Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Volume 2010, Article ID 539087,23pages doi:10.1155/2010/539087

Research Article

Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Josef Dibl´ık,

Denys Ya. Khusainov,

Irina V. Grytsay,

and Zden ˘ek ˇSmarda

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick´a 8, 616 00 Brno, Czech Republic

2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Veveˇr´ı331/95, 60200 Brno, Czech Republic

3Department of Complex System Modeling, Faculty of Cybernetics, Taras,

Shevchenko National University of Kyiv, Vladimirskaya Str., 64, 01033 Kyiv, Ukraine

Correspondence should be addressed to Josef Dibl´ık,diblik@feec.vutbr.cz Received 28 January 2010; Accepted 11 May 2010

Academic Editor: Elena Braverman

Copyrightq2010 Josef Dibl´ık et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ1 of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

1. Introduction

The main results on the stability theory of difference equations are presented, for example, by Agarwal 1, Agarwal et al. 2, Chetaev 3, Elaydi 4, Halanay and R˘asvan 5, Lakshmikantham and Trigiante6, and Martynjuk7. Instability problems are considered, for example, in8–10by Slyusarchuk. Note that stability and instability results often have a local character and are usually obtained without any estimation of the stability domain, or without investigating the character of instability. Moreover, it should be emphasized that global instability questions have only been discussed for linear systems.

Many processes and phenomena are described by differential or difference systems with quadratic nonlinearities. Among others, let us mention epidemic and populations models, models of chemical reactions, and models for describing convection currents in the atmosphere.

The stability of a zero solution of difference systems

xk1 fxk, 1.1

wherek 0,1, . . ., and x x1, . . . , x_n^T with differentiablef f1, . . . , f_n^T : Rⁿ → Rⁿ, is very often investigated by linearly approximating system1.1in question by using the matrix of linear terms

xk1 Axk gxk, 1.2

whereAf0,0, . . . ,0is the Jacobian matrix offat0,0, . . . ,0, andgx fx−Ax. This approach becomes unsuitable in what is called a critical case, that is, when the spectral radius of the matrixρA 1 because, among all systems1.2, there are classes of stable systems as well as classes of unstable systems. Concerning this, we formulate the following known resultssee, e.g., Corollary 4.344, page 222and Theorem 4.384, page 226.

Theorem 1.1. 1IfρA<1, then the zero solution of 1.2is exponentially stable.

2IfρA 1, then the zero solution of 1.2may be stable or unstable.

3IfρA>1 andgxisoxasx → 0, then the zero solution of 1.2is unstable.

In this paper, we consider a particular critical case when there exists a simple eigenvalue λ 1 of the matrix of linear terms and the remaining eigenvalues lie inside a unit circle centered at origin. The purpose of this paper is to obtain using the method of Lyapunov functionsconditions for the stability of a zero solution of difference systems with quadratic nonlinearities in the above case and derive classes of stable systems. In addition to the stability investigation, we estimate the stability domains as well. The domains of stability obtained are also called guaranteed domains of stability. Preliminary results in this direction were published in11.

1.1. Quadratic System and Preliminary Consideration In the sequel, the norms used for vectors and matrices are defined as

x

_n

i1

x²_{i 1/2}

1.3

for a vectorx x1, . . . , xn^Tand

A

λmaxA^TA_1/2

1.4

for anym×nmatrixA. Here and in the sequel,λ_max· orλ_min·is the maximalor minimal eigenvalue of the corresponding symmetric and positive-semi- definite matrixsee, e.g., 12.

