IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993 157

On Stability of Cellular Neural Networks with Delay Pier Paolo Civalleri, Marco Gilli, and Lucian0 Pandolfi

Abstract-It is known that symmetric cellular neural networks (CN”s) are completely stable. In this paper it is shown that CN”s with delay @CN”s), though symmetric, can become unstable if the delay if suitably chosen: actually such networks can exhibit periodic cycles. Moreover, a suflicient condition is presented to ensure complete stability: such a condition estab- lishes a relation between the delay time and the parameters of the network.

1. INTRODUCTION

ELLULAR neural networks (CNN’s) were introduced by C L. 0. Chua and L. Yang [l], [2] in 1988: in comparison with general neural networks, they have the most appealing property of being easily realizable either as planar or as multilayer structures. They have found important applications in signal processing, especially in static image treatment. The stability of such networks has been investigated in [l], [3], and [4]; in [l] it was proved that symmetric CNN’s are completely stable; in [3] a weaker property (stability almost everywhere) was established for the class of the positive cell linking templates and this result has been extended in [4], by means of equivalent transformations.

Processing of moving images requires the introduction of delay in the signals transmitted among the cells [6]. The study of stability in this case is much more difficult than for conventional CNN’s. In [7] it has been proved that positive cell linking templates are stable almost everywhere. In this paper, we show that a CNN with delay (DCNN), though symmetric, can become unstable if the delay is suitably chosen, and we give a sufficient condition in order that complete stability is ensured.

The elements of CNN theory, which are needed to make the paper self-contained, are briefly resumed in Section II. The main stability results, namely the possibility that a symmetric DCNN could become unstable and a sufficient condition for global stability, are presented in Section III. The theoretical results are illustrated by a simple example, consisting of a two-cell DCNN, in Section IV. Finally, Section V is devoted to the conclusions.

We use capital letters to denote matrices, and lower case letters to denote vectors or scalars; the difference will result from the context. Transposition is indicated by an apex. Dif-

Manuscript received May 18, 1992. This work was supported in part by Consiglio Nazionale delle Richerche, Rome, Italy, under Grant 89.04977.07.

P. Civalleri and M. Gilli are with Politecnico di Torino, Dipartimento di Elettronica, 1-10129 Torino, Italy.

L. Pandolfi is with Politecnico di Torino, Dipartimento Di Matematica, I- 10129 Torino, Italy.

IEEE Log Number 9208121.

ferentiation with respect to a variable is occasionally denoted by a dot.

11. CNN’S AND DCNN’S The state equation of a CNN, composed by M x N cells, [ 11,

after having ordered the cells in some way (e.g., by columns or by rows), can be written as

j : = - Z + A g + B U + I (1)

where 5, X€RMxN state vector and its derivative,

g € R M x N output vector, depending by 2 through the saturation function defined in [l].

UE RM input vector, I € R M x N vector, representing the bias current,

Rx, C resistance and capacitance, respectiyely: of each cell are assumed equal to 1 A, B E RM N , M depend on the established order among the cells and on the cloning templates.

A CNN is said symmetric or reciprocal if A is a symmetric matrix.

CNN’s with delay T (DCNN’s) were introduced in [6]. By assuming that the input of each cell is constant, they are described by state equations of the form:

c ( W ) € N 7 ( & d

by output equations:

(3) 1

Vy, i j = +,ij + 11 - lVx, i j - 11)

and input equations:

= Eij = const. (4)

Nr(i, j ) represents the neighborhood of order T of the cell c( i , j ) defined in [l]. For DCNN’s, the space invariance property is expressed by

A(i , j ; k, I ) = A(i - I C , j - 1 )

B(2, j ; I C , 1) = B(i - I C , j - 1) AT(Z, j ; k, 1) = A7(i - k, j - 1).

( 5 ) (6)

The latter matrix A‘, in the space-invariant case, defines the delay cloning template [6].

