A NEW LYAPUNOV-KRASOVSKII METHODOLOGY FOR COUPLED DELAY DIFFERENTIAL DIFFERENCE EQUATIONS (2006).pdf

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A NEW LYAPUNOV-KRASOVSKII METHODOLOGY

FOR COUPLED DELAY DIFFERENTIAL DIFFERENCE EQUATIONS @

P. Pepe ;+ Z.-P. Jiang # E. Fridman

Dipartimento di Ingegneria Elettrica, Universit a degli Studi dell'Aquila,Monteluco di Roio, 67040 L'Aquila, Italy, e-mail: [email protected]

# Department of Electrical and Computer Engineering, Polytechnic University,Six Metrotech Center, Brooklyn, NY 11201, USA, e-mail: [email protected]

Department of Electrical Engineering Systems, Tel Aviv University,Tel Aviv 69978, Israel, e-mail: [email protected]

ABSTRACT In this paper a new Lyapunov-Krasovskii methodology for nonlinear coupled delay dif-ferential dierence equations is proposed. This method-ology is based on the concept of input-to-state stabilityapplied to the dierence equation, for which a sucientLyapunov criterion is given, and on previous method-ologies developed in the literature for linear delay de-scriptor systems.

Keywords : Continuous Time Dierence Equations,Nonlinear Time Delay Systems, Lyapunov's SecondMethod, Input-to-State Stability.

I. INTRODUCTION

Coupled delay dierential dierence equations (forshort, here indicated as CDDDEs) describe, for in-stance, lossless propagation phenomena (see [9, 15]) andinternal dynamics of recently studied nonlinear delaycontrol systems (see [3] and references therein). Neutralequations in the Hale's form, which describe many engi-neering systems (consider for instance the model of Par-tial Element Equivalent Circuits in [1]), can be rewrit-ten as coupled delay dierential dierence equations (see[2,9,14] and references therein). Therefore stability cri-teria for coupled delay dierential dierence equationscan also be successfully used for neutral equations in theHale's form. Recently, a Lyapunov-Krasovskii method-

ology for general coupled delay dierential d ierenceequations has been proposed in [13,14]. The method-ology presented there consists of two steps: the rststep leads to the L2 stability and the second one leadsto the Lyapunov stability.

+ Corresponding author.@ The work of P. Pepe has been supported by Ital-

ian MIUR Project PRIN 05. The work of Z.P. Jianghas been supported by U.S. NSF grants ECS-0093176,OISE-0408925 and DMS-0504462. The work of E. Frid-man has been supported by Kamea fund of Israel.

In [2] a Lyapunov-Krasovskii methodology for lin-ear delay descriptor systems is proposed. Since coupleddelay dierential dierence equations can be written as

descriptor systems, the methodology proposed there canbe applied for studying the stability of the class of sys-tems considered in this paper, at least in the linear case.

In the context of linear systems, dierent conditions,in terms of Linear Matrix Inequalities, for the delay-independent asymptotic stability have been obtained bythe Lyapunov-Krasovskii methodologies proposed in [2]and in [13,14] respectively (see [2], Theorem 1, and [13],Corollary 3.4, [14], Corollary 5).

The purpose of this paper is to extend the method-ology p roposed in [2] from the linear case to the generalnonlinear case. In order to carry out the extension, Son-tag's concept of input-to-state stability (ISS) for nite-dimensional continuous-time systems [16] will be bor-rowed and our main results are based on the discrete-time version of ISS and its Lyapunov characterizationin [7]. In particular, a Lyapunov criterion for the input-to-state stability of continuous time dierence equations(for short, here indicated as CTDEs), based on the onefor nonlinear nite dimensional discrete time s ystemsgiven in [7], is rst proposed. Then, using this criterion,it can be stated that the variable of the continuous timedierence part of the equations can be guaranteed ar-bitrarily small if the variable of the dierential part of the equations is suciently small. After this result is

established, the descriptor methodology in [2] can beadapted to the current case of general nonlinear cou-pled delay dierential dierence equations, l eading t oa new Lyapunov-Krasovskii methodology. It is worthnoting that the ISS property of CTDEs is automaticallyguaranteed by the asymptotic stability of the unforceddierence equation, as long as the system in question islinear (see [2,5]).

We believe that the presented stability results to-gether with [13,14] will provide a solid foundation foranalysis and synthesis of nonlinear coupled delay dier-ential dierence equations. In particular, they should

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prove useful for delay-dependent stability analysis.

II. CTDEs

Consider the following system of nonlinear continuoustime dierence equations:

x( t) = f (x (t 1 ); x (t 2 ); : : : ; x(t m ); u(t )) ;t 0; x( ) = x 0 ( ) ; 2 [ ; 0]

(1)where the (continuous) time variable t 2 [0; + 1 ), x(t ) 2IR n , n is a positive integer, 0 < 1 < 2 < : : : < m = are the arbitrary (non commensurate) delays, f is acontinuous function dened on IR nm IR p and takingvalues in IR n , p is a positive integer, the input u is alocally essentially bounded function dened on [0 ;+ 1 )

and taking values in S , a subset of IR p

, the initial con-dition x0 is a continuous function dened on [ ; 0]and taking values in IR n . In the following we indicatewith M S the set of the input functions u . Assume thatf (0; : : : ; 0; 0) = 0, thus ensuring that x(t ) = 0 is the so-lution of the sy stem (1) with zero input and zero initialconditions.

Note that system (1) admits a unique solution in[0; + 1 ) for any input function u and any initial condi-tion x0 .

It was shown in [3] that continuous time dierenceequations can be rewritten as discrete time systems ona suitable Banach space. In [11] such transformationhas been g iven in the case of multiple and not commen-surate delays. In this general case, the involved Ba-nach space is the linear space F ((0 ; min ]; IR (k m +1) n )of the essentially bounded functions from (0 ; min ] toIR ( k m +1) n , endowed with the essential supremum normthat we will indicate with k kF , where:

min = min f 1 ; 2 1 ; : : : ; m m 1 g; (2)

i = ki min + i ; i = 1; 2; : : : ; m ; (3)

ki ; i = 1 ; 2; : : : ; m ; are suitable positive integers, i ; i =1; 2; : : : ; m are reals such that 0 i < min . Morespeci cally, system (1) can be rewritten as a discretetime system in the space F ((0; mi n ];IR (k m +1) n ) as fol-lows (see [3,11] for the details):

X (k + 1) = G (X (k) ; U (k)) ; k = 0 ; 1; : : : (4)

where X (k) 2 F ((0 ; min ]; IR (k m +1) n ), U (k ) 2B ((0; mi n ];S ) is dened as

U (k)( ) = u(k min + ); k = 0 ; 1 : : : ; (5)

B((0 ; mi n ];S ) is the set of essentially bounded func-tions dened on (0 ; mi n ] and taking values in S , G isa suitable function dened on

F ((0 ; min ];IR (k m +1) n ) B((0 ; mi n ];S )

and taking values inF ((0 ; min ]; IR ( k m +1) n )

. This transformation can be performed if the delays areexactly known, and is global with respect to the statevariables. Due to the particular discrete time dynamicsof system (4), which involves the function f just forthe last components of the state X (k), it is useful, forthe application of the Lyapunov's second method, toconsider the following discrete time system, obtainedby (4),

X (k + 1) = F (X ( k); U (k)) ; k = 0; 1; : : : (6)

where X ( k) = X (k( km + 1)) 2 F ((0 ; min ];IR (k m +1) n ),

U ( k) =2664

U ((k(km + 1))U ((k(km + 1) + 1)

...U (k( km + 1) + km )

3775

, and the function F is

suitably obtained by the function G in (4). The input of system (6), U ( k), belongs to B ((0 ; min ]; S (k m +1) ), thelinear space of essentially bounded functions dened in(0; min ] and taking values in S (k m +1) , endowed withthe essential supremum norm that we will indicate withk kB .

Remark 1: In the case of one single delay the abovedicult procedure leading to the discrete time system(6) is not necessary. In the case of one single delay ,the discrete time equation (6) is obtained with (k) 2F ((0 ;]; IR n ); and U (k) 2 B ((0 ; ]; S ) (see [3]). Whenthe delays are multiple but commensurate, the equation(1) can be transformed into an equation with one singledelay, by a state extension.

