Exponential Stability of Linear Discrete Systems with Multiple Delays
Abstract
The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with single delay $x\left( {k+1} \right)=Ax\left( k \right)+\sum_{i=1}^sB_ix\left( {k-m_i} \right)$, $k=0,1,\dots$ where $s\in \mathbb{N}$, $A$ and $B_i$ are square matrices and $m_i\in\mathbb{N}$. 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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Research Article Exponential Stability of Linear Discrete Systems with Multiple Delays J. Baštinec,1H. Demchenko,2J. Diblík ,1and D. Ya. Khusainov1 1Brno University of Technology, CEITEC-Central European Institute of Technology, Brno, Czech Republic 2Brno University of Technology, Faculty of Electrical Engineering and Communication Brno, Czech Republic Correspondence should be addressed to J. Dibl´ ık; [email protected]tbr.cz Received 14 January 2018; Accepted 8 April 2018; Published 1 August 2018 Academic Editor: Pasquale Candito Copyright © 2018 J. Baˇ stinec 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. The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays 𝑥(𝑘+1)=𝐴𝑥(𝑘)+∑𝑠 𝑖=1 𝐵𝑖𝑥(𝑘−𝑚𝑖),𝑘=0,1,...,where𝑠∈N,𝐴and 𝐵𝑖are 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. 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 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑠 ∑ 𝑖=1𝐵𝑖𝑥(𝑘−𝑚𝑖), 𝑘=0,1,... (1) where 𝑠∈N,𝐴and 𝐵𝑖are 𝑛×𝑛matrices, and 𝑚𝑖∈N.For (1) exponential-type stability and exponential estimate of the rate of convergence of solutions are derived. Set 𝑚flmax{𝑚1,...,𝑚𝑠}.TheinitialCauchyproblemfor system (1) is as follows: 𝑥(𝑘)=𝑥𝑘∈R, 𝑘=−𝑚,−𝑚+1,.... (2) For a vector 𝑥=(𝑥 1,...,𝑥𝑛)𝑇, we define |𝑥|2fl∑𝑛 𝑖=1 𝑥2 𝑖. Let 𝜌(𝐴) be the spectral radius of the matrix 𝐴.Denote by 𝜆max(A)and 𝜆min(A)the maximum and the minimum eigenvalues, respectively, of a symmetric matrix Aand define 𝜑(A)fl𝜆max(A)𝜆−1 min(A).ForagivenmatrixB,weusethe norm defined by |B|2fl𝜆max(B𝑇B). In the paper, assume |𝐴|+∑𝑠 𝑖=1 |𝐵𝑖|>0. The trivial solution 𝑥(𝑘)=0,𝑘=−𝑚,−𝑚+1,...of (1) is called Lyapunov exponentially stable if there exist constants 𝑁>0and 𝜃∈(0,1)such that, for an arbitrary solution 𝑥= 𝑥(𝑘)of (1), |𝑥(𝑘)|≤𝑁‖𝑥(0)‖𝑚𝜃𝑘, 𝑘=1,2,... (3) where ‖𝑥(0)‖𝑚flmax {|𝑥(𝑖)|,𝑖=−𝑚,−𝑚+1,...,0}.(4) Forthefoundationsofstabilitytheorytodifferenceequations, 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+1)𝑛.Forlarge𝑚and 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 thesecondLyapunovmethodandthefollowingwell-known result is utilized: if 𝜌(𝐴) < 1, then the Lyapunov matrix equation 𝐴𝑇𝐻𝐴−𝐻=−𝐶 (5) has a unique solution, a positive definite symmetric matrix 𝐻 for an arbitrary positive definite symmetric 𝑛×𝑛matrix 𝐶(we refer, for example, to [16]). Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 9703919, 7 pages https://doi.org/10.1155/2018/9703919
2 Discrete Dynamics in Nature and Society In Section 2, 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 Let 𝛾>1be a parameter. Define auxiliary numbers 𝐿1fl𝛾[𝜆max (𝐻)−𝜆min (𝐶)+𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖], 𝐿2fl𝜆min (𝐻)−1 2𝛾𝜑(𝐻)[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ], 𝐿3fl𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖−𝛾−1 𝛾𝜆max (𝐻)−1 2 ⋅𝜑2(𝐻)[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]. (6) Theorem 1. Let 𝜌(𝐴) < 1,𝐶be a fixed positive definite symmetric 𝑛×𝑛matrix, let matrix 𝐻solve the equation (5), and, for a fixed 𝛾>1,let𝐿1>0,𝐿2>0,𝐿3≥0.Then, system (1) is exponentially stable and, for an arbitrary solution 𝑥=𝑥(𝑘),theestimate |𝑥(𝑘)|≤√𝜑(𝐻)‖𝑥(0)‖𝑚𝛾−𝑘/2, 𝑘≥1 (7) holds. Proof. For the Lyapunov function 𝑉(𝑥,𝑘) fl𝛾𝑘𝑥𝑇𝐻𝑥, inequalities 𝛾𝑘𝜆min (𝐻)|𝑥|2≤𝑉(𝑥,𝑘)≤𝛾𝑘𝜆max (𝐻)|𝑥|2(8) hold. Let 𝛿fl𝜀/√𝜑(𝐻)where 𝜀>0is given. Let a solution 𝑥(𝑘)of (1) satisfy ‖𝑥(0)‖𝑚=𝛿.Then,for𝑘=−𝑚,−𝑚+ 1,...,0, 𝑉(𝑥(𝑘),𝑘)≤𝛾𝑘𝜆max (𝐻)|𝑥(𝑘)|2 ≤𝛾𝑘𝜆max (𝐻)‖𝑥(0)‖2 𝑚≤𝛾𝑘𝜆max (𝐻)𝛿2 =𝛾𝑘𝜆max (𝐻)𝜀2 𝜑(𝐻)=𝛾𝑘𝜀2𝜆min (𝐻) ≤𝜀2𝜆min (𝐻), (9) i.e., 𝑉(𝑥(𝑘),𝑘)≤𝜀2𝜆min (𝐻).(10) 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∗>0exists such that, for 𝑘=−𝑚,−𝑚+1,...,𝑘∗,(10)holds, and, for 𝑘=𝑘∗+1, 𝑉(𝑥(𝑘∗+1,𝑘∗+1))>𝜀2𝜆min (𝐻).(11) Inequality (11) implies that, for 𝑘=−𝑚,−𝑚+1,...,𝑘∗, 𝛾𝑘𝜆min (𝐻)|𝑥(𝑘)|2≤𝑉(𝑥(𝑘),𝑘)≤𝜀2𝜆min (𝐻) <𝑉(𝑥(𝑘∗+1),𝑘∗+1) ≤𝛾𝑘∗+1𝜆max (𝐻)𝑥(𝑘∗+1)2 (12) and |𝑥(𝑘)|<𝛾(𝑘∗+1−𝑘)/2√𝜑(𝐻)𝑥(𝑘∗+1), 𝑘=−𝑚,−𝑚+1,...,𝑘∗.(13) Now compute Δ𝑉(𝑥(𝑘∗),𝑘∗)=𝑉(𝑥(𝑘∗+1),𝑘∗+1) −𝑉(𝑥(𝑘∗),𝑘∗)=𝛾𝑘∗+1𝑥𝑇(𝑘∗+1)𝐻𝑥(𝑘∗+1) −𝛾𝑘∗𝑥𝑇(𝑘∗)𝐻𝑥(𝑘∗) =𝛾𝑘∗+1 [𝐴𝑥(𝑘∗)+𝑠 ∑ 𝑖=1𝐵𝑖𝑥(𝑘∗−𝑚𝑖)]𝑇 ⋅𝐻[𝐴𝑥(𝑘∗)+𝑠 ∑ 𝑖=1𝐵𝑖𝑥(𝑘∗−𝑚𝑖)]−𝛾𝑘∗𝑥𝑇(𝑘∗) ⋅𝐻𝑥(𝑘∗). (14) Rearranging this computation, we derive Δ𝑉(𝑥(𝑘∗),𝑘∗) =−𝛾𝑘∗+1𝑥𝑇(𝑘∗)[𝐻−𝐴𝑇𝐻𝐴]𝑥(𝑘∗) +2𝛾𝑘∗+1𝑥𝑇(𝑘∗)𝐴𝑇𝐻𝑠 ∑ 𝑖=1𝐵𝑖𝑥(𝑘∗−𝑚𝑖) +𝛾𝑘∗+1 𝑠 ∑ 𝑖,𝑗=1𝑥𝑇(𝑘∗−𝑚𝑖)𝐵𝑇 𝑖𝐻𝐵𝑗𝑥(𝑘∗−𝑚𝑗) +𝛾𝑘∗(𝛾−1)𝑥𝑇(𝑘∗)𝐻𝑥(𝑘∗). (15)
Discrete Dynamics in Nature and Society 3 We estimate the first difference and use the assumption that the matrix 𝐻is a solution of equation (5); therefore, Δ𝑉(𝑥(𝑘∗),𝑘∗) ≤−𝛾𝑘∗+1𝜆min (𝐶)𝑥(𝑘∗)2 +2𝛾𝑘∗+1 𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖𝑥(𝑘∗)𝑥(𝑘∗−𝑚𝑖) +𝛾𝑘∗+1 𝑠 ∑ 𝑖,𝑗=1 𝐵𝑇 𝑖𝐻𝐵𝑗𝑥(𝑘∗−𝑚𝑖)𝑥(𝑘∗−𝑚𝑗) +𝛾𝑘∗(𝛾−1)𝜆max (𝐻)𝑥(𝑘∗)2 (16) and Δ𝑉(𝑥(𝑘∗),𝑘∗)≤−𝛾𝑘∗+1𝜆min (𝐶)𝑥(𝑘∗)2 +𝛾𝑘∗+1 𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖[𝑥(𝑘∗)2+𝑥(𝑘∗−𝑚𝑖)2] +1 2𝛾𝑘∗+1 𝑠 ∑ 𝑖,𝑗=1 𝐵𝑇 𝑖𝐻𝐵𝑗 ⋅[𝑥(𝑘∗−𝑚𝑖)2+𝑥(𝑘∗−𝑚𝑗)2]+𝛾𝑘∗(𝛾−1) ⋅𝜆max (𝐻)𝑥(𝑘∗)2. (17) Now we apply inequality (13) to get Δ𝑉(𝑥(𝑘∗),𝑘∗)≤−𝛾𝑘∗+1𝜆min (𝐶)𝑥(𝑘∗)2 +𝛾𝑘∗+1 𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖 ⋅[𝑥(𝑘∗)2+𝛾𝑚𝑖+1𝜑(𝐻)𝑥(𝑘∗+1)2]+1 2 ⋅𝛾𝑘∗+1 𝑠 ∑ 𝑖,𝑗=1 𝐵𝑇 𝑖𝐻𝐵𝑗[𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝜑(𝐻) ⋅𝑥(𝑘∗+1)2+𝛾𝑘∗(𝛾−1)𝜆max (𝐻)𝑥(𝑘∗)2 (18) and Δ𝑉(𝑥(𝑘∗),𝑘∗)≤−𝛾𝑘∗+1 [𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖 −𝛾−1 𝛾𝜆max (𝐻)]𝑥(𝑘∗)2+1 2𝛾(𝑘∗+2)𝜑(𝐻) ⋅[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]𝑥(𝑘∗+1)2. (19) Inequality 𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖−𝛾−1 𝛾𝜆max (𝐻)>0 (20) can be deduced from the assumption 𝐿3≥0. Therefore, utilizing (8), Δ𝑉(𝑥(𝑘∗),𝑘∗)≤−𝛾[𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖 −𝛾−1 𝛾𝜆max (𝐻)]𝑉(𝑥(𝑘∗),𝑘∗) 𝜆max (𝐻)+1 2𝛾𝜑(𝐻) ⋅[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ] ⋅𝑉(𝑥(𝑘∗+1),𝑘∗+1) 𝜆min (𝐻). (21) Since Δ𝑉(𝑥(𝑘∗),𝑘∗)=𝑉(𝑥(𝑘∗+1),𝑘∗+1)−𝑉(𝑥(𝑘∗),𝑘∗),we get [ [1−1 2𝛾𝜑(𝐻) 𝜆min (𝐻)[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]] ]𝑉(𝑥(𝑘∗+1), 𝑘∗+1)≤[1− 𝛾 𝜆max (𝐻)[𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖 −𝛾−1 𝛾𝜆max (𝐻)]]𝑉(𝑥(𝑘∗),𝑘∗). (22) This inequality can be rewritten as 𝐿2 𝜆min (𝐻)𝑉(𝑥(𝑘∗+1),𝑘∗+1) ≤𝐿1 𝜆max (𝐻)𝑉(𝑥(𝑘∗),𝑘∗)(23) or as 𝑉(𝑥(𝑘∗+1),𝑘∗+1)≤Θ⋅𝑉(𝑥(𝑘∗),𝑘∗)(24) where Θfl L1 𝐿2𝜑(𝐻)>0. (25) Now we prove that Θ≤1. (26)
4 Discrete Dynamics in Nature and Society Inequality (26) is equivalent with an inequality 𝜆max (𝐻)−𝛾[𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖−𝛾−1 𝛾 ⋅𝜆max (𝐻)]≤[ [𝜆min (𝐻)−1 2𝛾𝜑(𝐻) ⋅[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]] ]𝜆max (𝐻) 𝜆min (𝐻). (27) After some simplification, we get 𝜆min (𝐶)−𝑠 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖−𝛾−1 𝛾𝜆max (𝐻)≥1 2𝜑2(𝐻) ⋅[ [2𝑠 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +𝑠 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ], (28) which is equivalent with the inequality 𝐿3≥0. Then (24), (26), and (10) imply 𝑉(𝑥(𝑘∗+1),𝑘∗+1)≤Θ⋅𝑉(𝑥(𝑘∗),𝑘∗) ≤𝑉(𝑥(𝑘∗),𝑘∗)≤𝜀2𝜆min (𝐻).