Consider a nonlinear autonomous discrete system with a quadratic right-hand side

x_ik1 ⁿ

s1

a_isx_sk ⁿ

s,q1

bⁱ_sqx_skxqk, i1. . . , n, 1.5

wherek0,1, . . .and the coefficientsaisandbⁱ_sqwe assume thatbⁱ_sq bⁱ_qsare constant. As emphasized, for example, in3,7,12, system1.5can be written in a general vector-matrix form

xk1 Axk X^TkBxk, k0,1, . . . , 1.6 where

aA{ais}, i, s1,2. . . , n, is ann×nconstant square matrix,

bmatrixX^T{X₁^T, X^T₂, . . . , X^T_n}isn×n²rectangular and all the elements of then×n matricesX_i^T, i 1, . . . , n, are equal to zero except the ith row with entriesx^T x1, x₂, . . . , x_n, that is,

X^T_ik

⎛

⎜⎜

⎜⎜

⎜⎝

0 0 · · · 0

· · · · x₁ x₂ · · · x_n

· · · · 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠, 1.7

cmatrix B^T {B1, B₂, . . . , B_n}isn²×nrectangular and then×nconstant matrices B_i{bⁱ_sq}, i, s, q1, . . . , n, are symmetric.

The stability of the zero solution of1.6depends on the stability of the matrixA. IfρA<1, then the zero solution of1.6is exponentially stable for an arbitrary matrixBbyTheorem 1.1.

In this case, matrixBonly impacts on the shape of the stability domain of the equilibrium state. If the zero solution of1.6is investigated on stability by the second Lyapunov method and an appropriate Lyapunov function is taken as the quadratic formVx x^THxwith a suitablen×nconstant real symmetric positive-definite matrixH, which is defined below, then the first differenceΔV along the trajectories of1.6equals

ΔVxk Vxk1−Vxk x^Tk1Hxk1−x^TkHxk

Axk X^TkBxk_T H

Axk X^TkBxk

−x^TkHxk

x^TkA^Tx^TkB^TXk H

Axk X^TkBxk

−x^TkHxk x^Tk

A^THA−H

A^THX^TkBB^TXkHAB^TXkHX^TkB xk x^Tk

A^THA−H

2B^TXkHAB^TXkHX^TkB xk

1.8

sinceA^THX^TkB^T B^TXkHA.

Since ρA < 1, for arbitrary positive-definite symmetric matrix C, the matrix Lyapunov equation

A^THA−H−C 1.9

has a unique solutionH—a positive-definite symmetric matrixe.g.,4, Theorem 4.30, page 216. We use such matrixHto estimate the stability domain. Then, as follows from1.8,

ΔVxk≤ −

λ_minC−2B · HA · xk − B²· H · xk²

· xk². 1.10

Analysing1.10, we deduce that the first differenceΔVxkwill be negative definite if B²· H · xk²2B · HA · xk ≤λ_minC, 1.11 that is, it will be negative definite in a neighborhoodU_δ {x∈Rⁿ:x< δ}of the steady- statexk ≡ 0, k 0,1, . . . ,ifδ is sufficiently small. In the case considered, the domain of stability can be described by means of two inequalities. The first inequality 1.11defines a part of the spaceRⁿ, where the first differenceΔVxkis negative definite. The second inequality

Vx≤r, x∈Rⁿ, r >0, 1.12 describes points inside a level surface. The guaranteed domain of stability is given by inequality1.12ifr is taken so small that the domain described by1.12is embedded in the domain described by inequality1.11.

Considering the investigated critical case, we will deal with a different structure of the right-hand side of the inequality from that seen in 1.10. Namely, we will show that, unlike the right-hand side of the inequality forΔVxkthat is multiplied byxk²with dimxk nin1.10, in the critical case considered, the right-hand side of the inequality or equalityforΔVxkwill be multiplied only by a termx_n−1k² with dimx_n−1k n−1< nsee2.21in the casen2 and2.69in the general case below.

2. Main Results

In this section we derive the classes of the stable systems1.6in a critical case when the matrixAhas one simple eigenvalueλ1.

2.1. Instability in One-Dimensional Case

We start by discussing a simple scalar equation with the eigenvalue of matrixAequaling one, that is,a₁₁ 1. Then1.6takes the form

xk1 xk bx²k, k0, . . . , 2.1

and it is easy to see that the trivial solution is unstable for an arbitraryb /0to show this, we can apply, e.g., Theorem 1.154, page 29.

