Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 13 1-14; http://www.math.u-szeged.hu/ejqtde/

Stability Results for Cellular Neural Networks with Delays

István Győri and Ferenc Hartung Department of Mathematics and Computing University of Veszprém H-8201 Veszprém, P.O.Box 158, Hungary gyori@almos.vein.hu and hartung@szt.vein.hu

This paper is dedicated to Prof. László Hatvani on the occasion of his 60th birthday.

Abstract In this paper we give a sufficient condition to imply global asymptotic stability of a

delayed cellular neural network of the form

ẋ_{i}(t) = −d_{i}x_{i}(t) + ∑_{j=1}^{n} a_{ij}f(x_{j}(t)) + ∑_{j=1}^{n} b_{ij}f(x_{j}(t − τ_{ij})) + u_{i}, t ≥ 0, i = 1, …, n,

where f(t) = \frac{1}{2}(|t + 1| − |t − 1|). In order to prove this stability result we need a sufficient condition which guarantees that the trivial solution of the linear delay system

ż_{i}(t) = ∑_{j=1}^{n} a_{ij}z_{j}(t) + ∑_{j=1}^{n} b_{ij}z_{j}(t − τ_{ij}), t ≥ 0, i = 1, …, n

is asymptotically stable independently of the delays τ_{ij}.

keywords: delayed cellular neural networks, global asymptotic stability, M-matrix

1 Introduction

The notion of cellular neural networks (CNNs) was introduced by Chua and Yang ([5]), and since then, CNN models have been used in many engineering applications, e.g., in signal processing and especially in static image treatment [6]. As a generalization of CNNs, cellular neural networks with delays (DCNNs) were introduced by Roska and Chua [14].

In this paper we study the asymptotic stability of the DCNN model described by the system of nonlinear delay differential equations

ẋ_{i}(t) = −d_{i}x_{i}(t) + ∑_{j=1}^{n} a_{ij}f(x_{j}(t)) + ∑_{j=1}^{n} b_{ij}f(x_{j}(t − τ_{ij})) + u_{i},  t ≥ 0, i = 1, …, n. (1.1)

This research was partially supported by Hungarian National Foundation for Scientific Research Grant No. T031935.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Here n is the number of cells; x_{i}(t) denotes the potential of the ith cell at time t; d_{i} represents the rate with which the ith unit resets its potential to the resting state when it is isolated from other cells and inputs; a_{ij} and b_{ij} denote the strengths of the jth unit on the ith unit at time t and t − τ_{ij}, respectively; τ_{ij} corresponds to transmission delay between the ith and jth cells; f denotes an output function; u_{i} is an external input to the ith cell.

The stability of (1.1) and more general classes of DCNNs has been intensively studied, see, e.g., [2]–[4], [11]–[13], [15]–[18], and the references therein. We will assume throughout this paper that the output function f: ℝ → ℝ is defined by

f(t) = \frac{1}{2}(|t + 1| − |t − 1|) = \{ \begin{matrix}1, & t > 1, \\ t, & −1 ≤ t ≤ 1, \\ −1, & t < −1. \end{matrix}  (1.2)

This function is widely used in CNN and DCNN models.

In a recent paper Mohamad and Gopalsamy ([13]) have shown using fixed point method that if f is defined by (1.2) and

d_{i} > ∑_{j=1}^{n}(|a_{ij}| + |b_{ij}|),  i = 1, 2, …, n,  (1.3)

then (1.1) has a unique fixed point which is globally exponentially stable. In our Theorem 4 (see below) we show that the weaker assumption

d_{i} − a_{ii} > ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}| + ∑_{j=1}^{n} |b_{ij}|,  i = 1, 2, …, n,  (1.4)

together with another condition (see (3.11) below) implies the global asymptotic stability of the unique equilibrium of (1.1). We also conjecture (see Conjecture 1 below) that assumption (3.11) can be omitted, (1.4) itself, or even a weaker condition implies the global asymptotic stability of the equilibrium.

We remark that condition (1.4) is equivalent to saying that the matrix K = (k_{ij}) with elements

k_{ij} = \{ \begin{matrix}d_{i} − a_{ii} − |b_{ii}|, & if i = j, \\ −|a_{ij}| − |b_{ij}| & otherwise \end{matrix}

is diagonally dominant and it has positive diagonal elements. We recall that an n × n matrix K = (k_{ij}) is (row) diagonally dominant, if

|k_{ii}| > ∑_{{j=1, \atop j≠i}}^{n} |k_{ij}|, i = 1, …, n.

Our condition (1.4) is similar to that given by Takahashi in [15], where it was shown that if d_{1} = d_{2} = ⋯ = d_{n} = 1 and the n × n matrix W = (w_{ij}) with elements

w_{ij} = \{ \begin{matrix}a_{ii} − 1 − |b_{ii}|, & if i = j, \\ −|a_{ij}| − |b_{ij}| & otherwise \end{matrix}

is a nonsingular M-matrix (see definition below), then every solution of (1.1) tends to a constant equilibrium, i.e., the system is completely stable. Clearly, condition (1.4) implies that d_{i} −a_{ii} > |b_{ii}|, so in this case W can not be an M-matrix. Similarly, if W is an M-matrix, then (1.4) can not hold, therefore the two conditions cover disjoint cases. We comment that despite the similarities of the two conditions, the proof of our result requires a different technique than that used in [15]. Our results were motivated by the monotone technique we used in [9], where we studied the scalar version of (1.1) with f defined by (1.2), and showed that the scalar version of (1.4) implies the global asymptotic stability of the unique equilibrium.

In Section 2 we give a sufficient condition which implies asymptotic stability of a linear delay system for all delays. Such stability is called absolute stability in the engineering literature. We extend a known result [3] for the case we use in Section 3 to prove our stability results for (1.1). In Section 4 we give an example to illustrate the main result and we formulate a conjecture to generalize the result.

First we introduce some notations. Let ℝ_{+} be the set of positive real numbers. We use the relation x ≤ y (x < y, respectively) for vectors x, y ∈ ℝ^{n}, if x_{i} ≤ y_{i} (x_{i} < y_{i}, respectively) for all i = 1, …, n, where x = (x_{1}, …, x_{n})^{T} and y = (y_{1}, …, y_{n})^{T}. We introduce the vectors 0 = (0, 0, …, 0)^{T} ∈ ℝ^{n} and 1 = (1, 1, …, 1)^{T} ∈ ℝ^{n}.

For an n × n matrix B the symbol |B| denotes the corresponding n × n matrix with ijth element |b_{ij}|. Similarly, |u| = (|x_{1}|, …, |x_{n}|)^{T}.

We say that an n × n matrix K is an M-matrix, if all of its diagonal elements are nonnegative, and its off-diagonal elements are nonpositive, and all of its principal minors are nonnegative (see, e.g., [1], [3] or [7]). It is known (see, e.g., [1]) that if K is a nonsingular M-matrix, then x ≤ y implies K^{−1}x ≤ K^{−1}y.

Remark 1 Let K be a matrix such that the diagonal elements of K are all positive and the off-diagonal elements are all nonpositive. Then it is known (see, e.g., Theorem 2.3 in [1]) that if K is a diagonally dominant, then it is a nonsingular M-matrix, as well. Moreover, K is a nonsingular M-matrix, if and only if, there exists a positive diagonal matrix D such that KD is a diagonally dominant matrix. We note that there are 50 conditions listed in [1] which are all equivalent to that a matrix is a nonsingular M-matrix.

2 Absolute Stability of a Linear System

Consider the autonomous linear delay system

ż_{i}(t) = ∑_{j=1}^{n} a_{ij}z_{j}(t) + ∑_{j=1}^{n} b_{ij}z_{j}(t − τ_{ij}),  t ≥ 0, i = 1, …, n,  (2.1)

where τ_{ij} ≥ 0 for i, j = 1, …, n.