1057-7122/93$03.00 0 1993 IEEE

~

158 IEEE TRANSACTIONS ON CIRCUITS AND S Y S T E M S 1 FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993

The state equations (2), by ordering the cells and assuming R, = C = 1 can be rewritten in the more compact form:

k( t ) = - ~ ( t ) + A o ~ ( t ) + - T ) + BU + I (7)

where matrices A. and A1 can be easily calculated from the cloning templates and the delay cloning template.

We say that a DCNN is symmetric if the correspondent CNN, obtained by putting T = 0 is symmetric. This requires the symmetry of the matrix A0 + Al but not the symmetry of matrices A0 and A1 separately.

Equation (7) is a particular case of the general functional differential equation [ 121:

x = f(t, X t )

where: zt E C([--7, 01, R M x N ) is a continuous mapping of the interval [-T, 01 into R M x N according to the following definition: x t ( e ) = x ( t + 0); and f E C(C([-T, 01, RMxN), R M x N ) is a continuous mapping of the space of functions xt into RMxN.

Using the above notation, (7) is rewritten as:

X = - X t ( O ) + Aoyt(0) + AlYt(-T) + Bu (8)

where yt(e) is the image of .,(e) under mapping (3).

m. STABLlTYPROPERTIES

We study the stability properties of symmetric DCNN’s described by (7). To simplify the proofs, we assume at the outset that U = 0; at the end, we will show that our results are still valid if U = const, according to (4).

De$nition I: A dynamical autonomous system is said to be completely stable if and only if, for any initial point in the state space 20, the (unique) forwards trajectory ~ ( t , 20) tends to a constant, i.e.,

limt+.mx(t, 2 0 ) = const.

According to our previous notations, the state space is R M X for a system without delay and C([--7, 01, R M x N ) for a system with a T-delay.

The stability of a symmetric CNN without delay, described by (l), has been proven in [ l ] by introducing a suitable Lyapunov function, i.e., a mapping of RMxN into R. When (1) is replaced by (7), such a function can be replaced by a suitable Lyapunov functional, i.e., by a mapping of C([--7, 01, R M x N ) into R [12].

De$nition 2: We define the following Lyapunov functional: 0

V ( 4 = Y’(t>PY(t> - J [Y’O + e> - Y’(t)l -7

.A”Y(t + e) - Y(t)l de (9)

where f(e) is any scalar function continuous with its deriva- t iveon[--7,0] , fEC1([-~,O], R ) a n d P = - I + A o + A l .

By means of such a functional, we can find a sufficient condition for the stability of DCNN’s in terms of the euclidean norm of AI,

and of the delay T . Its proof, however, needs the introduction of some lemmas.

L e m I: If P is a symmetric matrix and if l[AlII < ( 2 / 3 ~ ) , it is possible to find a function f(e) such that V ( x t ) 2 OVt.

Pro08 Consider the expression (9) of V(z t ) . After a change of variable w = t + 8, V ( z t ) is rewritten as

V ( 4 = Y’WYW - J’ [Y’(W> - Y W l t-7

.A:f(w - t)Al[Y(w) - Y ( 4 l dw. (10)

We differentiate (10) with respect to t:

i . ( X t > = Y’WY(t) + Y‘(WY(t) + [y’(t - T ) - y’(t)]A’,f(-~)Ai

. [Y(t - .) - Y(t)l

. [ Y ( 4 - Y@)l dw

+ L,[$(w) - y’(t)lA’,f(w - 4 - 4 1

+ L + 11,

Y’(t)A’,f(w - t)Al[Y(w) - Y(t) l dw

[y’(w) - y’(t)]A’lf(w - t)AlY(t) dw.

(11)

Equation (11) contains the time derivative of y(t). Let yh be a component of the output y and xh the corresponding component of the state 2. Because of (3), whenever Ixhl 7 1, yh is constant. Thus, we have

The equations above can be expressed in a synthetic form by means of a matrix N ( z ) , depending on the state, defined as follows:

{ withnh = 1 ifflzhl < 1, nh = Oifflshl > 1.

$( t ) = NHy(t) + NAly(t - T )

N ( x ) = diag {nh}

Using N ( x ) , (7) can be rewritten as

(12)

with

H = - I + A o .

Substitution of (12) into (1 1) yields:

V ( q ) = y’(t)H’NPy(t) + y’(t - .r)A:NPy(t) + y’(t)PNHy(t) + Y’(t)PNAiy(t - T )

+ [Y’(t - T ) - y’(t)lA’,f(--7)A1

. [Y(t - .> - Y ( 4 l

. [Y(W) - Y(t)l dw

[ Y ’ W - y’(t)lA’,f(w - t)Al + L, + 11,y’(t)H’NA:f(w - t)Al

CIVALLHU et d: STABILITY OF CNN’S WITH DELAY 159

After a simple manipulation, by inserting all terms under the integral sign, we obtain

~ ’ ( t , e) = (y’(t) y ~ t - 7) - y ~ t ) Y’O + e) - Y W .

Thus, for ensuring positivity of V ( x t ) for any possible state, it is sufficient to prove that QN(t9) is positive definite for all 0’s in the interval [-7, 01 and for all values of matrix N , that is, for all combinations of cells working in the linear and nonlinear region of the state-output relation. The representation of the quadratic form S ( t , S) can be simplified if P is symmetric and the follow&g new variables are introduced:

We obtain S( t , 0) = [’(t , e ) M ~ ( e ) < ( t , 0) where: E NAif(6)

) MN(6) = - df! NAif(0) . (15) ( f ( O z 1 i V f(efA1N f(e)

The proof proceeds in two steps, presented as Lemmas 1.1 and 1.2. Lemma 1.1: If MI(0) (that is, k f N ( e ) for N = I ) is not

negative VB E [-7, 01 and (f(-~)/.) > 0, then MN(O) is not negative VN.

Proof: In the following we omit to indicate the depen- dence of E on t and 8. Consider any vector 11, E R M x N . It can be represented as the direct sum of a vector in ker N and of a vector in the orthogonal complement of ker N:

{ where N 1 1 , ~ = 0 and N11,1 = $1, with $1 I 11,~. $ = $ K + $ I

Let us decompose accordingly vectors (14). We obtain:

and 52 in equations

f(-) S( t , e) = &MI(e)(A + ( ; K 7 ‘ $ 2 K (l6)

where

ra = ( G I G I $1. If f( -T)/T is chosen positive and MI is not negative, then the quadratic form based on M N , as the sum of two nonnegative numbers, is nonnegative, and therefore, MN is nonnegative definite. Q.E.D Lemma 1.2: If llAlll < 2/37, then there exists an

f(e) such that MI(6) is nonnegative VB E [-7, 01; and r>MI(O)<A = 0 implies E3 = 0.

Proof: With reference to ( 15) and (16) we obtain

E > M I ( W A

= t;ITtiI + & 7 1 < 2 1 + ~if(e)1t3

+ F;1;121+ &;&I + E;IA;f(0)<3

21 f(--7)

I I

+ <;f(e)A1<lI + &A’lf(e)r3

, I

IEEE TRANSAClTONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993

If there exists f(0) E C1[--7, 01 such that

L

(19) then the thesis of Lemma 1.2 is true. In fact, Mr(8) is nonnegative; if <LMr(e)<A = 0, the last term of the sum (17) must be zero; thus from (19) derives that t3 = 0.

It is therefore sufficient to prove that the hypothesis of the lemma implies inequality (19) and that inequalities (18) and (19) are mutually compatible. The left-hand term of inequality (19) satisfies the following inequalities:

Thus to prove inequality (19), it is sufficient to verify that

We assume

where c > 0 to insure the continuity of function (24) in [-T, 01. The derivative of function (24) shows that inequality (23) is always verified for such a function if 1 < 1, with equality when 1 = 1. Thus, the only constraint c > 0 has to be satisfied; it implies for 0 = -T:

r > 0. 1 --

kf (-TI

By replacing the expression of k, after some manipulations, the following is obtained:

