A converse to global exponential stability of a class of difference equations

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R E S E A R C H

Open Access

A converse to global exponential stability of a

class of difference equations

Yinghua Fu

1,2*

_{, Qunfei Zhao}

2

_{and Lisheng Wang}

2

*_{Correspondence: [email protected]} 1_{School of Optical-Electrical and}

Computer Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, P.R. China

2_{Department of Automation,} Shanghai Jiao Tong University, Shanghai, 200240, P.R. China

Abstract

This paper discusses the global exponential stability of a class of difference equations. Sufficient and necessary conditions for the global exponential stability are derived. Particularly, the equivalent relationship between the global exponential stability of difference equations and the contractive property of the nonlinear operator of the systems is shown.

Keywords: global exponential stability; exponential bound; diﬀerence equations; strongly equivalent metric

1 Introduction

In mathematics and automatic control, an equation system is discussed as discrete or con-tinuous one with the type of input variables and linear or nonlinear one according to the relationship between the input and output. The stability of the equation systems always catches researchers’ eyes. The Lyapunov function is important for stability theory and con-trol theory, and its existence is a necessary and sufficient condition of stability for many differential equation systems. Linear systems received more attention than nonlinear sys-tems because of simplicity and controllability, and many good results are shown in the textbooks and literature. Bastinec [] gives the sufficient condition for exponential stabil-ity and estimation of solutions of linear differential systems with delay-dependence. Non-linear systems are of interest to researchers and engineers because most physical systems are inherently nonlinear in nature and difficult to solve. The converse Lyapunov problem on the stability of nonlinear discrete dynamical systems is developed and studied by many researchers (Gordon []; Ortega []; Mousaet al.[]; Jiang and Wang []; Wanget al.[]). Diblik [] considers a particular critical case and gives conditions for the stability of a zero solution of difference systems with quadratic nonlinearities using the method of Lyapunov functions and derives classes of stable systems. Furthermore, the author estimates the sta-bility domains as well. Berezansky [] considers the asymptotic convergence of solutions in discrete and difference equations with delay and gives strong sufficient conditions of the asymptotic convergence of all solutions without computing the limits of the solutions as

n→ ∞. It demonstrates that a diﬀerence equation is of exponential stability inRn_{if and}

only if there is a Lyapunov function which is an abstract presentation of system energy (Mousaet al.[]; Jiang and Wang []; Wanget al.[]; Wanget al.[]). The inverse prob-lem of exponential stability instead of the inverse Lyapunov function has been considered from a diﬀerent view in the recent literature (Wanget al.[]; Wang and Xu []).

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Consider the following diﬀerence equation (Wanget al.[]; Wang and Xu []):

xk+=Txk, x∈D,k= , , , . . . , ()

whereD⊂Rn_{is a bounded closed set,}_T_:_D_→_D_{is nonlinear and continuously}

differen-tiable inD. ThenT is a Lipschitz continuous operator.x∗∈Dis the unique equilibrium point (fixed point). Wanget al.[], Wang and Xu [] show that system () is of expo-nential stability inDif and only ifT is contractive inDwith a certain strongly equivalent metric and also present a converse not only to the exponential stability of system () dif-ferent from the Lyapunov function, but also to the Banach contraction theorem different from the ones in Meyers [], Leader [], Opoitsev []. In other words, the two works show thatTis contractive inDwith a certain strongly equivalent metric if its iterative se-quence{xk}exponentially converges tox∗for any initial pointx∈D. The result is helpful to understand the exponential stability ofx*_{from a different view. However, Wang}_{et al.} [], Wang and Xu [] only consider nonlinear discrete dynamical systems in the bounded setD. The problem whether the result can be extended to difference equations in an un-bounded set still remains open. It is hard to answer for general difference equations.

This paper focuses on a class of important diﬀerence equations inRn_{shown as follows:}

xk+=Txk, x∈Rn,k= , , , . . . , ()

where x∗∈Rnis the unique equilibrium point andT :Rn→Rnis a continuous diﬀer-entiable nonlinear operator inRn_{, and}_lim

x→∞supT(x)=λ<  holds for a positive constantλ(e.g.,  <λ< ). System () includes the discrete-time neural network model in Wang and Xu [] and diﬀerence equations in Wanget al.[], Wang and Xu [] as the spe-cial cases. The exponential stability of system () is developed and necessary and suﬃcient conditions to be exponentially stable inRn_{(namely, globally exponentially stable) are}

de-rived. Particularly, we give the result that system () is of global exponential stability if and only ifTis contractive inRn_{with a certain strongly equivalent metric. A simple method}

to distinguish global exponential stability from global asymptotic stability is proposed. The results in this paper are helpful to discover some properties of the global exponential stability of system ().