We put the coefficients to the n × n matrices A = (a_{ij}) and B = (b_{ij}). For the matrix A we associate the n × n diagonal matrix A_{0} = diag(a_{11}, a_{22}, …, a_{nn}), i.e., the diagonal part of A, and let A_{1} = A − A_{0} be the off-diagonal part of A. Then with this notation, which we use throughout this paper, we can rewrite A as A = A_{0} + A_{1}. Similarly, let B_{0} be the diagonal part of B, and denote B_{1} = B − B_{0}.

In the case when A_{1} = 0 and B_{0} = 0 the necessary and sufficient condition for the stability and asymptotic stability of (2.1) for all selection of the delays τ_{ij} was established in [10]. Following the methods of [10] this result was extended in [3] for the special case when only A_{1} = 0, i.e., A is a diagonal matrix in (2.1), and B is an arbitrary matrix.

Theorem 1 (see Theorem 2.6 in [3]) Suppose A = A_{0}. Then the trivial solution of (2.1) is asymptotically stable for all delays τ_{ij} ≥ 0, if and only if −A − |B| is an M-matrix and A + B is a nonsingular matrix.

Note that in the case when B is a nonnegative matrix, this result follows from a more general theorem in [7], where such result was proved for quasilinear delay differential equations. In the case when B is a nonnegative matrix, Theorem 1 also follows from an other generalization of it given in [8], where it was shown that if τ_{k} ≥ 0, (k = 1, …, p), D_{k} ≥ 0 are diagonal matrices for k = 1, …, p such that ∑_{k=1}^{p} D_{k} is invertible, B_{ℓ} are nonnegative n × n matrices for ℓ = 1, …, r, and equation

u̇(t) = − ∑_{k=1}^{p} D_{k}u(t − τ_{k})

has a positive fundamental solution, then the trivial solution of

ẋ(t) = − ∑_{k=1}^{p} D_{k}x(t − τ_{k}) + ∑_{ℓ=1}^{r} B_{ℓ}x(t − σ_{ℓ})

is asymptotically stable for all σ_{1}, …, σ_{ℓ} ≥ 0, if and only if

∑_{k=1}^{p} D_{k} − ∑_{ℓ=1}^{r} B_{ℓ}

is a nonsingular M-matrix.

We extend the sufficient part of Theorem 1 for the case which we will need later. We assume A ≠ A_{0} , i.e., there are nonzero off-diagonal parts of A. The proof follows that of Theorem 1 (see [3]).

Theorem 2 Suppose −A_{0} − |A_{1}| − |B| is a nonsingular M-matrix. Then the trivial solution of (2.1) is asymptotically stable for all delays τ_{ij} ≥ 0.

Proof Finding the solution of (2.1) in the form e^{λt}v (v ≠ 0) leads to the characteristic equation

det ( \begin{matrix}a_{11} + b_{11}e^{−λτ_{11}} − λ & a_{12} + b_{12}e^{−λτ_{12}} & ⋯ & a_{1n} + b_{1n}e^{−λτ_{1n}} \\ a_{21} + b_{21}e^{−λτ_{21}} & a_{22} + b_{22}e^{−λτ_{22}} − λ & ⋯ & a_{2n} + b_{2n}e^{−λτ_{2n}} \\ ⋮ & ⋮ & & ⋮ \\ a_{n1} + b_{n1}e^{−λτ_{n1}} & a_{n2} + b_{n2}e^{−λτ_{n2}} & ⋯ & a_{nn} + b_{nn}e^{−λτ_{nn}} − λ \end{matrix} ) = 0 (2.2)

of (2.1). It is known that the asymptotic stability of the trivial solution of (2.1) is equivalent to that all roots of (2.2) have negative real parts. Let λ be a root of (2.2), then λ is an eigenvalue of the matrix