This elementary example shows that stability in the case of system 1.6 has an extraordinary significance and the results on stabilityforn /1lose their meaning forn1 when we deal with instability. We show that, ifn /1 andBsatisfies certain assumptions, the zero solution is stable. Moreover, the shape of the guaranteed domain of stability will be given.

We divide our forthcoming analysis into two parts. In the first one we give an explicit coefficient criterion in the subcase ofn2. Then we consider the generaln-dimensional case.

2.2. Stability in the General Two-Dimensional Case

Letn 2. Then system1.6with the matrixAhaving a simple eigenvalueλ 1 reduces after linearly transforming the dependent variables if necessaryto

x₁k1 ax₁k

b¹₁₁x²₁k 2b¹₁₂x₁kx2k b¹₂₂x²₂k , x₂k1 x₂k

b²₁₁x²₁k 2b²₁₂x₁kx2k b₂₂² x²₂k .

2.2

We will assume that|a|<1. Define auxiliary numbers as follows:

αh 1−a²

, β₁hab¹₁₁, β₂2hab¹₁₂b₁₁² , γ₁ h

b¹₁₁2

b²₁₁2

, γ₂ 4h b₁₂¹ 2

, δ₁2hb¹₁₁b₁₂¹ ,

2.3

wherehis a positive number.

Theorem 2.1. Lethandr be positive numbers. Assume that|a|<1 andb²₁₂ b¹₂₂ b₂₂² 0. Then the zero solution of system2.2is stable in the Lyapunov sense and a guaranteed domain of stability is given by the inequality

hx²₁x₂²≤r² 2.4

ifris taken so small that the domain described by2.4is embedded in the domain

γ₁x²₁2δ₁x₁x₂γ₂x²₂2β₁x₁2β₂x₂ ≤α. 2.5

If, moreover,b²₁₁·b¹₁₂/0, then a guaranteed domain of stability can be described using inequality

hx²₁x²₂≤r^∗2 2.6

with

r^∗ min

x1,x2

hx²₁x²₂, 2.7

wherex1, x₂runs over all real solutions of the nonlinear system with unknownsx₁andx₂:

γ1x²₁2δ1x1x2γ2x²₂2β1x12β2x2α, hx₁

δ₁x₁γ₂x₂β₂

−x₂

γ₁x₁δ₁x₂β₁ 0.

2.8

Proof. Define

B1

b₁₁¹ b¹₁₂ b₁₂¹ b¹₂₂

, B2

b²₁₁ b²₁₂ b²₁₂ b²₂₂

, x2y, x x₁

y

. 2.9

We rewrite system2.2as

x1k1 ax1k x^TkB1xk,

yk1 yk x^TkB2xk. 2.10

To investigate the stability of the zero solution, we use, in accordance with the direct Lyapunov method, an appropriate Lyapunov functionV. Let a matrixH, defined as

H

h h12

h12 h22

, 2.11

where instead of the entryh₁₁we put the numberh, be positive definite. We set

Vxk V

x₁k, yk

:x^TkHxk

x₁k, yk h h₁₂ h12 h22

x₁k yk

hx²₁k 2h12x1kyk h22y²k.

2.12

The first difference of the functionV along the trajectories of system2.10equals

ΔVxk Vxk1−Vxk

hx²₁k1 2h₁₂x₁k1yk1 h₂₂y²k1

−hx²₁k−2h12x1kyk−h22y²k h

ax₁k x^TkB1xk₂ 2h₁₂

ax₁k x^TkB1xk

yk x^TkB2xk h22

yk x^TkB2xk2

−hx²₁k−2h12x1kyk−h22y²k h

a²−1

x²₁k 2h12a−1x1kyk h

2ax₁k

x^TkB1xk

x^TkB1xk₂ 2h₁₂

ax₁k

x^TkB2xk

yk

x^TB₁xk

x^TkB1xk

x^TkB2xk h22

2yk

x^TkB2xk

x^TkB2xk₂ .

2.13

It is easy to see thatΔV does not preserve the sign ifh₁₂/0. Therefore, we puth₁₂ 0 and ΔVreduces to

ΔVxk h a²−1

x²₁k h

2ax1k

x^TkB1xk

x^TkB1xk2 h₂₂

2yk

x^TkB2xk

x^TkB2xk₂ .

2.14

In the polynomial ΔV, with respect to x1 and y, we will put together the third-degree terms the expression F₃x1k, yk below and the fourth-degree terms the expression F₄x1k, ykbelow. In the computations we use the formulas

x^TkBixk bⁱ₁₁x₁²k 2bⁱ₁₂x₁kyk bⁱ₂₂y²k, i1,2, x^TkBixk2

bⁱ₁₁2

x⁴₁k 4 bⁱ₁₂2

x²₁ky²k bⁱ₂₂2

y⁴k

4bⁱ₁₁bⁱ₁₂x₁³kyk 2bⁱ₁₁bⁱ₂₂x²₁ky²k 4bⁱ₁₂bⁱ₂₂x1ky³k, i1,2.

2.15

We get

ΔVxk h a²−1

x²₁k F₃

x₁k, yk F₄

x₁k, yk

, 2.16

where

F₃

x₁k, yk

2hab¹₁₁x₁³k 2

2hab¹₁₂h₂₂b₁₁²

x²₁kyk 2

hab¹₂₂2h₂₂b²₁₂

x₁ky²k 2h₂₂b₂₂² y³k, F4

x1k, yk

h

b¹₁₁₂ h22

b²₁₁₂

x⁴₁k 4

hb¹₁₁b¹₁₂h22b²₁₁b²₁₂

x³₁kyk 2

2h

b¹₁₂2

2h₂₂ b²₁₂2

hb₁₁¹ b¹₂₂h₂₂b²₁₁b₂₂²

x²₁ky²k 4

hb¹₁₂b₂₂¹ h₂₂b²₁₂b₂₂²

x₁ky³k

h b¹₂₂₂

h₂₂ b²₂₂₂

y⁴k.

2.17

Analysing the increment of V, we see that, if |a| < 1, ΔV will be nonpositive in a small neighborhood of the zero solution if the multipliers of the termsx1y²,y³andx1y³are equal to zero and the multiplier of the termy⁴is nonpositive, that is, if

hab¹₂₂2h22b₁₂² 0, h₂₂b²₂₂0, hb₁₂¹ b¹₂₂h22b₁₂² b²₂₂ 0, h

b₂₂¹ ₂ h₂₂

b₂₂²₂

≤0.

2.18

As long as the Lyapunov function is positive definite,h >0 andh₂₂ >0. Therefore, conditions 2.18hold if and only if

b¹₂₂0, b²₁₂0, b₂₂² 0. 2.19

Then, system2.2turns into

x₁k1 ax₁k

b¹₁₁x²₁k 2b₁₂¹ x₁kx2k , x2k1 x2k b²₁₁x²₁k

2.20

andΔVwithout loss of generality, we puth22 1, i.e.,Vx1, y hx²₁y²into ΔVxk −

h 1−a²

−2hab¹₁₁x1k−2

2hab¹₁₂b₁₁²

yk− h

b¹₁₁2

b²₁₁2

x₁²k

−4hb¹₁₁b¹₁₂x₁kyk−4h b₁₂¹ ₂

y²k

x²₁k −

α−2β1x1k−2β2yk−γ1x²₁k−2δ1x1kyk−γ2y²k x²₁k.

2.21 The first difference of the Lyapunov function is nonpositive in a sufficiently small neighborhood of the originthis is becauseh > 0, |a|< 1, andα h1−a² > 0. In other words, the zero solution is stable in the Lyapunov sense.

Now we will discuss the shape of the guaranteed domain of stability. It can be defined by the inequalities

γ₁x²₁2δ₁x₁yγ₂y²2β₁x₁2β₂y≤α, hx²₁y²≤r²,

2.22

where r > 0. This means that inequalities 2.4 and 2.5 are correct. Both inequalities geometrically express closed ellipses ifb²₁₁·b¹₁₂/0. For the second inequality, this is obvious.