2 Deﬁnition and main results

A nonlinear operatorT is Lipschitz continuous ifTx–Ty ≤M· x–yis satisﬁed for anyx,y∈Dand the given constantM(M> ), and contractive inE⊂Rnwith a strongly equivalent metricd(·,·) if there is a constant  <β<  such thatd(Tx,Ty)≤β·d(x,y) holds for anyx,y∈E. A metric functiond(·,·) :E×E→R+_{is a strongly equivalent metric with} the vector norm · (or strongly equivalent with · ) in the setE⊂Rn_{if there are two}

constantsC≥C>  such that for anyx,y∈E, the following inequalities hold (Wang and Xu []):

C·d(x,y)≤ x–y ≤C·d(x,y). ()

The equilibrium point of some operator (for example, () or ()) is globally asymptotically stable inRn_{if it is stable in the sense of Lyapunov and}_lim

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And it is exponentially stable in the setE⊂Rn_{if there are constants}_M_{>  and  <}_α_{< }

such that for any initial pointxinEand the positive integerk,

xk–x∗≤M·αk·x–x∗. ()

Letbe the set of all equivalent norms of · ,be the set of all strongly equivalent metrics of · inE⊂Rnandbe the set of all topologically equivalent metrics of · inE. Then the inclusion relationship⊂⊂is satisﬁed. In other words, each equiva-lent norm of · is also a strongly equivalent metric, and each strongly equivalent metric is a topologically equivalent metric. Generally,Tmaybe is not contractive inRn_{with any}

equivalent norm even if system () is of exponential stability. Besides, system () is possibly not of exponential stability inRn_{even if}_T_{is contractive with certain topologically}

equiv-alent metric (Wang and Xu []). In this paper, we show that strongly equivequiv-alent metrics are a class of appropriate equivalent metric functions describing the exponential stability of system ().

SupposeF:E→Eis a Lipschitz continuous nonlinear operator in the arbitrary subset

E⊂Rn_{. Then the lub Lipschitz constants of}_F_with_{· ∈}_and_d₍_·_,_·₎_∈_{are deﬁned as}

follows, respectively (Wang and Xu []; Soderlind []):

L (F,E) = sup

x,y∈E,x=y

Fx–Fy

x–y , Ld(F,E) =x,ysup∈E,x=y

d(Fx,Fy)

d(x,y) , ()

whereL (F,E) andLd(F,E) are two functions determined byF,Eand·ord(·,·). Hence,

for anyx,y∈E, one obtains

Fx–Fy ≤L (F,E)· x–y, d(Fx,Fy)≤Ld(F,E)·d(x,y).

Wang and Xu [] give the result thatlimk→∞L (Fk,E)/k andlimk→∞Ld(Fk,E)/k exist

and the following formulas hold:

lim

k→∞Ld

Fk,E/k= lim

k→∞L

Fk,E/k=inf

k L

Fk,E/k.

DenoteLip(F,E) =lim_k_→∞L (Fk_,_E₎/k_{. Then}_Lip₍_T_,_E_{) is a constant independent of}

dif-ferent equivalent norms or strongly equivalent metrics. L (F,E) andLip(T,E) can be regarded as nonlinear generalizations of the matrix norm and matrix spectral radius. By Wang and Xu [], one can have

Lip(F,E) = inf

d(·,·)∈Ld(F,E). ()

Namely,Lip(T,E) is a minimum Lipschitz constant. Furthermore,F is contractive in

Ewith respect to a certain strongly equivalent metric inif and only ifLip(F,E) <  or

L (Tm_,_E_{) <  holds for a positive integer}_m_.

Lemma  In system(),L (T,Rn_{) <}_∞_holds_._{And for any x}_∈_Rn_{and a positive integer j}_,

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Hence, the nonlinear operatorTin system () is Lipschitz continuous inRn_{. The main}

results of this paper are presented below.

Theorem  In system(),the following ﬁve conditions are equivalent:

(A) x*_{is globally exponentially stable in}_Rn_.

(B) x*_{is globally asymptotically stable in}_Rn_and_ρ₍_T₍_x∗_{)) < }_.

(C) Lip(T,Rn_{) < }_.