(llAlllTf(-T>>2 + U(-.)( ~ I I A ~ I I ~ T ~ - 2

Inequality (25) is satisfied if

Since the upper bound of 1 is 1, (26) can be rewritten as

Thus we have proven that the assumption of the lemma (equivalent to inequality (27)) implies inequality (19). Finally, ' we must verify that inequalities (25) and (18) are mutually compatible; we find that this happens when

(28) 41

llA1Il2T2 < 5

llA11I2T2 < 5.

and considering the upper limit of 1

4 (29)

Comparing inequalities (27) and (29), we find that the thesis of the lemma holds true if

This completes the proof of Lemma 1.2. Q.E.D The proof of Lemma 1 follows immediately from Lemmas

1.1 and 1.2. Q E D Lemma 2: V ( z t ) is bounded for every t 2 0.

Proof: It is sufficient to observe that the integrand func- tion in Definition 1 of V ( z t ) only depends on the output vector y, whose components are bounded due to equation (3), and that the integral is taken over a finite interval. Q.E.D Lemma 3: The Lyapunov functional has a constant limit

when time tends to infinity:

limt++,V(zt) = const.

Proof: The statement follows from the fact that V(zt) 2 Q.E.D

Lemma 4: If A1 is invertible, the output vector y has a 0 (Lemma 1) and V(zt) is bounded (Lemma 2).

constant limit for time t tending to infinity:

limt,+,y(t) = const.

Proof: An intuitive, but not rigorous proof of the state- ment is the following. From the fact that V(zt) is bounded for every t (Lemma 2) and has a constant limit at infinity (Lemma 3), it follows that limt++, V(zt) = 0. But we know that V(zt ) = J f 7 S ( t , 4) de. From expression (16) of S(t , e) we see that it must be limt++, [LMI(e)& = 0 and by using Lemma 1.2 one derives limt4+, &(t) = 0. The third of equations (14) gives limt++, Al[y(t + e) - y ( t ) ] = 0, or, under the assumption that A1 is invertible, limt++, y ( t ) = const. In Appendix I a more rigorous proof is given. Q.E.D

We can now prove the first of our main results. Theorem I: A sufficient condition for limt+, y ( t ) =

const for any initial condition E C([--7, 01, R M x N ) , is that A1 is invertible, llAlll < 2/37 and P symmetric.

Pmof: The statement follows immediately from Lemmas 1 4 .

We must now consider the more general case in which the components of the input vector U of the DCNN are nonzero constants, according to (4). To this aim, rewrite (7) as

k ( t ) = -z( t ) + Aoy(t) + Al?/(t - T) + 2,

CIVALLERI er al.: STABILITY OF CN"S WITH DELAY

with w = Bu. The above equation can be transformed in the following way:

2 = -(x + P - ~ v ) + Ao(y + P - ~ w ) +Ai [y ( t - T ) + P - ~ W ]

since P = -I + A0 + Al. By introducing the new variables

Zl(t) = z(t) + P-lu 4 t ) = y(t) + P-lu

the equation takes the form

which is identical to (7). Thus Theorem 1 still holds for the case of constant input.

The case in which the delays among the cells are not all equal is treated in much the same way: the statement of Theorem 1 remains valid by assuming for T the largest delay in the network.

Theorem 1 ensures simply the fact that the output vector is asymptotically constant. In Theorem 2 it is claimed that such a condition implies also the state stability.

Theorem2: A sufficient condition for limt+oo z ( t ) = const for any initid condition E C([-T, 01, PxN), is that A1 is invertible, llAlII < 2/37 and P symmetric.

Pro08 see Appendix 11.

N. EXAMPLE We have found a sufficient condition to ensure the complete

stability of a delay symmetric DCNN. The condition requires the euclidean norm of the matrix A1

to be less than 2/37, where T is the delay. Now we want to develop a simple example to show that

a symmetric CNN can become unstable and to show that our condition preserves the complete stability of the system. Consider the following dynamical equations of a (2 x 1)-

where

and h = uyl - 1. The eigenvalue equation for an arbitrary delay 7:

det (H - X I + A1 exp (-AT)) = 0

yields

[(h - A) + ut1 exp ( - - X T > ] ~

161

(32)

-[uy2 + ui2 exp ( -xT) ]~ = o which can be rewritten as

f l ( X ) . f2(A) = 0

~ I ( x ) = ( h - up2 - A) + (utl - ai2) exp ( -AT) f 2 ( ~ ) = ( h + uy2 - A) + (utl + ai2) exp (-AT).

If there exists a value, X = ju0, WO E R for which either f l ( X ) = 0 or f2(X) = 0, then a solution of equations (30) is a sinusoid with radian frequency equal to WO. In other words the system admits a closed orbit, that can be either stable or unstable depending on whether the signs of the real parts of the infinitely many eigenvalues, solutions of (32), are all negative or at least one of them is positive. In both cases the system is not completely stable, but in practice only the first must be avoided to assure a correct behavior of a DCNN. It is quite difficult to find all solutions of (32), but it is possible to find conditions to ensure that no eigenvalue has positive real part. Equations of the type f l ( X ) = 0 or f2(X) = 0 have been studied by many authors; a detailed discussion is reported in [12]. Consider an eigenvalue equation of the type:

(33) X = -a - bexp (-AT). The region, in the plane a, b, where the real part of each eigenvalue, solution of (33), is negative, is shown in Fig. 1; the upper boundary of this region is given parametrically by the equations:

where

U = -bcos<T C = b sin Cr

x O < < < - .

r f l ( X ) and f2(X) can be rewritten, according to (33):

f l ( X ) = X + a1 + bl exp(--XT) f2(X) = X + u2 + bZexp(---XT)

where

a1 = ay2 - h 1 1 bl = U 1 2 - a,,

= - 4 2 - h

162 EEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 3, MARCH 1993

In order to show the existence of a stable closed cycle we choose parameters a2, bz of the equation f2(A) = 0, as shown in Fig. 1, where az has to satisfy the constraint

1 -- < a2 < 0. 7

In order to assure h > 0, a:2 has to be chosen so that

1 a& < -.

7-

The choice of a2, bz ensures that there exists an imaginary solution of f 2 ( A ) = 0 and there is no solution of f 2 ( A ) = 0 with positive real part. From the above equations, a1 and bl are defined in terms of a2 and bz as follows:

a1 = a2 + 2ay2 bl = b2 + 2&.

By choosing values of ay2 and ai2 not too large and such that:

ayz > 0

ai2 < o also equation f l ( A ) = 0 has no solution with positive real part. In fact it is shown in Fig. 1 that a1 and bl are located in the region of the plane a, b, where all the eigenvalues have negative real part. Therefore, the imaginary value of A, for which f 2 ( X ) = 0, ensures the presence of a cycle, which is stable, because all the other solutions of (32) have non positive real part. The above value of A can be easily found, by solving the following equation:

j w o = -a2 - b2 cos WOT + j b p sin wor

which implies

W O 7 = arccos (2) WO" = bq - a;. (34)

It can be verified that the previous equations hold if

arccos (2) 1 7 - =

bi - a i ' & and since the norm of the matrix A1 is equal to b2, this contradicts, as one expects, the condition of stability given in Theorem 1.

We recapitulate: an initial condition of the type zO(0) = vcos(woO), where Y is the eigenvector associated to the imaginary eigenvalue, generates a solution of equations (31) which oscillates in time: the amplitude of the vector v can continuously vary from 0 to 1.

In particular: a symmetric DCNN can have either unstable closed orbits (it is sufficient to choose a value of r and WO. satisfying (34)) or stable closed orbits (if uy2 and aip are suitable chosen, according to the previous example); a symmetric DCNN, satisfying the conditions stated in Theorem 2, can have neither unstable closed orbits nor stable closed orbits;

"1 /

/ /

Fig. 1. Region of the plane a, b, where all the eigenvalues of (33) have negative real part.

a positive irreducible symmetric DCNN, belonging to the class of the positive irreducible networks [7], can have unstable closed orbits, but not stable ones (in our example this corresponds to choose ai2 positive).