For the first one, this follows from the following inequalities:γ₁>0, γ₂>0 and

γ₁γ₂−δ²₁

h b¹₁₁₂

b²₁₁₂

·

4h b¹₁₂₂

−4

hb¹₁₁b₁₂¹ ₂ 4h

b²₁₁b₁₂¹ ₂

>0. 2.23

Moreover, forr → 0, the ellipse2.4

hx²₁y²≤r² 2.24

is contained because it shrinks to the originin the ellipse 2.5, that is, there exists such rr^∗that, forr ∈0, r^∗, the ellipse2.4lies inside the ellipse2.5without any intersection points and, forr r^∗, there exists at least one common boundary point of both ellipses. Let us find the valuer^∗. It is characterized by the requirement that the slope coefficientsk1and k₂of both ellipses are the same at the point of contact. Therefore

k₁−γ₁x₁δ₁yβ₁

δ₁x₁γ₂yβ₂, k₂−hx₁

y , 2.25

where we assumewithout loss of generalitythat the denominators are nonzero. Thus, we get a quadratic system of two equations to find the contact pointsx1, y:

γ₁x²₁2δ₁x₁yγ₂y²2β₁x₁2β₂yα, hx1

δ1x1γ2yβ2

−y

γ1x1δ1yβ1

0. 2.26

For the corresponding values ofr, we have

hx²₁y²r². 2.27

In accordance with the geometrical meaning of the above quadratic system, we take such a solutionx1, yas a defintion of the minimal positive value ofrand setr^∗r.

Example 2.2. Consider a system

xk1 0.5xk x²k−4xkyk,

yk1 yk x²k. 2.28

In our case,n 2,a0.5 <1, andb¹₂₂ b²₁₂ b²₂₂ 0. Therefore, byTheorem 2.1, the zero solution of system2.28is stable in the Lyapunov sense. We will find the guaranteed domain of stability. We have

b¹₁₁1, b¹₁₂ −2, b²₁₁1, 2.29

andb²₁₁·b¹₁₂−2/0. Seth2. Then

αh 1−a²

21−0.25 1.5, β1hab₁₁¹ 2·0.5·11,

β₂2hab₁₂¹ b²₁₁2·2·0.5·−2 1−3, γ1h

b₁₁¹ ₂

b₁₁² ₂

2·1²1²3, γ₂4h

b¹₁₂₂

4·2·−2²32, δ₁2hb¹₁₁b¹₁₂2·2·1·−2 −8.

2.30

That is, the guaranteed domain of stability is given by the inequalities

3x²−16xy32y²2x−6y≤1.5, 2.31

2x²y²≤r² 2.32

if r is so small that the domain described by inequality2.32is embedded in the domain described by inequality2.31. We consider the case when the ellipse2.32is embedded in

−1

−0.5 0 0.5 1

−1.5 −1 −0.5 0 0.5 1 1.5 Figure 1: Graphical solution of system2.33and2.34.

the ellipse2.31and the boundaries of both ellipses have only one intersection point. Solving the system2.8, that is, the system

3x²−16xy32y²2x−6y1.5, 2.33

2x

−8x32y−3

−y

3x−8y1

0, 2.34

with Mathematica software, we get the solutionsseeFigure 1where thex-axis is identified with the horizontal line and the y-axis is identified with the vertical line, the blue ellipse graphically depicts equation 2.33, and the red hyperbola graphically depicts equation 2.34:

x, y

x₁, y₁

−1.60766,−0.31220, x, y

x₂, y₂

−0.03568,−0.32187, x, y

x3, y3

0.01952,−0.13664, x, y

x₄, y₄

1.10728,0.37750.