(D) Tis contractive inRnwith certain strongly equivalent metrics of .

(E) The power ofT is contractive inRn_{with the norm}_,_i.e._,_{there exist a positive}

integermand a constant <β< such thatTm_x_–_Tm_y_≤_β_·_x_–_y_{holds for}

anyx,y∈Rn.

As long as system () is of global exponential stability, we haveLip(T,Rn_{) <}_α_{, and for}

any suﬃciently smallε∈(,  –Lip(T,Rn_)),_Lip₍_T_,_Rn_{) +}_ε_{is an exponential bound. Here,}

α=max{λ,ρ(T(x∗))}andρ(T(x∗)) is the spectral radius of the Jacobian matrixT(x∗). Theorem  shows that we can distinguish global exponential stability from global asymp-totic one for system () by calculating the spectral radius of the Jacobian matrix at the equilibrium pointx∗, namely,ρ(T(x∗)). It tells us that the global exponential stability of system () is equivalent to the contractive property of the nonlinear operatorTwith cer-tain strongly equivalent metrics of the norm · . It actually describes a new converse to the Banach contraction theorem diﬀering from the ones in Meyers [], Leader [], Opoitsev [].

Theorem  In system(),assumelimx→∞supT(x) ≤ρ(T(x∗)).If x∗is globally

ex-ponentially stable in Rn_,_then_Lip₍_T_,_Rn_{) =}_ρ₍_T₍_x∗₎₎_and_ρ₍_T₍_x∗₎₎_{is the inﬁmum of all}

exponential bounds of convergent trajectories.Particularly,

xk–x∗≤

C(ε)

C(ε) ·

ρTx∗+εk·x–x∗

holds for anyε> and x∈Rn,where C(ε)>C(ε)> are the same constants as in

The-orem.

Theorem  shows that exponential bounds of convergent trajectories describing the global property of trajectory motion in the whole space actually can be determined by the local information at the equilibrium point, namelyρ(T(x∗)).

3 Proofs

In this section, the proofs of results and theorems will be given.

Proof of Lemma Sincelimx→∞supT(x)=λ, for anyε> , there isb>  such that

T₍_x₎_≤_λ₊_ε_as_x_–_x∗_≥_b_{. Let}_F₍_b_{) =}_{x_∈_Rn_:_x_–_x∗_≤_b}_{. Since}_T _{is continuously}

diﬀerentiable inRn_,_T₍_x₎_{is a continuous function in the compact set} _F₍_b_{). Thus, we}

have sup_z_∈_F₍_b₎T(z)<∞. This implies that M=sup_z_∈_RnT(z)<∞holds. From the mid-value theorem, for anyx,y∈Rn, one obtains

Tx–Ty ≤sup

z∈Rn

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This implies thatL (T,Rn₎_≤_M_<_∞_{holds. Since}_Tj_{is continuously diﬀerentiable, with}

Proof of Theorem The equivalent relationships between (C), (D) and (E) can be easily gotten from Wang and Xu []. And it is also easy to deduce (B) from (A). Next, we mainly prove (C)⇒(A) and (B)⇒(C).

Thus, system () is of global exponential stability in Rn _{with the exponential bound}

Lip(T,Rn_{) +}_ε_.

Secondly, prove (B)⇒(C); namely, if system () is of global exponential stability inRn

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is a continuous function ofx∈Rn_{. It implies that there exists a spherical neighborhood}

U(r) ={x∈Rn_:_x_–_x∗_ε_≤_r_}_{such that for any}_x_∈_U₍_r_),

T(x)_ε≤Tx∗_ε+ε/≤ρTx∗+ε.

By the mid-value theorem, for anyx,y∈U(r), we have

Tx–Tyε≤ sup

z∈U(r)

T(z)_ε· x–yε≤ρTx∗+ε· x–yε. ()

If system () is of global exponential stability inRn_{, it is globally uniformly asymptotically}

stable inRnas well (Elaydi []). Therefore, there exists a positive integerNsuch that for any positive integerk≥Nand anyx∈B(b),Tk_x_∈_U₍_r_{) holds. With equation (), for any}

x,y∈B(b), ifk>N, we have

Tkx–Tky_ε≤ρTx∗+ε·Tk–x–Tk–y_ε≤ · · · ≤ρTx∗+εk–N·TNx–TNy_ε ≤ρTx∗+εk–N·L ε

T,RnN· x–yε. It implies that for any positive integerk>N, one obtains

L ε

Tk,B(b)≤ρTx∗+εk–N·L ε

T,RnN.