III. ISS of CTDEs

In the following, for a given positive integer k, we willindicate with U

[k ]

the truncation of U (k) at k, that isthe sequence which is equal to U ( k), for k = 0; 1; : : : ; k ,and is null for k > k . We will indicate with kU [k ]kB ; 1the quantity sup 0 k k kU (k)kB . Recall that a function : IR + ! IR + is said to be of class K if it is continuous,strictly increasing and satises (0) = 0. It is of classK 1 if, additionally, it is unbounded. A function :IR + IR + ! IR + is of class KL if for each xed t, thefunction (; t ) is of class K and for each xed s, thefunction (s; t ) decreases to 0 as t ! + 1 .Denition 2: System ( 1), or its equivalent dierenceequation ( 6), is said to be input-to-state stable (for

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short, ISS) if there exist a function of class KL and a function of class K such that, for any essentially bounded initial condition x0 and for any input function u 2 M S , the following inequality holds for the discretetime solution of the equivalent system ( 6)

kX (k)kF (kX (0) kF ; k) + (kU [k ]kB ; 1 ) (7)

Theorem 3: Let V : F ((0 ; min ];IR (k m +1) n ) ! IR + bea continuous functional such that:i) there exist functions 1 and 2 , of class K 1 , such

that, for any X 2 F ((0 ; min ]; IR (k m +1) n ), the fol-lowing inequalities hold

1 (kX kF ) V (X ) 2 (kX kF ) ; (8)

ii) there exist a function 3 of class K 1 and a function of class K , such that, for any X 2

F ((0 ; mi n ];IR(k m +1) n

), and any U 2 B((0 ; mi n ];S ( km +1) );

the following inequality holds

V (F (X ; U )) V (X ) 3 ( kX kF ) + (kUkB ) (9)

Then, system ( 1) is input-to-state stable .Proof. The same proof given in [7] for Lemma 3.5,concerning nite dimensional discrete time systems, isapplicable to the present case of innite dimensional dis-crete time system (6). ut

IV. CDDDEs

The following system of time invariant nonlinear cou-pled delay dierential dierence equations is considered

_( t) = A (x(t 1 ); : : : ; x(t m ); (t );(t 1 ); : : : ; ( t m )) ; t 0;

x(t ) = B(x( t 1 ); : : : ; x (t m );(t ); ( t 1 ); : : : ; (t m )) ;

(10)

( ) = 0 ( ); x( ) = x 0 ( ) ; 2 [ ; 0]; (11)

where: m is a positive integer; 0 < 1 < 2 < : : : < m = are the arbitrary (non commensurate) delays;t 2 [0; + 1 ); x(t ) 2 IR n ; (t) 2 IR d ; n; d are positiveintegers; x0 and 0 are functions in C ([ ; 0]; IR n ) andC ([ ; 0]; IR d ), respectively; A is a continuous functionfrom IR d(m +1)+ nm to IR d ; B is a continuous functionfrom IR d (m +1)+ nm to IR N . Assume that A (0; : : : ;0) = 0,and B(0; : : : ; 0) = 0 ; thus ensuring that (t ) = 0 ; x(t ) =0; for every t 0, is the trivial solution of the system

(10)-(11) corresponding to zero initial conditions. Wealso impose the following hypothesis (see Remark 2.1 in[13]).

H 1 ) The functional A : C ([ ; 0]; R n ) C ([ ; 0];R d ) !R d , given, for 2 C ([ ; 0];R n ), 2 C ([ ; 0]; R d ),by

A(; ) = A (( 1) ; : : : ; ( m );(0) ; ( 1) ; : : : ; ( m ))

(12)

is such that, for any ( ; ) 2 C ([ ; 0];R n ) C ([ ; 0];R d ), there exist a neighborhood of ( ; )and a positive real L( ; ) such that, for all (; 1 ); (; 2 ) in that neighborhood, the inequality holds

j A( ;1 ) A (; 2 )j L ( ; ) k1 2 k1 (13)

From the hypothesis H 1 it follows that the system(10)-(11) admits a unique solution ( t)x (t ) on a maxi-mal time interval [0 ; b), 0 < b + 1 , with ( t) locallyabsolutely continuous and x(t ) continuous. Moreover, if b < + 1 , then (t) is unbounded in [0 ; b).

In the following it will be useful to consider the sec-ond equation in (10) rewritten as

x(t) = B(x(t 1 ); : : : ; x (t m ) ; u(t )) ; (14)

where the input u( t) 2 IR (m +1) d takes the place of the terms (t ); ( t 1 ); : : : ; (t m ). Moreover,for 0 < s < + 1 , we will indicate with M s the setf u : [0; + 1 ) ! IR ( m +1) d such that kuk1 < s g.