2.35

Then, in accordance with2.7,

r^∗ min

i1,2,3,4

2x_i²y_i²

2x₃²y₃² . 0.1369, 2.36

−0.4

−0.2 0 0.2 0.4

−1.5 −1 −0.5 0 0.5 1 Figure 2: The guaranteed domain of stability.

and the guaranteed domain of stability

2x²y²≤r^∗2 0.1369² 2.37

obtained from2.31,2.32is depicted inFigure 2as an ellipsoidal domain shaded in red and bounded by the thick red ellipse, with the identification ofx-axis andy-axis being the same as before. Here, the domain2.31is bounded by the blue ellipse2.33.

2.3. Stability in the Generaln-Dimensional Case

Consider system1.6inRⁿ. Assume that the matrixAhas a simple eigenvalue that is equal to unity with the others lying inside the unit circle. After linearly transforming the dependent variables if necessary, we can assume, without loss of generality, that the matrix Aof the linear terms in a block form, that is,

A A₀ θ

θ^T 1

, A0 aij

, i, j1,2, . . . , n−1, 2.38

whereθ 0,0, . . . ,0^T, is then−1-dimensional zero vector and all the eigenvalues of the matrix A₀ lie inside the unit circle. In order to formulate the next result and its proof, we

have to introduce some new definitionsthey copy the ones used inSection 1.1, but we use dimension or sizen−1 instead ofnand note this change as a subscript if necessary:

x_n−1 x₁, x₂, . . . , x_n−1^T, yx_n,

B⁰_i

⎛

⎜⎜

⎜⎜

⎜⎝

b₁₁ⁱ b₁₂ⁱ · · · bⁱ_1,n−1 b₂₁ⁱ b₂₂ⁱ · · · bⁱ_2,n−1

· · · · bⁱ_n−1,1 bⁱ_n−1,2 · · · bⁱ_n−1,n−1

⎞

⎟⎟

⎟⎟

⎟⎠, i1,2, . . . , n, B

⎛

⎜⎜

⎝

b¹_1n · · · b_n−1,n¹

· · · · bⁿ⁻¹_1n · · · b_n−1,nⁿ⁻¹

⎞

⎟⎟

⎠,

B^T

⎛

⎜⎜

⎝

b¹₁₁ · · · b¹_1,n−1 b₂₁¹ · · · b¹_2,n−1 · · · b¹_n−1,1 · · · b¹_n−1,n−1

· · · · b₁₁ⁿ⁻¹ · · · bⁿ⁻¹_1,n−1 bⁿ⁻¹₂₁ · · · bⁿ⁻¹_2,n−1 · · · bⁿ⁻¹_n−1,1 · · · bⁿ⁻¹_n−1,n−1

⎞

⎟⎟

⎠.

2.39

MatricesB⁰_i, i1,2, . . . , n, are symmetric sincebⁱ_sq bⁱ_qs,i, s, q1,2, . . . , nseeSection 1.1.

Moreover, we assume that there exists a symmetric positive definiten−1×n−1matrix Hsuch that the symmetric matrix

CH−A^T₀HA0 2.40

is positive definite. Leth >0 be a positive number and

αλ_minC, β₁ 1

2

A^T₀HB^T , β2 1

2

BHA03A^T₀HB^T2h B⁰_nT

, γ₁

BHB^T

hB⁰_n²,

γ₂4BHB^T, δ₁2B· H ·B.

2.41

Theorem 2.3. Lethandrbe positive numbers. Assume that

b_nn¹ b_nn² · · ·bⁿ_nn0, b_1nⁿ b_2nⁿ · · ·bⁿ_n−1,n0. 2.42

Then the zero solution of system1.6is stable by Lyapunov and the guaranteed domain of stability is described by the inequalities

γ1x²2δ1xyγ2y²2β1x2β2y≤α, 2.43

x^T_n−1Hx_n−1hy²≤r² 2.44

ifris so small that the domain described by inequality2.44is embedded into the domain described by inequality2.43.

Proof. We will perform auxiliary matrix computations. With this in mind, we have defined ann−1²×n−1matrixX_n−1as

X^T_n−1

X_1n−1^T , X^T_2n−1, . . . , X_n−1n−1^T

, 2.45

where all the elements of then−1×n−1matricesX^T_in−1, i1,2, . . . , n−1 are equal to zero except the rowi, which equalsx^T_n−1, that is,

X^T_in−1

⎛

⎜⎜

⎜⎜

⎜⎝

0 0 · · · 0

· · · · x₁ x₂ · · · x_n−1

· · · · 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠. 2.46

Moreover, we define

avectorsY_i, i 1,2, . . . , n−1, as a rown−1-dimensional vector with coordinates equal to zero except theith element, which equalsxn, that is,

Y_i 0,0, . . . ,0, x_n,0, . . . ,0, 2.47

b n−1×n−1zero matrixΘ,

cvectorsbi bⁱ_1n, bⁱ_2n, . . . , bⁱ_n−1,n^T,i1,2, . . . , n, dvectorb b¹_nn, b²_nn, . . . , b_nn^n−1T.

It is easy to see that

X^Tk

⎛

⎝X_1n−1^T k Y₁^Tk · · · X^T_n−1n−1k Y_n−1^T k Θ θ

θ^T 0 · · · θ^T 0 x^T_n−1k yk

⎞

⎠,

B

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝ B⁰₁ b₁ b^T₁ b¹_nn

· · · · B_n⁰ b_n b^T_n bⁿ_nn

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ .

2.48

Now we are able to rewrite system1.6in an equivalent form x_n−1k1

yk1

A₀ θ

θ^T 1

x_n−1k yk

X_1n−1^T k Y₁^Tk · · · X^T_n−1n−1k Y_n−1^T k Θ θ

θ^T 0 · · · θ^T 0 x^T_n−1k yk

×

⎛

⎜⎜

⎜⎜

⎜⎝ B₁⁰ b₁ b^T₁ b¹_nn

· · · · B_n⁰ b_n b^T_n bⁿ_nn

⎞

⎟⎟

⎟⎟

⎟⎠

x_n−1k yk

A₀r11 r12

r₂₁ 1r₂₂

x_n−1k yk

,

2.49

where

r11 r11

x_n−1k, yk ⁿ⁻¹

j1

X_jn−1^T kB⁰_jY_j^Tkb_j^T

B^TX_n−1k Byk,

r12r12

x_n−1k, yk ⁿ⁻¹

j1

X^T_jn−1kbjY_j^Tkb^jnn

B^Tx_n−1k byk,

r₂₁r₂₁

x_n−1k, yk

x^T_n−1kB⁰_nykb^T_n, r₂₂ r₂₂

x_n−1k, yk

x^T_n−1kbnykb_nnⁿ .

2.50

Before the following computations, for the reader’s convenience, we recall that for then− 1×n−1matricesA,A1, 1×n−1vectors , ₁,n−1×1 vectorsC,C1and 1×1 “matrices”

m,m₁, we have A C

m

×

A1 C1 1 m₁

A × A1C × 1 A× C1C ×m1

× A1m× 1 × C1m×m₁

. 2.51

To investigate the stability of system1.6, we use the Lyapunov function

V

x_n−1k, yk

x_n−1^T kHxn−1k hy²k

x^T_n−1k, ykH θ θ^T h

x_n−1k yk

,

2.52

whereHH_n−1is ann−1×n−1constant real symmetric and positive definite matrix.

Let us find the first difference of the Lyapunov function2.52along the solutions of2.49.

We get

ΔV

x_n−1k, yk

x^T_n−1k1, yk1H θ θ^T h

x_n−1k1 yk1

−

x_n−1^T k, ykH θ θ^T h

x_n−1k yk

x^T_n−1k, ykA0r11 r12

r₂₁ 1r₂₂ T

H θ θ^T h

A0r11 r12

r₂₁ 1r₂₂

x_n−1k yk

−

x_n−1^T k, ykH θ θ^T h

x_n−1k yk

x^T_n−1k, ykA^T₀ r₁₁^T r₂₁^T r₁₂^T 1r₂₂

H θ θ^T h

×

A₀r₁₁ r₁₂ r₂₁ 1r₂₂

− H θ θ^T h

×

x_n−1k yk

2.53

or, using formula2.51, ΔV

x_n−1k, yk

x_n−1^T k, ykA^T₀Hr₁₁^TH hr₂₁^T r₁₂^TH h1r₂₂

×

A₀r₁₁ r₁₂ r₂₁ 1r₂₂

− H θ

θ^T h

×

x_n−1k yk

x_n−1^T k, yk

×

⎛

⎝

A^T₀r₁₁^T

HA0r₁₁ r₂₁^Thr₂₁−H

A^T₀ r₁₁^T

Hr₁₂r₂₁^Th1r₂₂ r₁₂^THA0r₁₁ 1r₂₂hr21 r₁₂^THr₁₂ 1r₂₂h1r₂₂−h

⎞

⎠

×

x_n−1k yk

.

2.54 Using formulas2.50, we have

ΔV

x_n−1k, yk

x_n−1^T k, yk

× c₁₁ c₁₂ c₂₁ c₂₂

x_n−1k yk

, 2.55

where

c11 c11

x_n−1k, yk

A0B^TX_n−1k Byk T

H

A0B^TX_n−1k Byk h

B_n⁰T

x_n−1k b_nyk, x^T_n−1kB_n⁰ykb^T_n

−H,

c₁₂ c₁₂

x_n−1k, yk

A₀B^TX_n−1k Byk T

H

B^Tx_n−1k byk h

1x^T_n−1kbnbⁿ_nnyk B_n⁰T

x_n−1k bnyk , c21 c21

x_n−1k, yk c^T₁₂,

c22 c22

x_n−1k, yk

B^Tx_n−1k byk T

H

B^Tx_n−1k byk h

1x^T_n−1kbnb_nnⁿ yk₂

−h.

2.56

Then ΔV

x_n−1k, yk

x^T_n−1kc11x_n−1k 2x^T_n−1kc12yk ykc22yk x^T_n−1

A0B^TX_n−1k Byk T

H

A0B^TX_n−1k Byk h

B⁰_nT

x_n−1k b_nyk

x^T_n−1kB_n⁰ykb^T_n

−H

x_n−1k 2x_n−1^T k

A₀B^TX_n−1k Byk T

H

B^Tx_n−1k byk h

1x_n−1^T kbnbⁿ_nnyk B_n⁰_T

x_n−1k bnyk

yk yk

B^Tx_n−1k byk_T H

B^Tx_n−1k byk h

1x_n−1^T kbnbⁿ_nnyk₂

−h

yk.

2.57

After further computation, we get ΔV

x_n−1k, yk x^T_n−1k

A^T₀HA₀−HA^T₀HB^TX_n−1k X_n−1^T BHA₀ X_n−1^T kBHB^TX_n−1k

X_n−1^T kBHBB^THB^TX_n−1k yk

B^THA0A^T₀HB^T

yk B^THBy ²k h B_n⁰_T

x_n−1kx^T_n−1kB⁰_n 2h

B_n⁰_T

x_n−1kb^T_nyk hb_ny²kb^T_n

x_n−1k 2x^T_n−1k

A^T₀HB^Tx_n−1k A^T₀Hbyk X_n−1^T kBHB^Tx_n−1k X^T_n−1kBHbyk BH B^Tx_n−1kyk BH by ²k h

B_n⁰T

x_n−1k b_nyk

hx_n−1^T kbn

B_n⁰T

x_n−1k b_nyk

hbⁿ_nnyk B⁰_nT

x_n−1k b_nyk

yk

x^T_n−1kBH

B^Tx_n−1k byk b^TH

B^Tx_n−1k byk

yk h

x^T_n−1kbn

2

h

b_nnⁿ yk₂

2hx^T_n−1kbn2hb_nnⁿ yk 2hx^T_n−1kbnbⁿ_nnyk y²k.

2.58