# Exponential Stability of Linear Discrete Systems with Multiple Delays

Authors: J. Baštinec; H. Demchenko; J. Diblík; D. Ya. Khusainov
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delaysxk+1=Axk+∑i=1sBixk-mi, k=0,1,…, where s∈N, A and Bi are square matrices, and mi∈N. New criterion for exponential stability is proved by the Lyapunov method. An estimate of the norm of solutions is given as well and relations to the well-known results are discussed.


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## Body

## 1. Preliminaries

The investigation of the stability of linear difference systems with delay is a constant priority of research. We refer, for example, to [1–14] and to the references therein.The paper considers the exponential stability of linear discrete systems with multiple delays(1)xk+1=Axk+∑i=1sBixk-mi,k=0,1,…where s∈N, A and Bi are n×n matrices, and mi∈N. For (1) exponential-type stability and exponential estimate of the rate of convergence of solutions are derived.Setm≔max{m1,…,ms}. The initial Cauchy problem for system (1) is as follows: (2)xk=xk∈R,k=-m,-m+1,….For a vector x=(x1,…,xn)T, we define |x|2≔∑i=1n‍xi2. Let ρ(A) be the spectral radius of the matrix A. Denote by λmaxA and λminA the maximum and the minimum eigenvalues, respectively, of a symmetric matrix A and define φA≔λmaxAλmin-1A. For a given matrix B, we use the norm defined by |B|2≔λmaxBTB. In the paper, assume |A|+∑i=1s‍|Bi|>0.The trivial solutionx(k)=0, k=-m,-m+1,… of (1) is called Lyapunov exponentially stable if there exist constants N>0 and θ∈(0,1) such that, for an arbitrary solution x=x(k) of (1), (3)xk≤Nx0mθk,k=1,2,…where (4)x0m≔maxxi,i=-m,-m+1,…,0.For the foundations of stability theory to difference equations, we refer, e.g., to [15, 16].As it is customary, the asymptotic stability of (1) can be investigated by analyzing the roots of the related characteristic equation. The characteristic equation relevant to (1) is a polynomial equation of degree m+1n. For large m and n, it is impossible, in a general case, to solve such a problem. For example, the Schur-Cohn criterion [16, 17] is not applied because the computer calculation is too time-consuming.Below, the exponential stability of (1) is analyzed by the second Lyapunov method and the following well-known result is utilized: if ρ(A)<1, then the Lyapunov matrix equation(5)ATHA-H=-Chas a unique solution, a positive definite symmetric matrix H for an arbitrary positive definite symmetric n×n matrix C (we refer, for example, to [16]).In Section2, the exponential stability of system (1) and exponential estimates of solutions are investigated. Concluding remarks and relations to the well-known results are included in Section 3.


## 2. Exponential Stability

Letγ>1 be a parameter. Define auxiliary numbers (6)L1≔γλmaxH-λminC+∑i=1sATHBi,L2≔λminH-12γφH2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBj,L3≔λminC-∑i=1sATHBi-γ-1γλmaxH-12φ2H2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBj.Theorem 1.
Letρ(A)<1, C be a fixed positive definite symmetric n×n matrix, let matrix H solve the equation (5), and, for a fixed γ>1, let L1>0, L2>0, L3≥0. Then, system (1) is exponentially stable and, for an arbitrary solution x=x(k), the estimate(7)xk≤φHx0mγ-k/2,k≥1holds.Proof.
For the Lyapunov functionV(x,k)≔γkxTHx, inequalities(8)γkλminHx2≤Vx,k≤γkλmaxHx2hold. Let δ≔ε/φ(H) where ε>0 is given. Let a solution x(k) of (1) satisfy x(0)m=δ. Then, for k=-m,-m+1,…,0,(9)Vxk,k≤γkλmaxHxk2≤γkλmaxHx0m2≤γkλmaxHδ2=γkλmaxHε2φH=γkε2λminH≤ε2λminH,i.e.,(10)Vxk,k≤ε2λminH.Below, we prove that (10) is valid for k=1,2,…, too. Assume, on the contrary, that (10) is not always valid. Then, an integer k∗>0 exists such that, for k=-m,-m+1,…,k∗, (10) holds, and, for k=k∗+1,(11)Vxk∗+1,k∗+1>ε2λminH.Inequality (11) implies that, for k=-m,-m+1,…,k∗,(12)γkλminHxk2≤Vxk,k≤ε2λminH<Vxk∗+1,k∗+1≤γk∗+1λmaxHxk∗+12and(13)xk<γk∗+1-k/2φHxk∗+1,k=-m,-m+1,…,k∗.Now compute(14)ΔVxk∗,k∗=Vxk∗+1,k∗+1-Vxk∗,k∗=γk∗+1xTk∗+1Hxk∗+1-γk∗xTk∗Hxk∗=γk∗+1Axk∗+∑i=1sBixk∗-miTHAxk∗+∑i=1sBixk∗-mi-γk∗xTk∗Hxk∗.Rearranging this computation, we derive(15)ΔVxk∗,k∗=-γk∗+1xTk∗H-ATHAxk∗+2γk∗+1xTk∗ATH∑i=1sBixk∗-mi+γk∗+1∑i,j=1sxTk∗-miBiTHBjxk∗-mj+γk∗γ-1xTk∗Hxk∗.We estimate the first difference and use the assumption that the matrix H is a solution of equation (5); therefore,(16)ΔVxk∗,k∗≤-γk∗+1λminCxk∗2+2γk∗+1∑i=1sATHBixk∗xk∗-mi+γk∗+1∑i,j=1sBiTHBjxk∗-mixk∗-mj+γk∗γ-1λmaxHxk∗2and(17)ΔVxk∗,k∗≤-γk∗+1λminCxk∗2+γk∗+1∑i=1sATHBixk∗2+xk∗-mi2+12γk∗+1∑i,j=1sBiTHBjxk∗-mi2+xk∗-mj2+γk∗γ-1λmaxHxk∗2.Now we apply inequality (13) to get(18)ΔVxk∗,k∗≤-γk∗+1λminCxk∗2+γk∗+1∑i=1sATHBixk∗2+γmi+1φHxk∗+12+12γk∗+1∑i,j=1sBiTHBjγmi+1+γmj+1φHxk∗+12+γk∗γ-1λmaxHxk∗2and(19)ΔVxk∗,k∗≤-γk∗+1λminC-∑i=1sATHBi-γ-1γλmaxHxk∗2+12γk∗+2φH2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBjxk∗+12.Inequality (20)λminC-∑i=1sATHBi-γ-1γλmaxH>0can be deduced from the assumption L3≥0. Therefore, utilizing (8),(21)ΔVxk∗,k∗≤-γλminC-∑i=1sATHBi-γ-1γλmaxHVxk∗,k∗λmaxH+12γφH2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBjVxk∗+1,k∗+1λminH.Since ΔV(x(k∗),k∗)=V(x(k∗+1),k∗+1)-V(x(k∗),k∗), we get(22)1-12γφHλminH2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBjVxk∗+1,k∗+1≤1-γλmaxHλminC-∑i=1sATHBi-γ-1γλmaxHVxk∗,k∗.This inequality can be rewritten as (23)L2λminHVxk∗+1,k∗+1≤L1λmaxHVxk∗,k∗or as(24)Vxk∗+1,k∗+1≤Θ·Vxk∗,k∗where (25)Θ≔L1L2φH>0.Now we prove that(26)Θ≤1.Inequality (26) is equivalent with an inequality(27)λmaxH-γλminC-∑i=1sATHBi-γ-1γλmaxH≤λminH-12γφH2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBjλmaxHλminH.After some simplification, we get(28)λminC-∑i=1sATHBi-γ-1γλmaxH≥12φ2H2∑i=1sγmiATHBi+∑i,j=1sγmi+1+γmj+1BiTHBj,which is equivalent with the inequality L3≥0. Then (24), (26), and (10) imply (29)Vxk∗+1,k∗+1≤Θ·Vxk∗,k∗≤Vxk∗,k∗≤ε2λminH.This inequality contradicts (11). Then, inequality (11) is impossible and (10) holds for every k=1,2,…. Moreover, (8) and (10) imply (30)γkλminHxk2≤Vxk,k≤ε2λminH=δ2λmaxH=x0m2λmaxH,i.e., the inequality (31)γkλminHxk2≤x0m2λmaxH,k≥1,equivalent with (7).