At the end of this example we notice that the evaluation of the norm of the matrix A1 depends only on the delay template of the cell and not on the number of cells, because we choose the euclidean norm. A rigorous and simple proof of that is given in Appendix III. The assumption about the invertibility of the matrix A I , is verified in almost all network cases, proposed in the literature; only some pathological types do not satisfy this condition.

V. CONCLUSIONS An ordinary symmetric CNN is known to be completely

asymptotically stable. In this paper we have shown that, if the signals exchanged among cells are delayed, i.e., the network is a DCNN, then state and response can exhibit (bounded) oscillations. We have also shown that, for a network whose cells are specified, complete asymptotic stability is ensured provided the delay is less than a bound only depending on the cell parameters; this agrees with the fact that for zero delay a DCNN degenerates into a CNN, for which the property is true. The first result has been proven by an example while the second has been shown by actually constructing a suitable Lyapunov functional, which works both in linear and nonlinear region.

APPENDIX I

In this appendix we prove Lemma 4: that is, we prove that the limit limt--r+oo y ( t ) exists. In [l] it is shown that any solution z of (1) is bounded, the same result is easily extended to (7). From the latter, it then follows that also its derivative 2 is bounded. This implies that the incremental quotient of z is bounded. The function t + y ( t ) is not everywhere differentiable, but the definition of y ( t ) in terms of ~ ( t ) implies

I .

CIVALLERl et al.: STABILITY OF CNN‘S WlTH DELAY 163

that the incremental quotient of y is also bounded. This is important enough to be stated explicitly:

and the quotient &(tn, vn)-c(tn, eo)/(v, -6’0) is unbounded for t + -W. This quotient is equal to:

L - L This contradicts Lemma A1 and shows that a positive number L as above cannot be found. Hence, limt++, &(t, e ) = 0 for each 8. Under the assumption that A1 is invertible, Lemma 4 follows.

where 5 is defined in (14). BY writing down the

with (16)), we see that the following inequality holds for some a > 0:

Of (d /d t )V(z t )

APPENDIX

In this appendix we will prove the thesis of Theorem 2, i.e., that, We know that t + V(z,) is bounded so that its derivative

(being positive) is integrable. Consequently, if limt,, y ( t ) = const (the output of each cell tends to a constant:

admits of a finite limit for t + +W.

The derivative of the function F is the function t +

J!T 11 53 (t , e ) 11 d0 whose incremental quotient is bounded, from (35). Hence it is uniformly continuous and Barbalat’s Theorem ([ 1 1, p. 6201) implies that

limt++, ~ ’ ( t ) = Ilt3(t, e)l12 de = 0. Lr We recapitulate: we have proved that for each E > 0 there exists a number T, such that

Ile3(t, e)1I2 de < E fort > T,. (36) f r

This inequality has the following important consequence: let us assume that it is possible for find a number p, a divergent sequence {tn} and a number Bo E [--7, 01 such that [I&&,, &)(l > p. The continuity of the function 0 -+

&(tn, e ) implies that the inequality Il<3(tn, eo)ll 2 p/2 holds also over an interval (U,, w,) which contains 00. The length w, - v, of this interval tends to zero for n -+ +CO.

~n fact, f r l l < 3 ( t n , e)112 de 2 ~ . . ~ l b ( t n , e)ii2 de L ( ~ / 2 ) ~ ( w , , - v,). The left-hand side tends to zero, from the statement above.

With this observation in mind, we prove more. We prove that for each 8 E [-T, 01 we have limt,+, Il&(t, 6)ll = 0. The definition of ,5(t, e ) then implies that the limit of y ( t ) exists, if det A1 # 0.

Let us assume that there exists a value 00 such that lim,++, eo)ll does not exist or it is not zero. In this case we can

find a number L > 0 such that (123( tn , 0o)ll = L for each value of a sequence t, + +CO.