G(λ) = ( \begin{matrix}a_{11} + b_{11}e^{−λτ_{11}} & a_{12} + b_{12}e^{−λτ_{12}} & ⋯ & a_{1n} + b_{1n}e^{−λτ_{1n}} \\ a_{21} + b_{21}e^{−λτ_{21}} & a_{22} + b_{22}e^{−λτ_{22}} & ⋯ & a_{2n} + b_{2n}e^{−λτ_{2n}} \\ ⋮ & ⋮ & & ⋮ \\ a_{n1} + b_{n1}e^{−λτ_{n1}} & a_{n2} + b_{n2}e^{−λτ_{n2}} & ⋯ & a_{nn} + b_{nn}e^{−λτ_{nn}} \end{matrix} ) .

Since −A_{0} − |A_{1}| − |B| is a nonsingular M-matrix, it is known (see, e.g., Theorem 2.3 in [1]) there exist positive constants γ_{1}, …, γ_{n} > 0 such that

(−a_{ii} − |b_{ii}|)γ_{i} > ∑_{{j=1, \atop j≠i}}^{n}(|a_{ij}| + |b_{ij}|)γ_{j},  i = 1, …, n.  (2.3)

Let Γ = diag(γ_{1}, …, γ_{n}). Then Γ is nonsingular, therefore λ is an eigenvalue of the matrix Γ^{−1}G(λ)Γ, as well. Therefore an application of Gersgorin’s theorem for the matrix Γ^{−1}G(λ)Γ yields

|λ − a_{ii} − b_{ii}e^{−λτ_{ii}}| ≤ ∑_{{j=1, \atop j≠i}}^{n} γ_{i}^{−1}(|a_{ij}| + |b_{ij}||e^{−λτ_{ij}}|)γ_{j}

for some i. Therefore for this fixed i

Re(λ) ≤ Re(a_{ii} + b_{ii}e^{−λτ_{ii}}) + ∑_{{j=1, \atop j≠i}}^{n} γ_{i}^{−1}(|a_{ij}| + |b_{ij}|e^{−(Re λ)τ_{ij}})γ_{j}.

Suppose Re(λ) ≥ 0. Then (2.3) yields

Re(λ)γ_{i} ≤ (a_{ii} + |b_{ii}|)γ_{i} + ∑_{{j=1, \atop j≠i}}^{n}(|a_{ij}| + |b_{ij}|)γ_{j} < 0,

which contradicts to the assumption, therefore Re(λ) < 0 for all solutions of (2.2). □

The proof implies immediately the next technical result.

Corollary 3 If −A_{0} − |A_{1}| − |B| is a nonsingular M-matrix, then A + B is nonsingular, as well.

Proof Let A and B satisfy the assumption, pick any τ_{ij} ≥ 0 (i, j = 1, …, n), and consider the corresponding system (2.1). The proof of Theorem 2 shows that v is a nonzero constant solution of system (2.1) if and only if λ = 0 is a solution of (2.2). But under this assumption all solutions of (2.2) satisfy Re(λ) < 0, therefore the only constant solution of (2.1) is the zero solution. On the other hand, the constant v solutions of (2.1) satisfy (A + B)v = 0, hence A + B is nonsingular. □

3 Stability of a Delayed Neural Network System

Suppose n is a fixed positive integer,

d_{i} > 0, τ_{ij} ≥ 0, a_{ij}, b_{ij}, u_{i} ∈ ℝ (i, j = 1, …, n), and f(t) = \frac{1}{2}(|t + 1| − |t − 1|). (3.1)

We introduce the notations D = diag(d_{1}, …, d_{n}), A = (a_{ij}), B = (b_{ij}), u = (u_{1}, …, u_{n})^{T}. As in the previous section, we use the notation A = A_{0} + A_{1}, where A_{0} is the diagonal part, A_{1} is the off-diagonal part of A.

