Stability of a Time Discrete Perturbed Dynamical System with Delay

Let Cn be the set of n complex vectors endowed with a norm ‖⋅‖Cn. Let A,B be two complex n×n matrices and τ a positive integer. In the present paper we consider the nonlinear difference equation with delay of the type uk

Let C be the set of n complex vectors endowed with a norm II'llCn" Let A,B be two complex n x n matrices andbe a positive integer.Consider the perturbed difference equation with delay Uk+l Auk + Buk-+ Fk(uk, Uk-), k--0, 1,2,..., where Fk" C x Cn C satisfies the condition F(x, Y)[]c P[ x[ c + ql[ylIc,1,2,..., (2) where p and q are positive constants.In this paper, we will be concerned with the stability problem of Eq. (1) under the condition that all the zeroes of det(zI-A z-B) lie in the open unit disk in the complex plane C with center at the origin, where I stands for the identity matrix.In other words, we will assume that the greatest modulus p(A,B;-) of the roots of det(A + z-'-B-zI) is less than 1.Similar problems have been dealt with perturbed equations of the form Uk+l Auk +f(xk), k-O, 1,2,..., We remark that it would have been better to replace the last inequality by / 1/2 0 k=l k=-7as early as 1929 (see, e.g.Ortega, 1973), under the assumption that the spectral radius of A is less than 1.For recent and related investigations, the readers may consult Ortega (1973), Xie andCheng (1995, 1996) and others.We remark that in case n 2, our equation can be used to model the population growth of two species that are under delayed migration and interaction, see, e.g.Sandefur (1990, Chapter 7).
There are many concepts of stability for discrete time dynamical systems.Here we will adopt a specific one described as follows.First, note that a solution of (1) is a sequence {uk}kc*=_7of vectors in C such that it renders (1) into an identity after sub- stitution.Given initial vectors u_7-, u_7-+l,..., u0, it is easily seen that we can successively calculate Ua, u2,.., according to (1) in a unique manner.An existence and uniqueness theorem for (1) can thus be easily formulated and proved.Next, note that the assumption (2) implies that the zero sequence 0 {0}k_7is a solution of (1).Let us take 12(Cn) to be the Hilbert space of all complex sequences of the form v {vk}kc*__0 endowed with the usual inner product and norm c k=O in the above definition, but clearly the two concepts are equivalent and we will keep the present defi- nition for technical reasons (to be seen below).
Our main concern in this paper is to derive a stability criterion for the zero solution.To this end, let us first consider the following nonhomogeneous equation Uk+l Auk + Buk_7-+fk, k 0, 1,2,... (3) LEMMA Suppose Ilflll2 < oo and p(A, B; -) < 1.
Then the truncated sequence fi-{uk}k__ 0 of a solution {Uk}k_7of(3) will satisfy o Ilull2 Mllf [l + F Z Ilu c for some constant F, where M-max (zI-A z-7-B) (4) Proof First note that the sequence {fk}k0 is of exponential order.Thus the solution {uk}k_7is also of exponential order (see e.g., Gy6ri and Ladas, 1991, Lemma 1.4.2).Let u [--, oc) C and f: [0, oo) oc be the piecewise continuous step functions defined by We say that the zero solution of (1) is absolutely Z-stable (in the class of nonlinearities ( 2)) if there is a constant I' depending only on the numbers p and q such that for any solution u-{uk}k_7-, the trun- cated sequence fi {uk}kO satisfies and u(t)--Uk, k<_t<k+l, k-0,1,2,...; Then the Laplace transforms of the restricted functions u[{0,o) and f exist.Let their Laplace transforms be denoted by u* and f* respectively, that is, u*(z) e-Ztu(t)dt and f* (z) e-Zt f t) dt, where z is the dual complex variable.In view of (3), we see that u(t + l) Au(t) + Bu(t-7-) +f(t), t>_O, thus multiplying both sides of the above equation by e -zt and then integrating from 0 to oc, we  In view of (5) and the above calculations, we now (p + q)M < 1, where M is defined by (4).Then the zero solution of ( 1) is absolutely 12-stable.Proof Let {uk}k_ be a solution of (1).For each nonnegative integer rn, let the sequence h (m) (m) h(k m) be de i.ed by -, or and h(km)= 0 for k>m, and let v (m) -{ }k_ be the unique solution determined by the conditions v( m) u( m), k --, --+ 1,..., 0 (6) and Clearly, by uniqueness, Finally, letting m tend to oc, we see that our desired result holds.The proof is complete.
The constant M in the above Theorem is stated in terms of the inverse matrix of zI-A-z-B.This is inconvenient in general.In order to obtain a more convenient estimate, we proceed as follows.Let us first take II'llcn to denote the Euclidean norm in the sequel.Further, let AI(H),..., An(H) be the eigen- values of an n x n complex matrix H including their multiplicities.We will make use of the following quantity: g(H)--N2(H) Ak(H) k=l where N(H) is the Frobenius (Hilbert-Schmidt) norm of H, i.e.
Indeed, in view of ( 8) and ( 9), We may also show that M_< Ao.
In fact, if H is a linear operator on Cn, then the following estimate is true (see Gil', 1995, p.  where d(H, A) is the distance between the spectrum or(H) of H and the complex number A. Hence for any invertible matrix H, where po(H) is the smallest modulus of the eigenvalues of H. Relation (12) yields ]](A + z--B-zI) -IIc n--1 9'n,k <-Zgk(A + z--B-ZI) po+l(A + z--B zI) k=0 for regular z, so that, in view of (9), =max,=p+l lzl= (A + z-B-zI) as desired.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning (t) dt Z uke -zt dt k----T Jk e_zk e_z(k+l) =' z Ukfor z-i0, and since for k--7-, -7-+ 1, ...,-1 _< Zgk(H)pko+l(H), k=0

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation