# Exponential Stability of Linear Discrete Systems with Multiple Delays.

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.summation over (i=1)][B.sub.i]x(k-[m.sub.i]), k = 0, 1, ... (1)

where s [member of] N, A and [B.sub.i] are nxn matrices, and [m.sub.i] [member of] N. For (1) exponential-type stability and exponential estimate of the rate of convergence of solutions are derived.

Set m := max{[m.sub.1], ..., [m.sub.s]}. The initial Cauchy problem for system (1) is as follows:

x(k) = [x.sub.k] [member of] R, k = -m, -m + 1, ... (2)

For a vector x = [([x.sub.1], ..., [x.sub.n]).sup.T], we define [[absolute value of x].sup.2] := [[summation].sup.n.sub.i=1][x.sup.2.sub.i]. Let [rho](A) be the spectral radius of the matrix A. Denote by [[lambda].sub.max](A) and [[lambda].sub.min](A) the maximum and the minimum eigenvalues, respectively, of a symmetric matrix A and define [phi](A) := [[lambda].sub.max](A)[[lambda].sup.-1.sub.min](A). For a given matrix B, we use the norm defined by [[absolute value of B].sup.2] := [[lambda].sub.max]([B.sup.T]B). In the paper, assume [absolute value of A] + [[summation].sup.s.sub.i=1][absolute value of [B.sub.i]] > 0.

The trivial solution x(k) = 0, k = -m, -m + 1, ... of (1) is called Lyapunov exponentially stable if there exist constants N > 0 and [theta] [member of] (0,1) such that, for an arbitrary solution x = x(k) of (1),

[absolute value of x(k)] [less than or equal to] [[parallel]x(0)[parallel].sub.m][[theta].sup.k], k = 1,2, ... (3)

where

[[parallel]x(0)[parallel].sub.m] := max{[x(i)], i = -m, -m + 1, ..., 0}. (4)

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 + 1)n. 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 [rho](A) < 1, then the Lyapunov matrix equation

[A.sup.T]HA - H = -C (5)

has a unique solution, a positive definite symmetric matrix H for an arbitrary positive definite symmetric nxn matrix C (we refer, for example, to ).

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 [gamma] > 1 be a parameter. Define auxiliary numbers

[mathematical expression not reproducible] (6)

Theorem 1. Let [rho](A) < 1, C be a fixed positive definite symmetric nxn matrix, let matrix H solve the equation (5), and, for a fixed [gamma] > 1, let [L.sub.1] > 0, [L.sub.2] > 0, [L.sub.3] [greater than or equal to] 0. Then, system (1) is exponentially stable and, for an arbitrary solution X = x(k), the estimate

[absolute value of x(k)] [less than or equal to] [square root of ([phi](H))][[parallel]x(0)[parallel].sub.m][[gamma].sup.-k/2], k [greater than or equal to] 1 (7)

holds.

Proof. For the Lyapunov function V(x, k) := [[gamma].sup.k][x.sup.T]Hx, inequalities

[[gamma].sup.k][[lambda].sub.min](H)[[absolute value of x].sup.2] [less than or equal to] [[gamma].sup.k][[lambda].sub.max](H)[[absolute value of x].sup.2] (8)

hold. Let [delta] := [epsilon]/[square root of ([phi](H))] where [epsilon] > 0 is given. Let a solution x(k) of (1) satisfy [[parallel]x(0)[parallel].sub.m] = [delta]. Then, for k = -m, -m + 1, ..., 0,

[mathematical expression not reproducible] (9)

i.e.,

V(x(k),k) [less than or equal to] [[epsilon].sup.2][[lambda].sub.min](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.sup.*] > 0 exists such that, for k = -m, -m + 1, ..., [k.sup.*], (10) holds, and, for k = [k.sup.*] + 1,

V(x([k.sup.*] + 1, [k.sup.*] + 1)) > [[epsilon].sup.2][[lambda].sub.min](H). (11)

Inequality (11) implies that, for k = -m, -m + 1, ..., [k.sup.*],

[mathematical expression not reproducible] (12)

And

[mathematical expression not reproducible]. (13)

Now compute

[mathematical expression not reproducible] (14)