## 3. Concluding Remarks

Based on the investigations on exponential stability published previously, the present paper brings in Theorem1 new results. The exponential rate of convergence of solutions is studied in [1] assuming that det⁡A≠0; therefore, the results are independent. Let us discuss the independence of the results of other sources listed in the references. The criteria for the exponential stability of nonlinear difference systems, for example, are proved in [11, 14]. The nonlinearities are estimated by some linear terms with matrices having nonnegative entries with the sums of such matrices being, for example, a constant nonnegative matrix with a spectrum less than 1. In general, an attempt to estimate the right-hand sides of the systems by a nonnegative matrix does not provide a matrix with a spectrum less than 1 and the results are independent. For special classes of equations, sharp criteria (depending on delay) for detecting asymptotic stability are proved in [2, 3]. The following example illustrates the above-mentioned independency of results.Example 2.
Letn=s=2 and let system (1) be of the form(32)x1k+1=x1k+x2k+μx2k-m1,(33)x2k+1=-x1k-x2k+νx1k-m2where k≥0 and μ and ν are constants. We show that Theorem 1 is applicable if |μ| and |ν| are sufficiently small. We have (34)A=11-1-1,B1=0μ00,B2=00ν0.Lyapunov equation (5) is satisfied, e.g., for (35)C=0.90.90.91,H=1111.1.Then, λmax(H)≐2.0512492, λmin(H)≐0.0487508, λmin(C)≐0.0486122, and φ(H)≐42.0762336. Simple computations result in (36)ATHB1=0,ATHB2=0.12ν,B1THB1=μ2,B2THB2=1.1ν2,B1THB2=μν,L1=γλmaxH-λminC+∑j=12ATHBj≐γ2.0026370+0.12ν,L2=λminH-12γφH2∑i=12γmiATHBi+∑i,j=12γmi+1+γmj+1BiTHBj≐0.0487508-γ42.0762336γm20.12ν+γm1+1μ2+γm1+1+γm2+1μν+γm2+11.1ν2and (37)L3=λminC-∑i=12ATHBi-γ-1γλmaxH-12φ2H2∑i=12γmiATHBi+∑i,j=12γmi+1+γmj+1BiTHBj≐0.0486122-0.12ν-γ-1γ2.0512492-42.07623362γm20.12ν+γm1+1μ2+γm1+1+γm2+1μν+γm2+11.1ν2.Theorem 1 is applicable if |μ| and |ν| are sufficiently small since this implies Li>0, i=1,2, and, if the expression (38)0.0486122-γ-1γ2.0512492=2.0512492γ-2.0026370is positive, provided that γ>1; that is, if (39)1<γ<2.05124922.0026370≐1.0242741,then L3>0 as well. In such a case, for an arbitrary solution xk=x1k,x2kT of system (32), (33), the estimate (40)xk≤φHx0mγ-k/2≐42.0762336x0mγ-k/2,k≥1holds.Sincedet⁡A=0 in the above example, the results of the paper [1] are not applicable to system (32), (33). Moreover, an attempt to apply results of [11, 14] is not successful since the sum of matrices A∗, B1∗, and B2∗, defined by replacing the entries in the previously given matrices A, B1, and B2 by their absolute values, leads to a matrix (41)U≔A∗+B1∗+B2∗=1111+0μ00+00ν0=11+μ1+ν1whose eigenvalues are λ1,2(U)=1±(1+|μ|)(1+|ν|), and, obviously, ρ(U)≥1.Finally, we compare the results published in [4–7] with Theorem 1. The assumptions of Theorem 1 are, for the reduced case s=1 of a single delay, weaker than those of Theorem 2 in [7]. In [4] an analysis of Theorem 2 is carried out. Although the results are independent, a limiting process (for γ→1+) indicates that the conditions of the main result in [7] are, in general, more restrictive. Now we will demonstrate that, with respect to the derived estimates of the norms of solutions, the situation is just the opposite and that the estimation (7) is, in general, better than that in [4, Theorem 2]. The last estimation mentioned says that (below, s, A, Bi, i=1…,s, H and C are the same as in the paper)(42)xk≤φHx0mΘk/2m+1H,k≥1,where (43)ΘH≔1λmaxHLH-∑i=1sLiH+sλminH,LH≔λmaxH-λminC+∑j=1sATHBj,LiH≔λminH-φHATHBi+∑j=1sBiTHBj,i=1,…,s,if ρ(A)<1, C is a fixed positive definite matrix, matrix H solves the corresponding Lyapunov matrix equation (5), and (44)LH-∑i=1sLiH<λmaxH-sλminH,LH>0.Assuming that |Bi|→0, i=1,…,n, we deduce that for (44) to hold, the following is necessary:(45)λmaxH-λminC>0,the limiting value of Θ(H) is (46)ΘH≐λmaxH-λminCλmaxH,and (42) can approximately be written as(47)xk≤φHx0mλmaxH-λminCλmaxHk/2m+1,k≥1.Considering the same limiting process as above, for the validity of (7), an analysis of Li, i=1,2,3 implies that inequality (45) must hold in addition to inequality(48)λminC-λmaxH+1γλmaxH>0,derived from the assumption L3≥0. Inequality (48), together with the assumption γ>1, yields (49)1<γ<λmaxHλmaxH-λminCand (7) can be approximatively written as(50)xk≤φHx0mλmaxH-λminCλmaxHk/2,k≥1.Obviously, estimation (50) is (due to the absence of the maximal delay m) better than estimation (47). We finish this part with a remark that the results of [5] are generalized in [4]. Results of [6] are on the exponential stability of linear perturbed systems with a single delay. Among others, it is proved [6, Theorem 3] that inequality (50) holds for nondelayed linear systems (51)xk+1=Axk,k=0,1,,….



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