Consider the DCNN model equations

ẋ_{i}(t) = −d_{i}x_{i}(t) + ∑_{j=1}^{n} a_{ij}f(x_{j}(t)) + ∑_{j=1}^{n} b_{ij}f(x_{j}(t − τ_{ij})) + u_{i},  t ≥ 0, i = 1, …, n (3.2)

with the initial conditions

x_{i}(t) = φ_{i}(t),  t ∈ [−r, 0], i = 1, …, n,  (3.3)

where r = max{τ_{ij}: i, j = 1, …, n}.

To (3.2) we associate an auxiliary system. For a given c > 0 and ψ_{i} : [−r, 0] → ℝ_{+} (i = 1, …, n) consider the system

ẏ_{i}(t) = −d_{i}y_{i}(t)+a_{ii}f(y_{i}(t))+ ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(y_{j}(t))+∑_{j=1}^{n} |b_{ij}|f(y_{j}(t−τ_{ij}))+c_{i}, t ≥ 0, i = 1, …, n

(3.4)

associated to (3.2), and the initial condition

y_{i}(t) = ψ_{i}(t)  t ∈ [−r, 0], i = 1, …, n.  (3.5)

Lemma 1 Suppose (3.1). Let ψ_{i}: [−r, 0] → ℝ_{+} (i = 1, …, n), c > 0, and let y_{1}, …, y_{n} be the corresponding solution of (3.4)-(3.5). Then there exists M > 0 such that

0 < y_{i}(t) < M, t ≥ 0, i = 1, …, n.

Proof Since y_{i}(0) > 0 and y_{i} is continuous on [0, ∞) for all i = 1, …, n, y_{i}(t) > 0 for small enough t ≥ 0. Suppose there exists i and T > 0 such that

y_{j}(t) > 0 for t ∈ [−r, T), j = 1, …, n, and y_{i}(T) = 0.

Then ẏ_{i}(T −) ≤ 0. On the other hand, (3.4) implies

ẏ_{i}(T) = ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(y_{j}(T)) + ∑_{j=1}^{n} |b_{ij}|f(y_{j}(T − τ_{ij})) + c_{i} > 0,

which is a contradiction. Therefore y_{i}(t) > 0 for all t > 0 and i = 1, …, n.

Fix i. To prove that y_{i} is bounded from above, assume that lim sup_{t→∞} y_{i}(t) = ∞. Then there exists a monotone increasing sequence t_{n} such that

lim_{n→∞} t_{n} = ∞,  lim_{n→∞} y_{i}(t_{n}) = ∞,  and  y_{i}(t_{n}) = max{y_{i}(t): t ∈ [−r, t_{n}]}.

Then ẏ_{i}(t_{n}−) ≥ 0, which contradicts to the relations

ẏ_{i}(t_{n}) = −d_{i}y_{i}(t_{n}) + a_{ii}f(y_{i}(t_{n})) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(y_{j}(T)) + ∑_{j=1}^{n} |b_{ij}|f(y_{j}(t_{n} − τ_{i})) + c_{i}

≤ −d_{i}y_{i}(t_{n}) + ∑_{j=1}^{n} |a_{ij}| + ∑_{j=1}^{n} |b_{ij}| + c_{i}
< 0

for large enough n. □

Remark 2 It is easy to check that the matrix D − A_{0} − |A_{1}| − |B| is a diagonally dominant matrix with positive diagonal elements, if and only if

0 < (D − A_{0} − |A_{1}| − |B|)1.

Lemma 3 Assume (3.1), D − A_{0} − |A_{1}| − |B| is a diagonally dominant matrix, and

0 < c < (D − A_{0} − |A_{1}| − |B|)1.  (3.6)

Let ψ_{i}: [−r, 0] → ℝ_{+} (i = 1, …, n), and let y(t) = (y_{1}(t), …, y_{n}(t))^{T} be the corresponding solution of (3.4)-(3.5). Then

lim_{t→∞} y(t) = (D − A_{0} − |A_{1}| − |B|)^{−1}c < 1.  (3.7)

Proof It follows from Lemma 1 that

M_{i} = lim sup_{t→∞} y_{i}(t) m_{i} = lim inf_{t→∞} y_{i}(t)

are finite and m_{i} ≥ 0. For a fixed i there exists a sequence t_{n} such that

t_{n} → ∞ as n → ∞,  ẏ_{i}(t_{n}) ≥ 0, n = 1, 2 …,  and  lim_{n→∞} y_{i}(t_{n}) = M_{i}.