SinceL ε(T,Rn) <∞, then

LipT,B(b)= lim

k→∞L ε

Tk,B(b)/k

≤ lim

k→∞

ρTx∗+ε(k–N)/k· lim

k→∞L ε

T,RnN/k ≤ρTx∗+ε.

Lip(T,B(b))≤ρ(T(x∗)) holds forεis an arbitrary positive number. Thus, there exists a positive integerNsuch thatL (Tk,B(b))≤(ρ(T(x∗)) +ε)kfor the fixedε>  as a positive integerk≥N. For any positive integerk≥N,Tkis continuously differentiable, and for anyx∈F(b)⊂B(b),xis an inner point ofB(b). By the definition of the Gateaux differential, for anyx∈F(b),k≥Nandh∈Rnwithh= , one can obtain

Tk(x)·h= lim

t→+

Tk₍_x₊_t_·_h_{) –}_Tk_x

t =tlim→+

Tk₍_x₊_t_·_h_{) –}_Tk_x

t·h ≤L Tk,B(b).

It implies that for anyx∈F(b) andk≥N,

Tk(x)= sup h=

Tk(x)·h≤L Tk,B(b)

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holds. For any z∈ Rn_{, either} _z _∈_F₍_b_{) or} _z _∈ _E₍_b_{), the result that if} _z_∈ _F₍_b_{), then}

(Tk₎₍_z₎_≤_β_{holds for any positive integer}_k_≥_N

is shown above. Next, we will prove that a similar result exists for anyz∈E(b). For any givenz∈E(b) and any given positive integerk≥N, there are the following two diﬀerent cases.

Case : Ti_z_∈_E₍_b_{) for any positive integer} _i_<_k_{. From equation (), one may have}

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Proof of Theorem Since system () is of global exponential stability inRn_{, there are} con-is globally exponentially stable inRn_{. And from equation (), for any}_ε_{>  and}_k_{, we have}

Tkx–x∗≤C(ε) is not less than the inﬁmum of exponential bounds of convergent trajectories. Moreover, from Wanget al.[], Wang and Xu [] if system () is of global exponential stability in

Rn _{with an exponential bound}_α_{, then}_α_≥_ρ₍_T₍_x∗_{)). Therefore,}_ρ₍_T₍_x∗_{)) is not larger}

than the inﬁmum of exponential bounds of convergent trajectories. Finally,ρ(T(x∗)) is the inﬁmum of exponential bounds of convergent trajectories.

4 Examples

In this section, we present two examples to clarify the concepts introduced in this paper and apply the obtained results to several concrete nonlinear discrete systems.

Example  Consider the following recurrent neural network model studied in Jin []:

xk+= –Axk+WF(Bxk) +u, x∈Rn, ()

wherex*_∈_Rn_{is an equilibrium point,}_A₌_diag_{_a

,a, . . . ,ann}is a diagonal state feedback

coeﬃcient matrix,B=diag{b,b, . . . ,bnn}withbii>  is the activation gain matrix; the

interconnection matrixWis the lower triangular given by

W=

In Jinet al.[], it has been shown that () will be of global asymptotical stability inRn

if|aii|+|wii|bii<  for any ≤i≤n. In Wang and Xu [], () is proven to be of the global

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stability in the whole space is more interesting than in a bounded region, we will focus on it in the following.

DenoteT(x) = –Ax+WF(B(x)) +u,x∈Rn_{. Then it is easy to see that the Jacobian matrix}

ofTatx*is given byT(x*) = –A+WF(B(x*))Band its eigenvalues are given by

–aii+wii·fi

bii·x*i

·bii: ≤i≤n.

Thus, whenever|aii|+|wii|bii< , we haveρ(T(x*)) <  andlimx→∞supT(x)< ; thus

system () is of global exponential stability.

Example  Consider a general class of discrete-time dynamic neural networks with con-tinuous states described in (Jin []) as follows:

x(k+ ) = –Ax(k) +Wσψx(k)+s, ()

wherex= (x,x, . . . ,xn)Tis the neural state vector,W= (wij)n×nis the synaptic weight

ma-trix,s= (s,s, . . . ,sn)Tis the constant threshold vector,A=diag{a,a, . . . ,an}with|ai|< 

is the feedback coeﬃcient matrix,ψ=diag{μ,μ, . . . ,μn}is the matrix of activation gains

for controlling state decaying, andσ(ψx) = (σ(μx),σ(μx), . . . ,σ(μnxn))Tis the

vector-valued activation function with a gain matrixψ.