The system (10) is a descriptor system with delays(see [2]). Actually it can be rewritten as

E _(t )_x(t ) = 264A (x(t 1 ); : : : ; x(t m );

(t ); (t 1 ); : : : ; ( t m )) x( t) + B(x (t 1 ); : : : ; x(t m ) ;

(t ); (t 1 ); : : : ; ( t m ))

375

;

(15)

where E = I d d 0d n0n d 0n n .In the following, the functions t 2 C ([ ;0]; R d )

and x t 2 C ([ ; 0]; R n ) are given, as usual (see [5]), byt ( ) = (t + ), xt ( ) = x (t + ), t 0; 2 [ ; 0].

V. STABILITY OF CDDDEs

For stability and asymptotic stability denitionsof coupled delay dierential dierence equations see[5,9,13,14,15]. For global asymptotic stability we mean,as usual, stability and global attractivity.


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For a continuous functional V : C ([ ; 0]; IR n + d ) !IR + , dene (see [2,5])

_V 12 = limsuph ! 0+ 1h V hx h V 12 ;(16)

where tx t ; t 0; is the solution of system (10)with initial conditions 0 = 1 2 C ([ ; 0]; IR d ), x0 =2 2 C ([ ;0]; IR n ). Since the equations of system (10)are satised also for t = 0, 1 and 2 must satisfy

2 (0) = B(2 ( 1 ); : : : ; 2 ( m );1 (0) ; 1 ( 1 ); : : : ; 1 ( m ))

(17)

Theorem 4: Assume that, for any given positive real s, the continuous time dierence equation ( 14) is input-to-state s table, with respect to inputs u 2 M s . Fur-ther assume there exist a continuous functional V :

C ([ ; 0]; IR n + d ) ! IR + , functions , and of classK 1 , such that:i) for every 1 2 C ([ ; 0]; IR d ) , every 2 2

C ([ ; 0]; IR n ) , with

2 (0) = B(2 ( 1 ); : : : ; 2 ( m ) ;1 (0) ; 1 ( 1 ); : : : ; 1 ( m )) ;

(18)

the following inequalities hold

(k1 (0) k) V 12

12

1;

_V 12

1 (0)2 (0)

;

( 19 )

ii) the function w(t) = V txt is locally absolutely continuous in [0; b) , for txt satisfying ( 10) in a maximal time interval [0; b), 0 < b + 1 .

Then, the origin of the system ( 10) is globally asymp-totically stable.Proof. By Lemma A.1 in [6], we may assume withoutany loss of generality that t he function is continu-ously dierentiable. From ( 19 ), taking into account ii ,

it follows that, for t 2 [0; b),

(k( t) k) V tx t V 0x 0 Z

t

0

(s)x(s )ds

0x0 1

(20)

From (20) it follows that b = + 1 (otherwise (t) wouldbe unbounded in [0 ; b)), and that ( t) can be as small

as desired provided the initial conditions are sucientlysmall. From the hypothesis o f input-to-state stabilityof (14), it follows that the origin of the system (10) isstable. As far as the global attractivity is concerned, letus note rst that, since the inequalities (20) hold glob-ally and since, for any given positive real s, the continu-

ous time dierence equation (14) is input-to-state stablewith respect to inputs u 2 M s , for any initial condi-tions in (10), the correspondent solution is bounded in[0; + 1 ). From (20) it follows that

limt ! + 1 Z

t

0

(s)x(s )ds V 0x 0 (21)

From (21) it follows that the function

t ! Z t

0(k[ ( s) ]k) ds (22)

admits a nite limit as t ! + 1 . We claim thatlim t !1 (k(t)k) = 0. For, let us consider the deriva-tive with respect to time of the function t ! ( k(t )k).The following equality/inequality hold

d(k(t) k)

dt =

d(k(t )k)dk(t )k

T ( t) _(t)

pT (t )(t)

d( k(t)k)

dk( t) k k _(t)k

(23)

From the boundedness of the solution and the continu-ity of the functional A, it follows that _( t) is bounded in[0; + 1 ). Since the function is continuously dieren-tiable, and the solution is bounded, it follows that thefunction t ! d (k ( t )k )d k( t )k is bounded in [0; + 1 ). There-fore, since its derivative is bounded in [0 ; + 1 ), the func-tion t ! ( k(t )k) is uniformly continuous in [0 ; + 1 ).From this fact, taking into account that the function(22) admits a nite limit, by invoking the Barbalat'sLemma, it follows that lim t ! + 1 (k( t) k) = 0 a n dtherefore that lim t ! + 1 k (t ) k = 0.