We may also assume that

lim_{n→∞} y_{j}(t_{n}) = m_{j}^{∗}  and  lim_{n→∞} y_{j}(t_{n} − τ_{ij}) = m_{ij}^{∗∗}

for all j = 1, …, n for some m_{j}^{∗}, m_{ij}^{∗∗} ∈ [m_{j}, M_{j}], since otherwise we can select a subsequence of t_{n} with this property. Then

0 ≤ lim_{n→∞} ẏ_{i}(t_{n})
= lim_{n→∞}(−d_{i}y_{i}(t_{n}) + a_{ii}f(y_{i}(t_{n})) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(y_{j}(t_{n})) + ∑_{j=1}^{n} |b_{ij}|f(y_{i}(t_{n} − τ_{ij})) + c_{i})

= −d_{i}M_{i} + a_{ii}f(M_{i}) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(m_{j}^{∗}) + ∑_{j=1}^{n} |b_{ij}|f(m_{ij}^{∗∗}) + c_{i}

≤ −d_{i}M_{i} + a_{ii}f(M_{i}) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(M_{j}) + ∑_{j=1}^{n} |b_{ij}|f(M_{j}) + c_{i}.

Therefore for all i = 1, …, n

c_{i} ≥ d_{i}M_{i} − a_{ii}f(M_{i}) − ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(M_{j}) − ∑_{j=1}^{n} |b_{ij}|f(M_{j})

≥ d_{i}M_{i} − a_{ii}f(M_{i}) − ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}| − ∑_{j=1}^{n} |b_{ij}|.  (3.8)

Suppose M_{i} ≥ 1 for some i. Then (3.8) implies

c_{i} ≥ d_{i} − a_{ii} − ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}| − ∑_{j=1}^{n} |b_{ij}|

which contradicts to assumption (3.6), which yields

0 < c_{i} < d_{i} − a_{ii} − ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}| − ∑_{j=1}^{n} |b_{ij}|.

Therefore 0 ≤ M_{i} < 1 for all i = 1, …, n. This means there exists t_{1} > 0 such that for t ≥ t_{1} (3.4) is equivalent to the linear system

ẏ_{i}(t) = (−d_{i} + a_{ii})y_{i}(t) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|y_{j}(t) + ∑_{j=1}^{n} |b_{ij}|y_{j}(t − τ_{ij}) + c_{i},  t ≥ t_{1}.  (3.9)

Define

e = (D − A_{0} − |A_{1}| − |B|)^{−1}c.

Then e = (e_{1}, …, e_{n})^{T} is the unique equilibrium of the system (3.9), and it follows from (3.6) that 0 ≤ e_{i} ≤ M_{i} < 1, so 0 ≤ e < 1. Introducing z(t) = y(t) − e we can rewrite (3.9) as

ż_{i}(t) = (−d_{i} + a_{ii})z_{i}(t) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|z_{j}(t) + ∑_{j=1}^{n} |b_{ij}|z_{j}(t − τ_{ij}),  t ≥ t_{1}.  (3.10)

Since D − A_{0} − |A_{1}| − |B| is a nonsingular M-matrix by Remark 1, Theorem 2 yields the trivial solution of (3.10) is asymptotically stable (independently of the size of the delays), therefore (3.7) holds. □

Theorem 4 Assume (3.1), D −A_{0} −|A_{1}|−|B| is a diagonally dominant matrix with positive diagonal elements, and u is such that

|u| < (D − A_{0} − |A_{1}| − |B|)1.  (3.11)

Then any solution x of (3.2)-(3.3) satisfies

lim_{t→∞} x(t) = (D − A − B)^{−1}u.  (3.12)

Proof Fix any initial functions ψ_{i}: [−r, 0] → ℝ_{+} such that

ψ_{i}(s) > |φ_{i}(s)|, s ∈ [−r, 0], i = 1, …, n,

and let c > |u| be such that c < (D − A_{0} − |A_{1}| − |B|)1. Let y denote the solution of the corresponding IVP (3.4)-(3.5). Since y(0) > |x(0)|, relation |x(t)| < y(t) holds for sufficiently small t > 0. Suppose there exists i and T > 0 such that

|x_{j}(t)| < y_{j}(t),  t ∈ [−τ, T), j = 1, …, n,  and |x_{i}(T)| = y_{i}(T).  (3.13)

It follows from Lemma 1 that |x_{i}(T)| = y_{i}(T) ≠ 0, therefore \frac{d}{dt}|x_{i}(t)| exists at T, and \frac{d}{dt}(|x_{i}(t)|)_{|t=T} = ẋ_{i}(T) sign x_{i}(T). Hence

\frac{d}{dt}(|x_{i}(t)|)_{|t=T}

= (−d_{i}x_{i}(T) + ∑_{j=1}^{n} a_{ij}f(x_{j}(T)) + ∑_{j=1}^{n} b_{ij}f(x_{j}(T − τ_{ij})) + u_{i}) sign x_{i}(T)

= −d_{i}|x_{i}(T)| + a_{ii}f(|x_{i}(T)|) + ∑_{{j=1, \atop j≠i}}^{n} a_{ij}f(x_{j}(T)) sign x_{i}(T)

+ ∑_{j=1}^{n} b_{ij}f(x_{j}(T − τ_{ij})) sign x_{i}(T) + u_{i} sign x_{i}(T)

< −d_{i}|x_{i}(T)| + a_{ii}f(|x_{i}(T)|) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(|x_{j}(T)|) + ∑_{j=1}^{n} |b_{ij}|f(|x_{j}(T − τ_{ij})|) + c_{i}

≤ −d_{i}y_{i}(T) + a_{ii}f(y_{i}(T)) + ∑_{{j=1, \atop j≠i}}^{n} |a_{ij}|f(y_{j}(T)) + ∑_{j=1}^{n} |b_{ij}|f(y_{j}(T − τ_{ij})) + c_{i}
= ẏ_{i}(T).

This contradicts to assumption (3.13), therefore |x_{i}(t)| < y_{i}(t) holds for all t > 0 and i = 1, …, n. Moreover, Lemma 3 yields

lim_{t→∞} y(t) = (D − A_{0} − |A_{1}| − |B|)^{−1}c < 1

holds, therefore there exists t_{1} > 0 such that |x(t)| < 1 for t ≥ t_{1}. Then (3.2) is equivalent to

ẋ_{i}(t) = −d_{i}x_{i}(t) + ∑_{j=1}^{n} a_{ij}x_{j}(t) + ∑_{j=1}^{n} b_{ij}x_{j}(t − τ_{ij}) + u_{i}, t ≥ t_{1}.

This implies (3.12) using an argument similar to that in the proof of Lemma 3. □

4 Examples

To illustrate our results consider the two-dimensional DCNN model equations

ẋ_{1}(t) = −x_{1}(t) − 6f(x_{1}(t)) + f(x_{2}(t)) − 3f(x_{1}(t − 1)) + f(x_{2}(t − 2)) + u_{1} (4.1)
ẋ_{2}(t) = −x_{2}(t) − f(x_{1}(t)) − 3f(x_{2}(t)) − f(x_{1}(t − 1)) + f(x_{2}(t − 2)) + u_{2}, (4.2)

where f is defined by (1.2). It is easy to see that

D − A_{0} − |A_{1}| − |B| = ( \begin{matrix}4 & −2 \\ −2 & 3 \end{matrix} )

is a diagonally dominant matrix. Therefore Theorem 4 yields that if |u_{1}| < 2 and |u_{2}| < 1 then the trivial solution of this system is asymptotically stable. In Figure 1 we have plotted the two components of the solutions corresponding to u_{1} = −1 and u_{2} = 0.5 and to the initial functions

