ON EXPONENTIAL STABILITY FOR LINEAR DIFFERENCE EQUATIONS WITH DELAYS

2018. Т. 18, No 3. С. 31–38 31


ON EXPONENTIAL STABILITY FOR LINEAR DIFFERENCE EQUATIONS WITH DELAYS
Leonid Berezansky 1 , brznsky@cs.bgu.ac.il, Elena Braverman 2 1 Ben-Gurion University of the Negev, Beer-Sheva, Israel, 2 University of Calgary, Calgary, Canada The article gives an overview of recent results on the stability of finite-difference equations with delay.
All results are compared with known signs of exponential stability of linear difference equations.
The results are obtained using the Bohl-Perron theorem and comparing the equation under study with an equation for which the Cauchy function is positive.
The Bohl-Perron theorem allows us to reduce the question of the exponential stability of a linear difference equation with delay to the solvability of an operator equation in one of the functional infinite-dimensional spaces.
That is, in fact, to an estimate of the norm or the spectral radius of a bounded linear operator in this space. For this estimation, different difference inequalities are used. One way to obtain such inequalities is to evaluate the fundamental solution in the event that this solution is positive.
The above scheme is used in this paper to obtain sufficient conditions for the exponential stability of the following equation provided that the coefficients and the lag are limited functions.
The main results of the paper are the following ones. Theorem 1. Suppose the fundamental function of (1) is eventually positive, i.e., for some n 0 ≥ 0 we have X(n, k) > 0, n ≥ k ≥ n 0 , and, in addition, a = lim →∞ ∑ ( ) > 0.
By this lemma to obtain stability conditions we need to estimate norms of a some linear operator. The problem is that this operator is given in an implicit form, since the fundamental function Y (n, k) of a model equation usually is not known. So we can only estimate this function in some particular cases.
Such estimations we cam obtain for linear delay difference equations with positive fundamental functions.
Then the fundamental function of (1) is eventually positive: ( , ) > 0, ≥ . Corollary 1. Suppose a l (n) ≥ 0, (4) holds and (5) is satisfied for some n 0 ≥ 0. Then (1) is exponentially stable. Consider together with (1) the following comparison equation where g l (n) ≤ n. Denote by Y (n, k) the fundamental function of equation (6). Lemma 3 [3]. Suppose ( ) ≥ ( ) ≥ 0, ( ) ≥ ℎ ( ), = 1, 2, … , , for sufficiently large n. If equation (1) has an eventually positive solution, then (6) has an eventually positive solution and its fundamental function Y (n, k) is eventually positive. Corollary 2. Suppose (4) and at least one of the following conditions hold: Then (1) is exponentially stable. Remark 3. By Theorem 3.1 in [3] it is enough to assume the existence of an eventually positive solution in the conditions of Theorem 1 rather than to require that the fundamental function is positive.
Now let us proceed to explicit stability conditions. (4) holds, the fundamental function X 1 (n,k) of the equation

Theorem 2. Suppose there exists a subset of indices
Then equation (1) is exponentially stable. Now let us proceed to explicit stability conditions. Now we will take general exponentially stable difference equations with a positive fundamental function as a class of comparison equations. Corollary 3. Suppose there exist a set of indices ⊂ {1,2, … , }, functions ( ) ≤ , ∈ , and positive numbers , < 1, such that for n sufficiently large the inequalities 0 < ≤ ∑ ( ) ∈ ≤ < 1, ∈ , hold and the difference equation then (1) is exponentially stable. Corollary 6. Suppose for some positive , , , ℎ < 1, < 1, the following inequalities are satisfied for n large enough (1) Then equation (19) is exponentially stable.

Discussion and Examples
Let us note that the approach using Bohl-Perron Theorem is similar to the method developed in [20] where stability is deduced based on the fact that some linear exponentially stable equation is close to the considered equation. Unlike the present paper, [20] considers nonlinear perturbations of stable linear equations as well. The main result (Theorem 2) of [20] is the following one.
then equation (1) is asymptotically stable. This result is also true for general equation (1)
Stability tests (22) and (23) are obtained for equations with positive coefficients. In the present paper we consider coefficients of arbitrary signs. The next interesting feature of the results obtained here is that some of the delays can be arbitrarily large (see for example, Parts 1 and 2 of Corollary 7). (25) are exponentially stable.
Two previous results of [9,19,23,24] and [10] fail to establish exponential stability for equation (24) with positive coefficients, as well as all parts of Corollary 3.10 in [4] cannot be applied to equation (25) with an oscillating coefficient. None of the inequalities in Corollary 8 of [2] can be applied to establish stability of (25).
We note that (24), (25) are special cases of equation with one nondelay term and two delay terms considered in [2], however none of the inequalities in Corollary 8 of [2] can be applied to establish stability of (25). Let us also note that for (25) we have where [13] π/2 is the best possible constant [19,22] in ∑ < which implies exponential stability of (19). Example 2. Consider equation (17) Then in (17) (17) is exponentially stable. None of the inequalities in Corollary 8 of [2] can be applied to establish stability of (17). We note that it would be harder to treat this example if the equation were written as high order equations with constant delays and variable coefficients.
Then equation (28) is exponentially stable if and only if p > q. Condition 1) of Corollary 8 is close to this result. It gives the same sufficient stability test for q of an arbitrary sign but does not involve the necessity part.
The paper [16] contains a nice review on stability results obtained for equations with oscillating coefficients. The results of [16] generalized the following stability test obtained in [14] for equation (28): If < 1, > then (28) is asymptotically stable.
By condition 2) of Corollary 8 equation (28) is asymptotically stable if It is easy to see that these two tests are independent. Let us discuss sharpness of conditions of Theorem 1 for exponential stability of (1), assuming the fundamental function is positive; in particular, we demonstrate sharpness of condition (4).
i.e., the equation is neither asymptotically nor exponentially stable. Let us demonstrate that the facts that the sum of coefficients ∑ ( ) in (1) is positive, exceeds a positive number and that the fundamental function is positive do not imply stability, in the case when coefficients have different signs.
Finally, let us formulate some open problems.
1. Under which conditions will exponential stability of (1) imply exponential stability of the equation with the same coefficients and smaller delays: b) delays are infinite but coefficients decay exponentially with memory, i.e., there exist positive numbers M and λ < 1 such that | ( )| ≤ ( ) . Let us note that Bohl-Perron type result in case b) was obtained in [5], Theorem 4.7. 4. Example 5 demonstrates that for equations with positive and negative coefficients and a positive fundamental function inequality (4) does not imply exponential stability. Is it possible to find such conditions on delays and coefficients of different signs that (4) would imply exponential stability? For instance, prove or disprove the following conjecture. Let us remark that conditions when the fundamental function of (30) is positive were obtained in [6].