. Main Result In this section, we establish our main result on the existence of ( ) bounded solution of the linear Lyapunov system (1.1) and deduce stability criteria for the system (1.1). We first note the following theorem Theorem 3.1: Let

, then (i) (ii) (iii)

Conversely, if * that

, , If *

( ) for all , , - for all and + is a non-decreasing sequence converging to

+ is a family of subsets of , -

, then , -

, -

⋂

.

satisfying the above three conditions, then there exists a

for

and , Lemma 3.1: For any , ̂( )-

,

.

- such that

and for any

, ̂( )-

Proof: Let

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. Then by hypothesis, it follows that , ̂( )-

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is contained in , ̂( )-

and hence

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such

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̂ ( )

( )

( )

www.irjet.net ( )

( )

( )

( )

̂ ( ). ̂ ( )

Thus, we have the selection of inclusions ̂( )

( ) ̂( ) ( )

̂( )

̂ ( ) and hence

( )

( ),

and further ( ) Thus ̂

̂

( )

( )

( )

( ). Further

and It follows that ̂

( )

̂

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

Let be a fundamental matrix solution of (2.1). Then it can easily be verifies that ( ) ( ) is a solution of the homogeneous Lyapunov system (1.2). Further, let the vector space be represented as direct sum of three subspaces say , , such that a solution of (1.2) is -bounded on R if and only if ( ) and bounded on , ) if and only if ( ) . Also let , , denote the corresponding projection on onto , , respectively for each , -. We are now in position to prove the following result.

Theorem 3.2: Let ( ) be a continuous ( ) matrix on . Then the system (1.2) has at least one for every continuous and -bounded function if and only if there exists a constant ( )

∫ ‖

∫ ‖ for

( )

( )

∫ ‖

( )

( )

( )

( ) (

( )

( ) ) ( )(

bounded solution on R

such that

( )

( )

( )(

( )

)(

) ( )(

( )

) ( )( ) ( )(

( )

) ( )(

) ( )‖

( )‖

) ( )(

) ( )‖

) ( )(

) ( )‖

,

(3.1)

and ∫ ‖

( )

( )

( ) ∫ ‖ ∫ ‖

for

( ) (

) ( )(

( )

( )

( )

( ) ( )

( )

) ( )( ( )( ( ) (

) ( )( )(

) ( )‖

) ( )( ) ( )(

) ( )(

) ( )(

) ( )(

) ( )‖

. (3.2)

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Proof: To make proof simple, we use the following notation ( ) ( ) atleast one with the norm

( ) and ( ) ( ) ( ) for each , -. First suppose that the Lyapunov linear system (1.1) has bounded solution on R for every , -. Let B denote the set of all bounded solutions and ‖

and let D denote the set of all such that ( ) ‖

‖

( ) ‖

‖

‖

‖

( )

( )‖

bounded and continuously differentiable functions

( )

and

. Clearly, D is a vector space and if we define norm on ( )

‖

( )

( )

by

( )‖

‖ ‖ ) is a Banach space.

Then it can easily be proved that (

For, let * + be a fundamental sequence of elements of . Then, ( ) is a fundamental sequence in B. Therefore, there ( ) ( ) converges to exists a continuous and -bounded function for each , - such that ( ) ( ) uniformly on . From the inequality ‖

( )

‖(

( )‖

( )

) ( )‖‖

It follows that

( )

( )

( )

( ) for each

,

( )‖

- uniformly on every compact subset of

. Thus ( )

.

Similarly, ( )

.

( )

( )

( ) ( )

( )

( ) ( )/ ( ),

( ), uniformly on every compact subset of ,

and hence

-.

Hence. ‖

( )

( )‖

. This proves that (

Differentiable functions ( )

( ) ( )‖

( )

‖ ‖ ) is a Banach space.

such that . Clearly,

Then, it can easily be prove that (

( )

and

is a vector space and if we define norm on

‖ ‖ ) is a Banach space. For, let *

, - be a fundamental sequence of elements of . Then * continuous -bounded function for each , - such that Thus,

( )

for each

,

+

+

by ‖

‖

‖

‖

‖

( )

,

is a fundamental sequence in . Therefore, there exists a ‖ ( ) ( )‖ converges to ( ) ( ) uniformly on .

-.

Similarly, ((

)( )

( )

( )

( )

( ))

( )

Uniformly on every compact subset of R and hence

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( )

( )

( )

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( ) for

and

,

-.

Hence ‖

( )

( )‖

for each

Step 2: There exists a positive constant say system (1.1), we have ‖ ( ) ( )‖

=

such that for every

of Lyapunov

such that

-A

-

.

This mapping us linear and bounded with ‖ ‖ =A

This shows that X is a

and for the corresponding solution

‖ ( ) ( )‖ (3.3)

For, define the mapping

Let XT = 0, then

-.

‖ ‖ ) is a Banach space.

This proves that (

X

,

,

1.

.

bounded solution on

on (1.2). Then ( )

(

)

* +.

Thus T=0, such that X is one-one and can easily be proved that T is onto. ̅​̅​̅​̅ is a solution of (1.2) with ( )

Then

From the definition of ̅

, it follows that

(

) ( )

( )

bounded on . Then ̅ belongs to D and

is

.

From a fundamental lemma of Banach space, it follows that ‖

‖

‖

‖‖ ‖ for all

Step 3: Let ( ) ,

be fixed points and let ). Then it is easily to see that the mapping

∫

( )( ∫

{

∫

( )

( )

( )

) ( )(

) ( )(

) ( ) (

) ( ) ( ( )

∫

( ) (

( ) ( ( )(

∫ ∫

( )

( )

∫

( )

and hence step 2 follows

( ) ( ) ( (

) ( ) ( )

) ( )(

) ( )(

) ( ) ( )

) ( ) ( )

) ( )(

(

-bounded functions which vanishes on

) ( ) ( )

) ( )(

) ( )(

be continuous and defined by

) ( ) ( ) ) ( ) ( )

) ( ) ( )

is the solution of D of Lyapunov system (1.1) Similarly, the other case follows. If we put

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( ) (

∫

Now, consider for (

∫

(

) ( )

(

) ( )

( ) (

( )(

)(

)( )( ) ( )(

( ) ( ) ( )( ( ) ( ) ( ( ) ( )( )( ( ) ( ) ( )(

)

{ we then have ( )

( )

,

and for each

,

) ( ) ) ( )

)(

) ( ) )( )( ) ( ) )( )( ) ( ) )( ) ( ) -.

,

) ( )

∫

(

∫

∫

( )

( ) (

) ( )(

∫

( )

( )(

)(

∫

( )

( ) (

) ( )(

) ( ) ( )

∫

( )

( ) (

) ( )(

) ( ) ( )

) ( )

) ( ) ( ) ) ( )(

) ( ) ( )

( ) Similarly, the other cases follow.

Now, to prove the other part, suppose the fundamental matrix solution of ( ) of (2.1) satisfies either condition (3.1) or (3.2) ( ) bounded solution on R of (1.1). for some . Then it can easily be proved that there exists a Theorem 3.2: Suppose that the homogeneous Lyapunov system has no non-trivial - bounded solution on R for each , - and suppose that the fundamental matrix solution satisfies the condition ∫

( )

‖

( ) (

) ( )(

) ( )‖

∫ ‖

( )

( ) (

and further assume that -bounded function is such that homogeneous Lyapunow system (1.1) has unique solution ( ) on ‖ Note that if ( ) and

( )‖

for each

,

) ( )(

) ( )‖

‖ ( )‖ such that

, for each

,

-. Then the non-

-.

( ) is a fundamental matrix solution of (2.1) if and only if ( ) is a fundamental matrix solution of (2.2) with ( ). If the system (2.1) has a ( )-bounded solution, then system (2.2) has -bounded solution for , - If the system (2.1) is stable then (2.2) is a stable with .

References: 1.

2.

3.

4.

O. Kaleva, “Fuzzy differential equations, Fuzzy sets and systems”, 24 p: 301-317 (1987) Kasi Viswanadh V. Kanuri, K. N. Murty, “Three-Point boundary value problems associated with first order matrix difference system-existence and uniqueness via shortest and closest Lattice vector methods”, Journal of Nonlinear Sciences and Applications, Volume 12, Issue 11, (2019) 720--727 Kasi Viswanadh V. Kanuri, Existence of -bounded solutions for fuzzy dynamical systems on time scales”, International Journal of Scientific & Engineering Research, 2020, Vol. 11, No. 5, 613--624.

Kasi Viswanadh V. Kanuri, R. Suryanarayana, K. N. Murty, Existence of Ψ-bounded solutions for linear differential systems on time scales, Journal of Mathematics and Computer Science, 20 (2020), no. 1, 1--13.

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5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

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Murty, K. N., Andreou, S., Viswanadh, K. V. K., Qualitative properties of general first order matrix difference systems.

Nonlinear Studies, 16(2009), no. 4, 359--370.

K. N. Murty, V. V. S. S. S. Balaram, K.V. K. Viswanadh, “Solution of Kronecker Product Initial Value Problems Associated with First Order Difference System via Tensor-based Hardness of the Shortest Vector Problem”, Electronic Modeling, vol. 6, p: 19--33 (2008).

K.N. Murty, K.V.K. Viswanadh, P. Ramesh, Yan Wu, “Qualitative properties of a system of differential equation involving Kronecker product of matrices”, Nonlinear Studies, Vol 20, No: 3, P: 459--467 (2013) K.N. Murty, Yan Wu, Viswanadh V. Kanuri, “Metrics that suit for dichotomy, well conditioning of object oriented design”, Nonlinear Studies, Vol 18, No: 4, P: 621--637 (2013) Kasi Viswanadh V Kanuri, Y. Wu, K.N. Murty, “Existence of ( ) bounded solution of linear first order Kronecker product of system of differential equations”, International Journal of Science and Engineering Research, 2020, Vol 11, No. 6, P: 721--730.

Viswanadh V. K, K.N. Murty, P. Sailaja, “A new approach to the construction of a transition matrix and applications to control systems”, Proc. of the International Conference on Material science, Dec 2019 (India) Martin Bohner, Allan Peterson, “Dynamical equations on time scales”, Birkhauser, Boston (2003) V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, “Dynamical systems on measure chains”, Kluwer Academic publishers, Dordrecht (1998).

V Lakshmikantham, K.N. Murty and J. Turner, “Two-Point boundary value problems associated with non-linear fuzzy differential equations”, Mathematical inequalities and applications, Vol 4, P: 527-533 (2001).

Charyulu L. N. Rompicherla, Sundaranand V. Putcha, Deekshitulu G.V.S.R, Existence of ( ) bounded solutions for linear first order kronecker product systems, International Journal of Scientific Research, Vol 11, No: 6, P: 3904739053 (2020) Kasi Viswanadh V. Kanuri, SriRam, Bhagavathula and K.N. Murty, “New Stability Analysis of Linear First order Sylvester system”, International Journal of Engineering and Computer Science., Vol 11, P: 25252-25259 (2020) Madan L Puri, Dan A Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications, Volume 114, Issue 2, 1986, Pages 409-422, Lakshmikantham, V. and Mohapatra, R. Theory of fuzzy differential equations and inclusions, Taylor - Francis, 2003.

Phil Diamond, Peter E. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, 1994.

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