Let [v,, wn] be the largest interval containing 00 and con- tained in [-T, 0) such that over this interval: 11t3(tn, e)ll > L/2. We proved already that the difference w, - v, tends to zero sothat also do-w, -+ 0. Moreover, Il&(tn, v,)ll = L/2 . Now,

Ilb(tn, vn) - h ( t n 7 e ~ ) l l L 1116(tn, un)ll- lI&(tn, e ~ ) l l l = L / 2

then limt+oo z(t) = const (the state of each cell tends

Consider the dynamic equation of cell i: it can be written as: to a constant).

dx, - = -zz(t) + & j Y i ( t ) + C”:,jyi(t - T )

j j d t

= -zi(t) + d ( t ) (37)

limt+,q5(t) = 1 (38)

zi(t) = e- tz i (0) + e-(t-s)q5(s) ds. (39)

where by theorem 1 it follows:

being 1 is a constant value. The formal solution of v is:

I’ We claim that z;(t) admits a limit as the time tends to infinity and such a limit is exactly 1.

To prove that, we have to show that for any positive U there exists a Tu so that for every t > Tu Izi(t) - 11 < U . To this purpose let us write the difference q ( t ) - 1:

q ( t ) - 1 = e-,zi(O) + e-(”-”)#(s) ds - 1 (40) I’ I’ = e-tzi(0) + e-(t-s)(q5(s) - 1) ds

(41)

e-(t-s)(q5(s) - 1) ds. (42)

The absolute value of the first term Ie-,(xi(O) - I ) ( is lower than u/3, by choosing

t > T,~ = Iln(lz;(o) - 1 1 ) - In (3) I. As far as the second term is concerned, by (38) we can state that for any positive E there exists a T, so that for every t > T,,

+ (1 - e-t)l - Z

= e-,(z;(O) - 1) + I’ U

IMt) - 11 < e . Thus for t > T, we can write

Jute-(”’(O(s) - 1) ds = e-(t-s)(q5(s) - 1) ds I” r t

e-(t-s)(q5(s) - 1) ds. (43) +

-

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Since +(t) is bounded, because it depends on the output functions which are bounded, there exists a positive constant k such that I+;(t) - 1 ) < IC for every t , 0 5 t 5 T,; therefore,

by choosing

a t > Tu2 = Iln (I IC I) + In I eT‘ - 1 I - In I 11

(44)

(45)

by providing E < a/3 (e can be chosen arbitrarily small) and of course t > T,.

We recapitulate: by choosing Tu = max{Tul, TUz, T,} it is verified that for any positive a there exists a Tu so that for every t > Tu Iz;(t) - 11 < a, which is what we want to prove.

APPENDIX n I In this Appendix we show that the norm of the matrix AI,

which derives from the delay cloning template is bounded. The euclidean norm of a vector U of R” is defined as follows:

where uj are the components of U with respect to the canonical basis of R”. It is possible to show [ 131 that, given an operator T from R” to R”, its norm satisfies the following inequality:

l l ~ l l 5 (maxj ri)t (maxk r:);

= CITjkl (47)

‘r: = XITjkl. (48)

(46)

k

j

For the delay matrix AI, by using the local connectivity property, is easy to show that the sum of the moduli of each row and column is bounded by the sum of the moduli of the elements of the cloning delay template. This proves that the norm of A1 does not depend on the number of cells of the DCNN.

ACKNOWLEDGMENT The authors thank Prof. L. 0. Chua, of the University of

California at Berkeley, and Prof. T. Roska, of the University of Budapest, for their friendly and deep advice.