Rearranging this computation, we derive

[mathematical expression not reproducible] (15)

We estimate the first difference and use the assumption that the matrix H is a solution of equation (5); therefore,

[mathematical expression not reproducible] (16)

and

[mathematical expression not reproducible] (17)

Now we apply inequality (13) to get

[mathematical expression not reproducible] (18)

and

[mathematical expression not reproducible] (19)

Inequality

[[lambda].sub.min](C)-[s.summation over (i=1)[absolute value of [A.sup.T]H[B.sub.i]]- [gamma]-1/[gamma] [[lambda].sub.max](H) > 0 (20)

can be deduced from the assumption [L.sub.3] [greater than or equal to] 0. Therefore, utilizing (8),

[mathematical expression not reproducible] (21)

Since [mathematical expression not reproducible], we get

[mathematical expression not reproducible] (22)

This inequality can be rewritten as

[mathematical expression not reproducible] (23)

or as

V(x([k.sup.*] + 1), [k.sup.*] + 1) [less than or equal to] [PHI] x V(x([k.sup.*]), [k.sup.*]) (24)

where

[PHI] := [L.sub.1]/[L.sub.2][phi](H) > 0. (25)

Now we prove that

[PHI] [less than or equal to] 1. (26)

Inequality (26) is equivalent with an inequality

[mathematical expression not reproducible] (27)

After some simplification, we get

[mathematical expression not reproducible], (28)

which is equivalent with the inequality [L.sub.3] [greater than or equal to] 0. Then (24), (26), and (10) imply

V(x([k.sup.*]+1), [k.sup.*]+1) [less than or equal to] [PHI] x V(x([k.sup.*]), [k.sup.*]) [less than or equal to] V(x([k.sup.*]), [k.sup.*]) [less than or equal to] [[epsilon].sup.2][[lambda].sub.min](H). (29)

This inequality contradicts (11). Then, inequality (11) is impossible and (10) holds for every k = 1,2,.... Moreover, (8) and (10) imply

[[gamma].sup.k][[lambda].sub.min](H)[[absolute value of x(k)].sup.2] [less than or equal to] V (x(k),k) [less than or equal to] [[epsilon].sup.2][[lambda].sub.min](H) = [[delta].sup.2][[lambda].sub.max](H) = [[parallel]x(0)[parallel].sup.2.sub.m] [[lambda].sub.max](H), (30)

i.e., the inequality

[[gamma].sup.k][[lambda].sub.min](H)[[absolute value of x(k)].sup.2] [less than or equal to] [[parallel]x(0)[parallel].sup.2.sub.m][[lambda].sub.max](H), k [greater than or equal to] 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  assuming that det A [not equal to] 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. Let n = s = 2 and let system (1) be of the form

[x.sub.1](k +1) = [x.sub.1](k) + [x.sub.2](k) + [mu][x.sub.2](k - [m.sub.1]), (32)

[x.sub.2](k +1) = -[x.sub.1](k) - [x.sub.2](k) + [vx.sub.1](k - [m.sub.2]) (33)

where k [greater than or equal to] 0 and [mu] and v are constants. We show that Theorem 1 is applicable if [absolute value of [mu]] and [absolute value of v] are sufficiently small. We have

[mathematical expression not reproducible] (34)

Lyapunov equation (5) is satisfied, e.g., for

[mathematical expression not reproducible] (35)

Then, [[lambda].sub.max](H) = 2.0512492, [[lambda].sub.min](H) [??] 0.0487508, [[lambda].sub.min](C) [??] 0.0486122, and [phi](H) [??] 42.0762336. Simple computations result in

[mathematical expression not reproducible], (36)

and

[mathematical expression not reproducible] (37)

Theorem 1 is applicable if [absolute value of [mu]] and [absolute value of v] are sufficiently small since this implies [L.sub.i] > 0, i = 1,2, and, if the expression

0.0486122 - [[gamma]-1/[gamma]] 2.0512492 = 2.0512492/[gamma] - 2.0026370 (38)

is positive, provided that [gamma] > 1; that is, if

1 < [gamma] < 2.0512492/2.0026370 [??] 1.0242741, (39)

then [L.sub.3] > 0 as well. In such a case, for an arbitrary solution x(k) = [([x.sub.1](k), [x.sub.2](k)).sup.T] of system (32), (33), the estimate

[absolute value of x(k)] [less than or equal to] [square root of ([phi](H))][[parallel]x(0)[parallel].sub.m][[gamma].sup.-k/2] [??] 42.0762336 [[parallel]x(0)[parallel].sub.m][[gamma].sup.-k/2], k [greater than or equal to] 1 (40)

holds.

Since det A = 0 in the above example, the results of the paper  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.sup.*], [B.sup.*.sub.1], and [B.sup.*.sub.2], defined by replacing the entries in the previously given matrices A, [B.sub.1], and [B.sub.2] by their absolute values, leads to a matrix

[mathematical expression not reproducible] (41)

whose eigenvalues are [[lambda].sub.1,2](U) = 1 [+ or -] [square root of (1 + [absolute value of [mu]])(1 + [absolute value of v]), and, obviously, [rho](U) [greater than or equal to] 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 . In  an analysis of Theorem 2 is carried out. Although the results are independent, a limiting process (for [gamma] [right arrow] [1.sup.+]) indicates that the conditions of the main result in  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, [B.sub.i], i = 1 ..., s, H and C are the same as in the paper)

[absolute value of x(k)] [less than or equal to] [square root of ([phi](H))][[parallel]x(0)[parallel].sub.m][[PHI].sup.k/2(m+1)](H), k [greater than or equal to] 1, (42)

where

[mathematical expression not reproducible] (43)

if [rho](A) < 1, C is a fixed positive definite matrix, matrix H solves the corresponding Lyapunov matrix equation (5), and

L(H) - [s.summation over (i=1)][L.sub.i](H) < [[lambda].sub.max](H) - S[[lambda].sub.min](H), L(H) > 0. (44)

Assuming that [absolute value of [B.sub.i]] [right arrow] 0, i = 1, ..., n, we deduce that for (44) to hold, the following is necessary:

[[lambda].sub.max](H) - [[lambda].sub.min](C) > 0, (45)

the limiting value of [THETA](H) is

[THETA](H) [??] [[lambda].sub.max](H)-[[lambda].sub.min](C)/[[lambda].sub.max](H), (46)

and (42) can approximately be written as

[absolute value of (x(k))] [less than or equal to] [square root of ([phi](H))][[parallel]x(0)[parallel].sub.m][[[[lambda].sub.max](H)- [[lambda].sub.min](C)/[[lambda].sub.max]].sup.k/(2(m+1))], k [greater than or equal to] 1. (47)

Considering the same limiting process as above, for the validity of (7), an analysis of [L.sub.i] = 1,2,3 implies that inequality (45) must hold in addition to inequality

[[lambda].sub.min](C)-[[lambda].sub.max](H) + [1/[gamma]][[lambda].sub.max](H) > 0, (48)

derived from the assumption [L.sub.3] [greater than or equal to] 0. Inequality (48), together with the assumption [gamma] > 1, yields

1 < [gamma] < [[lambda].sub.max](H)/[[lambda].sub.max](H)-[[lambda].sub.min](C) (49)

and (7) can be approximatively written as

[absolute value of x(k)] [less than or equal to][square root of [phi](H)][[parallel]x(0)[parallel].sub.m][[[[lambda].sub.max](H)- [[lambda].sub.min](C)/[[lambda].sub.max](H)].sup.k/2], k [greater than or equal to] 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  are generalized in . Results of  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)

https://doi.org/10.1155/2018/9703919

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 been supported by the Grant FEKT-S-17-4225 of Faculty of 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.

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J. Bastinec, (1) H. Demchenko, (2) J. Diblik (iD), (1) and D. Ya. Khusainov (1)

(1) Brno University of Technology, CEITEC-Central European Institute of Technology, Brno, Czech Republic

(2) Brno University of Technology, Faculty of Electrical Engineering and Communication Brno, Czech Republic

Correspondence should be addressed to J. Diblik; diblik@feec.vutbr.cz

Received 14 January 2018; Accepted 8 April 2018; Published 1 August 2018

Academic Editor: Pasquale Candito
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