The nonlinear neural activation functionσ(·) may be chosen as a continuous and diﬀer-entiable nonlinear sigmoid function satisfying the following conditions: ()σ(x)→ ± as

x→ ±∞; ()σ(x) is bounded with the upper bound  and the lower bound –; ()σ(x) =  at a unique pointx= ; ()σ(x) >  andσ(x)→ asx→ ±∞; and ()σ(x) has a global maximal value one.

Jin [] only shows the global asymptotic stability of system (). The global exponential stability is discussed as follows.

Denote T(x) = –Ax +Wσ(ψx) + s, x ∈ Rn, then T(x) = –A +Wσ(ψx)ψ. From

T₍_x₎_{≤ A}₊_W_σ₍_ψx₎_ψ _and _σ₍_x₎_→_{ as} _x _{→ ±∞}_{, it is easy to know that} limx→∞supT(x)<  as  <|ai|< . Ifρ(T(x*)) <  is obtained atx*, then system ()

is of global exponential stability.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LW gave the key idea about the connection between global exponential stability and global asymptotic stability out and participating the revision and illustration. QZ participated the proof of Lemma 1 and part of Theorem 1. YF carried out the left work including the proof of Theorem 1 and Theorem 2 gave the two examples. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the reviewers for their valuable comments and helpful suggestions that greatly improved the note’s quality.

Received: 5 May 2012 Accepted: 18 December 2012 Published: 11 January 2013 References

1. Bastinec, J, Diblik, J, Khusainov, DY, Ryvolova, A: Exponential stability and estimation of solutions of linear diﬀerential systems of neutral type with constant coeﬃcients. Bound. Value Probl.2010, Article ID 956121 (2010).

doi:10.1155/2010/956121

(10)

3. Ortega, JM: Stability of diﬀerence equations and convergence of iterative processes. SIAM J. Numer. Anal.10, 268-282 (1973)

4. Mousa, MS, Miller, RK, Michel, AN: Stability analysis of hybrid composite dynamical systems: descriptions involving operators and diﬀerence equations. IEEE Trans. Autom. Control31, 603-615 (1986)

5. Jiang, ZP, Wang, Y: A converse Lyapunov theorem for discrete-time systems with disturbances. Syst. Control Lett.45, 49-58 (2002)

6. Wang, LS, Xu, ZB, Zhao, QF: Quantitative characterizations of exponential convergent property of nonlinear discrete dynamical systems. IEEE Trans. Autom. Control52(11), 2129-2134 (2007)

7. Diblik, J, Khusainov, DY, Grytsay, IV, Smarda, Z: Stability of nonlinear autonomous quadratic discrete systems in the critical case. Discrete Dyn. Nat. Soc.2010, Article ID 539087 (2010). doi:10.1155/2010/539087

8. Berezansky, L, Diblik, J, Ruzickova, M, Suta, Z: Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case. Abstr. Appl. Anal.2011, Article ID 709427 (2011). doi:10.1155/2011/709427 9. Wang, LS, Heng, PA, Leung, KS, Xu, ZB: Global exponential asymptotic stability in nonlinear discrete dynamical

systems. J. Math. Anal. Appl.258, 349-358 (2001)

10. Wang, LS, Xu, ZB: Suﬃcient and necessary conditions for global exponential stability of a class of general discrete-time recurrent neural networks. IEEE Trans. Circuits Syst. I, Regul. Pap.53(6), 1373-1380 (2006)

11. Wang, LS, Xu, ZB: On characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions. IEEE Trans. Autom. Control52(10), 1871-1881 (2007)

12. Meyers, PR: A converse to Banach’s contraction theorem. J. Res. Natl. Bur. Stand.71(B), 73-76 (1967) 13. Leader, S: A topological characterization of Banach contractions. Pac. J. Math.69(2), 461-466 (1977) 14. Opoitsev, VI: A converse to the principle of contracting maps. Russ. Math. Surv.31(4), 175-204 (1976) 15. Soderlind, G: On nonlinear difference and differential equations. BIT Numer. Math.24, 667-680 (1984) 16. Elaydi, SN: An Introduction to Difference Equations. Springer, New York (1995)

17. Jin, L, Gupta, MM: Global asymptotic stability of discrete-time analog neural networks. IEEE Trans. Neural Netw.7(6), 1024-1031 (1996)

doi:10.1186/1687-1847-2013-9