As far as the proof that lim t ! + 1 k x( t) k = 0 is con-cerned, taking into account the time invariant characterof the equation (14) and its ISS property, the following

inequality holds with suitable function of class KLand function of class K

kX (k)kF (kX (k0 )kF ; k k0 ) + (kU [k 0 ;k ]kB ; 1 );(24)

where: k0 ; k are positive integers, k k0 ; X , U are thevariables obtained for the equation (14) as describedin (6); U [k 0 ;k ]() = U () for k0 k, and is = 0elsewhere. Now, let be a positive real. Since in theequation (14) the role of the input u is played by thesolution variable , it follows that there exist a k0 such


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that (kU [k 0 ;k ]kB; 1 ) < = 2, for any k k0 . Moreover,since is a KL function, it follows that there exist ak1 k0 such that (kX ( k0 )kF ;k k0 ) < = 2, for anyk k1 . Therefore, for k k1 , the inequality holds

kX (k)kF < = 2 + =2 = ; (25)

and the proof of the theorem is accomplished. ut

Remark 5: Note that the argument of the function in the inequalities ( 19 ) involves only 1 . This is veryhelpful in order to avoid, in the computation of _V , thederivative of the variable x(t ) which may well non exist,and in order to obtain the absolute continuity of thefunction t ! w(t) (see the example section). Moreover,only the value of 1 at 0 is involved, which weakensthe lesser bound condition as in the classical Lyapunov-Krasovskii Theorem for RFDEs.

Analogously, a version of Theorem 4 concerning local

asymptotic stability can be obtained (the proof is notreported here for lack of space).Theorem 6: Assume that there exist a positive real ssuch that the continuous time dierence equation ( 14)is input-to-state stable with respect to inputs u 2 M s .Further assume there exist a positive real , a continu-ous functional V : C ([ ; 0]; IR n + d ) ! IR + , functions, and of class K 1 , such that:i) for every 1 2 C ([ ; 0]; IR d ) with k1 k1 < , and

every 2 2 C ([ ; 0]; IR n ), with

2 (0) = B(2 ( 1) ; : : : ; 2 ( m );1 (0) ; 1 ( 1 ); : : : ; 1 ( m )) ;

(26)

the following inequalities hold

(k1 (0) k) V 12 12 1

;_V 12

1 (0)2 (0) ;

( 27 )

ii) the function w(t) = V txt is locally absolutely continuous in [0; b) , for txt satisfying ( 10) in a maximal time interval [0; b), 0 < b + 1 .

Then, the origin of the system ( 10) is asymptotically stable.

VI. ILLUSTRATIVE EXAMPLE

Let us consider the following coupled delay dierentialand continuous time dierence equation

_(t ) = 3 (t ) + ( t 1 )x( t 2 );x(t) = 0 :5x (t 1 ) + (t)x( t 2 ); (28)

where (t); x(t) 2 R.Let us prove rst that the dierence equation in (28)

is ISS with respect to suitable small (). Let us con-sider, for instance, the case min = 1 and k2 = 2.Other cases can be treated analogously. Since n = 1and k2 = 2, the state vector X of the system (6) con-

sists of three scalar functions, that is

X (k) = 24

1 ( k)2 ( k)3 ( k)

35

2 F ((0 ; min ];IR 3 ); (29)

and U ( k) 2 B ((0 ; mi n ];R 3 ) (take into account that therole of the input is here taken by (t)). Let us choosethe following Lyapunov functional

V (X ) = supi =1 ; 2;3

sup 2 (0 ; min ]

ki ( ) k; (30)

and let us apply Theorem 3. Let () 2 M s , with 0 p1 , 2q1 > p 1 , 3 p1 > 2q1 + 2s1 ; 32 p2 >q2 + s2 ; 12 p2 < q2 . A solution is p1 = 1, p2 = 3, q1 = 1,q2 = 2, s1 = 1 =3, s2 = 4 =3.

Therefore, the system (28) is asymptotically stable.Remark 7: The asymptotic stability of system (28)cannot be checked by means of methods based on rstorder approximations, since the linear approximation of system (28) is not asymptotically stable.

VII. CONCLUDING REMARKS

In this paper, we have employed the notion of input-to-state stability to establish a new Lyapunov-Krasovskiimethodology for a general nonlinear class of coupleddelay d ierential dierence s ystems. The obtained re-sults are a signicant extension of earlier results on lin-ear descriptor time-delay systems (see [2]). Nonlinearneutral systems in the Hale's form can be studied by

the methodology here proposed as well, by means of atransformation into coupled delay dierential dierencesystems (the linear case has been studied in [2] too).

REFERENCES

[1] A. Bellen, N. Guglielmi, A. E. Ruehli, Methods forLinear Systems of Circuit Delay Dierential Equa-tions of Neutral Type, IEEE Transactions on Cir-cuits and Systems - I: Fundamental Theory and Ap-plications , Vol. 46, No. 1, pp. 212{216, January1999.

[2] E. Fridman, Stability of linear descriptor systemswith delay: a Lyapunov-based approach, Journal of Mathematical Analysis and Applications , 273, pp.24{44, 2002.

[3] A. Germani, C. Manes, P. Pepe, Input-Output L in-earization with Delay Cancellation for Nonlinear De-

lay Systems: the Problem of the Internal Stability,International Journal of Robust and Nonlinear Con-trol , (Special Issue on Time Delay Systems), Vol. 13,No. 9, pp. 909{937, July 2003.

[4] K. Gu, V. L. Kharitonov, J. Chen, Stability of TimeDelay Systems , Birkhauser, 2003.

[5] J. K. Hale and S. M. Verduyn Lunel, Introduction toFunctional Dierential Equations , Springer Verlag,1993.

[6] Z.-P. Jiang, I.M.Y. Mareels, Y. Wang, A LyapunovFormulation of Nonlinear Small Gain Theorem forInterconnected ISS Systems, Automatica , Vol. 32,No. 8, pp. 1211{1215, 1996.

[7] Z.-P. Jiang, Y. Wang, Input-to-state stability fordiscrete-time nonlinear systems, Automatica , Vol.37, No. 6, pp. 857{869, 2001.

[8] V. Kolmanovskii and A. Myshkis, Introduction to theTheory and Applications of Functional Dierential Equations , Kluwer Academic Publishers, 1999.

[9] S.-I. Niculescu, Delay Eects on Stability, a Robust Control Approach , Lecture Notes in Control and In-formation Sciences, Springer, 2001.

[10] T. Oguchi, A. Watanabe and T. Nakamizo, Input-Output Linearization of Retarded Non-linear Sys-tems by Using an Extension of Lie Derivative, Int.J. Control , Vol. 75, No. 8, 582{590, 2002.

[11] P. Pepe, The Liapunov's Second Method for Contin-uous Time Dierence Equations, International Jour-nal of Robust and Nonlinear Control , Vol. 13, No.15, pp. 1389{1405, December 2003.

[12] P. Pepe, Adaptive Output Tracking for a Class of Nonlinear Time Delay Systems, International Jour-nal of Adaptive Control and Signal Processing , Vol.18, N. 6, pp. 489-503, August 2004.

[13] P. Pepe, E. I. Verriest, On the S tability of CoupledDelay Dierential and Continuous time DierenceEquations, IEEE Transactions on Automatic Con-trol , vol. 48, No. 8, pp. 1422{1427, August 2003.

[14] P. Pepe, On the Asymptotic Stability of CoupledDelay Dierential and Continuous Time DierenceEquations, Automatica , Vol. 41, No. 1, pp. 107{112, January 2005.

[15] V. Rasvan and S. -I. Niculescu, \Oscillations in Loss-less Propagation Models: a Liapunov-Krasovskii ap-proach", IMA Journal of Mathematical Control andInformation, Vol. 19, 157{172, 2002

[16] E.D. Sontag, Smooth Stabilization Implies CoprimeFactorization, IEEE Transactions on AutomaticControl, Vol. 34, pp. 435{443, 1989.


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A NEW LYAPUNOV-KRASOVSKII METHODOLOGY FOR COUPLED DELAY DIFFERENTIAL DIFFERENCE EQUATIONS (2006).pdf