( \begin{matrix}φ_{1}(t) \\ φ_{2}(t) \end{matrix} ) = ( \begin{matrix}t + 1 \\ −t \end{matrix} ) , ( \begin{matrix}sin 2t \\ t^{2} − 1 \end{matrix} ) , ( \begin{matrix}cos t + 1 \\ t + 2 \end{matrix} ) and ( \begin{matrix}t^{3} − 2 \\ −2 cos t \end{matrix} ) , (4.3)

respectively.

x_{1}(t)  x_{2}(t)

Figure 1. Case (u_{1}, u_{2}) = (−1, 0.5).

We can observe that all solutions tend to the unique equilibrium (−0.058824, 0.20588)^{T}. Note that the condition of Mohamad and Gopalsamy (1.3) is not satisfied for (4.1)-(4.2),

and also the condition of Takahashi gives the matrix

W = ( \begin{matrix}−7 & −2 \\ −2 & −4 \end{matrix} ) ,

which is not an M-matrix. Therefore none of this two conditions can be applied for system (4.1)-(4.2).

By checking other input values outside the region |u_{1}| < 2 and |u_{2}| < 1 we observed in every cases we tried all solutions tended to the unique equilibrium (v_{1}, v_{2})^{T} of the system (not necessary satisfying |v_{1}|, |v_{2}| < 1). In Figure 2 we can see the graphs of solutions of (4.1)-(4.2) corresponding to (u_{1}, u_{2}) = (3, 5) and to the initial functions (4.3). We can observe that all solutions tend to the unique equilibrium (0.5, 2)^{T}.

x_{1}(t)  x_{2}(t)

Figure 2. Case (u_{1}, u_{2}) = (3, 5).

Next we plotted the solutions corresponding to (u_{1}, u_{2}) = (−8.5, −5.5) and to the initial functions (4.3) in Figure 3. Again, all solutions tend to the unique equilibrium (−1.5, −1.5)^{T}.

Figure 3. Case (u_{1}, u_{2}) = (−8.5, −5.5).

Now change the coefficient of f(x_{2}(t − 2)) in (4.1) to 4, i.e., consider the system

ẋ_{1}(t) = −x_{1}(t) − 6f(x_{1}(t)) + f(x_{2}(t)) − 3f(x_{1}(t − 1)) + 4f(x_{2}(t − 2)) + u_{1} (4.4)
ẋ_{2}(t) = −x_{2}(t) − f(x_{1}(t)) − 3f(x_{2}(t)) − f(x_{1}(t − 1)) + f(x_{2}(t − 2)) + u_{2}. (4.5)

We plotted the solutions corresponding to (u_{1}, u_{2}) = (−6, 4) and to the initial functions (4.3) in Figure 4. As before, all solutions tend to the unique equilibrium, which is (−0.1, 2.2)^{T} in this case. On the other hand,

D − A_{0} − |A_{1}| − |B| = ( \begin{matrix}4 & −5 \\ −2 & 3 \end{matrix} )

is no longer a diagonally dominant matrix, but it is a nonsingular M-matrix.

x_{1}(t)  x_{2}(t)

Figure 4. Case (u_{1}, u_{2}) = (−6, 4).

Therefore our numerical experiments on these and other systems suggest the following conjecture.

Conjecture 1 Assume (3.1) and D − A_{0} − |A_{1}| − |B| is a nonsingular M-matrix. Then (3.2) has a unique equilibrium for any input vector u, and any solution of (3.2) tends to this equilibrium.

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(Received September 29, 2003)