(29) This inequality contradicts (11). Then, inequality (11) is impossible and (10) holds for every 𝑘=1,2,....Moreover,(8)and (10) imply 𝛾𝑘𝜆min (𝐻)|𝑥(𝑘)|2≤𝑉(𝑥(𝑘),𝑘)≤𝜀2𝜆min (𝐻) =𝛿2𝜆max (𝐻) =‖𝑥(0)‖2 𝑚𝜆max (𝐻), (30) i.e., the inequality 𝛾𝑘𝜆min (𝐻)|𝑥(𝑘)|2≤‖𝑥(0)‖2 𝑚𝜆max (𝐻), 𝑘≥1, (31) equivalent with (7). 3. Concluding Remarks Based on the investigations on exponential stability published previously, the present paper brings in Theorem 1 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 abovementioned independency of results. Example 2. Let 𝑛=𝑠=2and let system (1) be of the form 𝑥1(𝑘+1)=𝑥1(𝑘)+𝑥2(𝑘)+𝜇𝑥2(𝑘−𝑚1), (32) 𝑥2(𝑘+1)=−𝑥1(𝑘)−𝑥2(𝑘)+]𝑥1(𝑘−𝑚2)(33) where 𝑘≥0and 𝜇and ]are constants. We show that Theorem 1 is applicable if |𝜇|and |]|are sufficiently small. We have 𝐴=(11 −1 −1), 𝐵1=(0𝜇 00 ), 𝐵2=(00 ]0). (34) Lyapunov equation (5) is satisfied, e.g., for 𝐶=(0.9 0.9 0.9 1), 𝐻=(11 1 1.1). (35) Then, 𝜆max(𝐻) ≐ 2.0512492,𝜆min(𝐻) ≐ 0.0487508, 𝜆min(𝐶) ≐ 0.0486122,and𝜑(𝐻) ≐ 42.0762336.Simple computations result in 𝐴𝑇𝐻𝐵1=0, 𝐴𝑇𝐻𝐵2=0.1√2], 𝐵𝑇 1𝐻𝐵1=𝜇2, 𝐵𝑇 2𝐻𝐵2=1.1]2, 𝐵𝑇 1𝐻𝐵2=𝜇],
Discrete Dynamics in Nature and Society 5 𝐿1=𝛾[ [𝜆max (𝐻)−𝜆min (𝐶)+2 ∑ 𝑗=1 𝐴𝑇𝐻𝐵𝑗] ] ≐𝛾[2.0026370+0.1√2]], 𝐿2=𝜆min (𝐻)−1 2𝛾𝜑(𝐻)[ [22 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +2 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]≐0.0487508 −𝛾42.0762336[𝛾𝑚20.1√2]+𝛾𝑚1+1𝜇2 +(𝛾𝑚1+1 +𝛾𝑚2+1)𝜇]+𝛾𝑚2+11.1]2](36) and 𝐿3=𝜆min (𝐶)−2 ∑ 𝑖=1 𝐴𝑇𝐻𝐵𝑖−𝛾−1 𝛾𝜆max (𝐻)−1 2 ⋅𝜑2(𝐻)[ [22 ∑ 𝑖=1𝛾𝑚𝑖𝐴𝑇𝐻𝐵𝑖 +2 ∑ 𝑖,𝑗=1 [𝛾𝑚𝑖+1 +𝛾𝑚𝑗+1]𝐵𝑇 𝑖𝐻𝐵𝑗] ]≐0.0486122 −0.1√2]−𝛾−1 𝛾2.0512492−(42.0762336)2 ⋅[𝛾𝑚20.1√2]+𝛾𝑚1+1𝜇2+(𝛾𝑚1+1 +𝛾𝑚2+1)𝜇] +𝛾𝑚2+11.1]2]. (37) Theorem 1 is applicable if |𝜇|and |]|are sufficiently small since this implies 𝐿𝑖>0,𝑖=1,2, and, if the expression 0.0486122−𝛾−1 𝛾2.0512492 =2.0512492 𝛾−2.0026370 (38) is positive, provided that 𝛾>1;thatis,if 1<𝛾<2.0512492 2.0026370≐1.0242741, (39) then 𝐿3>0as well. In such a case, for an arbitrary solution 𝑥(𝑘)=(𝑥1(𝑘),𝑥2(𝑘))𝑇of system (32), (33), the estimate |𝑥(𝑘)|≤√𝜑(𝐻)‖𝑥(0)‖𝑚𝛾−𝑘/2 ≐42.0762336‖𝑥(0)‖𝑚𝛾−𝑘/2, 𝑘≥1 (40) holds. Since det 𝐴=0in 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∗,𝐵∗ 1,and𝐵∗ 2, defined by replacing the entries in the previously given matrices 𝐴,𝐵1,and𝐵2by their absolute values, leads to a matrix 𝑈fl𝐴∗+𝐵∗ 1+𝐵∗ 2=(11 11 )+(0𝜇 00 )+(00 |]|0) =( 11+ 𝜇 1+|]|1)(41) whose eigenvalues are 𝜆1,2(𝑈)=1±√(1+|𝜇|)(1+|]|),and, obviously, 𝜌(𝑈)≥1. Finally, we compare the results published in [4–7] with Theorem1.TheassumptionsofTheorem1are,forthereduced case 𝑠=1of 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,𝐻and 𝐶are thesameasinthepaper) |𝑥(𝑘)|≤√𝜑(𝐻)‖𝑥(0)‖𝑚Θ𝑘/2(𝑚+1) (𝐻),𝑘≥1, (42) where Θ(𝐻) fl1 𝜆max (𝐻)[𝐿(𝐻)−𝑠 ∑ 𝑖=1𝐿𝑖(𝐻)+𝑠𝜆min (𝐻)], 𝐿(𝐻)fl𝜆max (𝐻)−𝜆min (𝐶)+𝑠 ∑ 𝑗=1 𝐴𝑇𝐻𝐵𝑗, 𝐿𝑖(𝐻) fl𝜆min (𝐻)−𝜑(𝐻)[ [𝐴𝑇𝐻𝐵𝑖+𝑠 ∑ 𝑗=1 𝐵𝑇 𝑖𝐻𝐵𝑗] ], 𝑖=1,...,𝑠, (43) if 𝜌(𝐴) < 1,𝐶is a fixed positive definite matrix, matrix 𝐻 solves the corresponding Lyapunov matrix equation (5), and 𝐿(𝐻)−𝑠 ∑ 𝑖=1𝐿𝑖(𝐻)<𝜆max (𝐻)−𝑠𝜆min (𝐻), 𝐿(𝐻)>0. (44) Assuming that |𝐵𝑖|→0,𝑖=1,...,𝑛,wededucethatfor(44) to hold, the following is necessary: 𝜆max (𝐻)−𝜆min (𝐶)>0, (45)
6 Discrete Dynamics in Nature and Society the limiting value of Θ(𝐻)is Θ(𝐻)≐𝜆max (𝐻)−𝜆min (𝐶) 𝜆max (𝐻),(46) and (42) can approximately be written as |𝑥(𝑘)| ≤√𝜑(𝐻)‖𝑥(0)‖𝑚[𝜆max (𝐻)−𝜆min (𝐶) 𝜆max (𝐻)]𝑘/(2(𝑚+1)) , 𝑘≥1. (47) Considering the same limiting process as above, for the validity of (7), an analysis of 𝐿𝑖,𝑖 = 1,2,3implies that inequality (45) must hold in addition to inequality 𝜆min (𝐶)−𝜆max (𝐻)+1 𝛾𝜆max (𝐻)>0, (48) derived from the assumption 𝐿3≥0.Inequality(48), together with the assumption 𝛾>1,yields 1<𝛾< 𝜆max (𝐻) 𝜆max (𝐻)−𝜆min (𝐶)(49) and(7)canbeapproximativelywrittenas |𝑥(𝑘)| ≤√𝜑(𝐻)‖𝑥(0)‖𝑚[𝜆max (𝐻)−𝜆min (𝐶) 𝜆max (𝐻)]𝑘/2 , 𝑘≥1. (50) 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 𝑥(𝑘+1)=𝐴𝑥(𝑘), 𝑘=0,1,, .... (51) Data Availability No data were used to support this study. Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgments The first, third, and fourth authors have been supported by the Czech Science Foundation under Project 16-08549S. Their work has been realized in CEITEC-Central European Institute of Technology with research infrastructure supported by Project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. The second author has beensupportedbytheGrantFEKT-S-17-4225ofFacultyof Electrical Engineering and Communication, Brno University of Technology. An earlier presentation of preliminary results was introduced on Thursday (May 19, 2016) at Faculty of Physics and Mathematics, University of Latvia. References [1] A. S. Bychkov and D. Y. Khusainov, “Exponential convergence estimates for delay difference systems,” Journal of Differential Equations,vol.38,no.9,pp.1368–1370,2002,translationfrom Differ. Uravn. vol. 38, no. 9, pp. 1285–1287, 2002. [2] J. ˇ Cerm´ ak, “Stability conditions