REFERENCES

L. 0. Chua and L. Yang, “Cellular neural networks: Theory,” IEEE Trans. Circuits Syst., vol. 35, pp. 1257-1272, 1988. - , “Cellular neural networks: Applications,” IEEE Trans. Circuits,

L. 0. Chua and T. Roska, “Stability of a class of nonreciprocal cellular neural networks,” IEEE Trans. Circuits Syst., vol. 37, pp. 1520-1527, 1990. L. 0. Chua and C. W. Wu, “The universe of stable CNN templates,” Memo UCWERL, Univ. California at Berkeley, Apr. 1991. T. Roska, T. Boros, P. Thiran, and L. 0. Chua, “Detecting simple motion using cellular neural networks,” in Proc. CNNA-90, pp. 127-138, 1990. L. 0. Chua and T. Roska, “Cellular neural networks with nonlinear and delay-type template elements,” in Proc. CNNA-90, pp. 12-25, 1990. T. Roska, C. W. Wu, M. Balsi, and L. 0. Chua, “Stability of delay- type cellular neural networks,” Memo UCB/ERL, Univ. California at Berkeley, Nov. 1991. M. W. Hirsch, “Convergent activation dynamics in continuous time networks,” Neural Networks, vol. 2, pp. 331-349, 1989. -, “System of differential equations that are competitive and cooperative II: convergence almost everywhere,” SIAM J. Math. Anal., vol. 16, pp. 423-439, 1985. H. L. Smith, “Monotone semiflows generated by functional differential equations,” J. Differential Equations, vol. 66, pp. 420-442, 1987. R. Reissig, G. Sansone, and R. Conti, Nonlinear Differential Equations of Higher Order. Leyden: Noordhoff, 1974. J. K. Hale, Theory of Functional Differential Equations. New York Springer, 1977. T. Kato, Perturbation Theory for Linear Operators. Berlin: Springer- Verlag, pp. 28-29, 1976.

vol. 35, pp. 1273-1290, 1988.

Pier Paolo Civalleri received the degree in elec- trical engineering from the Polytechnic of Turin in 1959 and the Professorship (Libera Docenza) in Network Theory from Ministry of Public Education in 1966.

From 1960 to 1970 he was a researcher and from 1971 to 1975 a Research Director at Istituto Elet- trotecnico Nazionale Galileo Ferraris, Turin, Italy. From 1967 to 1986 he was Professor of Applied Mathematics and since 1975 is Professor of Elec- trotechnics in Polytechnic of Turin. From 1975 to

1981 he was Director of the Institute of Mathematics of the Polytechnic of Turin. In 1989 he was elected a corresponding Member of the Accademia delle Scienze di Torino. He was President of the North-Italy Section of IEEE (1978-1980), President of the Turin Section of Associazione Elettrotecnica ed Elettronica Italiana (AEI) (1981-1983). He is presently a member of the Board of Directors of the same Section. He was a cofounder of and is currently a member of the Board of Directors of Centro di Studi sui Sistemi, Turin, Italy. He has been Visiting Professor in Cornell University, Ithaca, NY, in 1977,1978,1982-1984. His research interests have covered network topology, analysis and synthesis of single-element-kind networks, multilayer n-port analysis, modeling of active and passive distributed circuits, controllability and observability of linear systems, modeling of dielectric guides, broadband matching. He is presently working mainly in the field of cellular neural networks. He is the author of more than 50 scientific papers.

Dr. Civalleri was awarded the IEEE Centennial Medal in 1984.

Marc0 Gilli received a degree in electronic engi- neering from the Polytechnic of Turin on October 1989.

Since 1991 he has been a Researcher at the Department of Electronics of Polytechnic of Turin. His research activity is mainly in the field of elec- tromagnetic theory and circuit theory, especially nonlinear systems and neural networks.

,, . CIVAUEIU et al.: STABILITY OF CNN'S WITH DELAY

Lucian0 Pandolfi received a degree in mathematics from the University of Florence in July 1970 and was with Florence and Siena University until 1980.

Since 1980 he has been professor of advanced calculus at the Politecnico di Torino. The research activity of Prof. Pandolfi is in the field of linear control systems, in particular systems with delays. He is associate editor of the journal Sysrem & Control Leners and is a member of the editorial board of the journal Applied Mathemutics Leners.

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