for linear delay difference equations: a survey and perspective,” Tatra Mountains Mathematical Publications,vol.63,pp.1–29,2015. [3] J. ˇ Cerm´ ak and J. J´ ansk´ y, “Stability switches in linear delay difference equations,” Applied Mathematics and Computation, vol. 243, pp. 755–766, 2014. [4] J. Dibl´ ık, “Exponential stability of linear discrete systems with variable delays via Lyapunov second method,” Discrete Dynamics in Nature and Society,vol.2017,ArticleID7417909,9pages, 2017. [5] J. Dibl´ ık, D. Y. Khusainov, J. Baˇ stinec, and A. S. Sirenko, “Exponential stability of linear discrete systems with constant coefficients and single delay,” Applied Mathematics Letters,vol. 51, pp. 68–73, 2016. [6] J. Dibl´ ık, D. Y. Khusainov, J. Baˇ stinec, and A. S. Sirenko, “Exponential stability of perturbed linear discrete systems,” Advances in Difference Equations,vol.2016,pp.1–20,2016. [7] J. Dibl´ ık,D.Y.Khusainov,J.Ba ˇ stinec, and A. S. Sirenko, “Stability and exponential stability of linear discrete systems with constant coefficients and single delay,” Applied Mathematics and Computation,vol.269,ArticleID21439,pp.9–16,2015. [8] E. Kaslik, “Stability results for a class of difference systems with delay,” Advances in Difference Equations,vol.2009,ArticleID 938492, 13 pages, 2009. [9] M. M. Kipnis and R. M. Nigmatulin, “Stability of trinomial linear difference equations with two delays,” Automation and Remote Control,no.11,pp.1710–1723,2004. [10] S. A. Kuruklis, “The asymptotic stability of 𝑥𝑛+1 −𝑎𝑥𝑛+𝑏𝑥𝑛−𝑘 = 0,” JournalofMathematicalAnalysisandApplications,vol.188, no.3,pp.719–731,1994. [11] E. Liz, “Stability of non-autonomous difference equations: simple ideas leading to useful results,” Journal of Difference Equations and Applications, vol. 17, no. 2, pp. 203–220, 2011. [12]E.LizandJ.B.Ferreiro,“Anoteontheglobalstabilityof generalized difference equations,” Applied Mathematics Letters, vol.15,no.6,pp.655–659,2002. [13] M. Medved’ and L. ˇ Skripkov´ a, “Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices,” Electronic Journal of Qualitative Theory of Differential Equations,vol.22,pp.1–13, 2012. [14] P. H. Ngoc and L. T. Hieu, “New criteria for exponential stability of nonlinear difference systems with time-varying delay,” International Journal of Control,vol.86,no.9,pp.1646– 1651, 2013.
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