Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems in R2

Copyright © 2017 Josef Dibĺık 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. Two-dimensional linear discrete systems x(k+1) = Ax(k)+∑nl=1 Blxl(k−ml), k ≥ 0, are analyzed, wherem1, m2, . . . , mn are constant integer delays, 0 < m1 < m2 < ⋅ ⋅ ⋅ < mn, A, B1, . . . , Bn are constant 2 × 2matrices, A = (aij), Bl = (bl ij), i, j = 1, 2, l = 1, 2, . . . , n, and x : {−mn, −mn + 1, . . .} → R. Under the assumption that the system is weakly delayed, the asymptotic behavior of its solutions is studied and asymptotic formulas are derived.

1.1.Weakly Delayed Systems.It is well-known that the characteristic equation to (1) is where  ∈ C, and the characteristic equation to a system without delays is det ( − ) = 0.
The following definition and lemma are taken from [1].
Definition 1. System (1) is called weakly delayed if equations ( 4) and ( 6) are equal, that is, if Lemma 2. If system (1) is a weakly delayed system, then its arbitrary linear nonsingular transformation () = S() with 2 × 2 matrix S is again a weakly delayed system.
In [1], the following necessary and sufficient conditions determining weakly delayed systems are also derived.
1.2.Problem under Consideration.In the paper (in Section 2), we are concerned with conditional stability of (1) with the results formulated in Theorems 11-18.In Section 3, asymptotic formulas describing the behavior of solutions (for  → ∞) of nontrivial solutions of (1) are derived with the results formulated in Theorems 19-24.
To prove such results, we use explicit analytic formulas on the representation of solutions, derived by the first two authors in [1] and, for the reader's convenience, we recall them in the following part.

Conditional Stability
In [ [1], Theorem 10], it is explained that the space of solutions of a weakly delayed system (1), depending initially on 2(  + 1) parameters (i.e., on the initial data (3)) is reduced (as  ≥   + 2) to a space of solutions depending either on   + 1 or even only on 2 parameters.This is also visible from an analysis of formulas describing the behavior of solutions for  ≥   + 2 given in Theorems 4-9.In this part, we explain how this property can be used when the stability of a weakly delayed system (1) is considered.Since not all of the initial data have an impact on the behavior of the solution  = () as  ≥   + 2, a part of them can be fixed and, under such an assumption, we define a so-called conditional stability below.The fixing of some of the initial data leads to an unexpected phenomenon; In the following definition, the notion of conditional stability is explained.The two first parts of it are the classical definitions of stability and can be found, for example, in [2,3].
Definition 10.The zero solution () = ,  ∈  ∞ −  , of ( 1) is said to be (a) stable if, given  > 0 and  0 ≥ 0, there exists  = (,  0 ) such that (),  ∈ , ‖‖   <  implies ‖(,  0 , )‖ <  for all  ≥  0 , uniformly stable if  may be chosen independently of  0 , and unstable if it is not stable; (b) asymptotically stable if it is stable and lim →∞ ‖()‖ = 0; (c) conditionally stable (conditionally asymptotically stable) if it is unstable (not asymptotically stable), but if it is stable (asymptotically stable) under the condition that there exists a fixed subspace  ⊂ R 2(  +1) , 1 ≤ dim  < 2(  + 1), and the initial data satisfy Utilizing the formulas on the representation of solutions of (1), we prove results on conditional stability (since the existence of the subspaces  in the proofs is obvious, we do not write them explicitly).
that is, the zero solution is conditionally asymptotically stable.If  * 2 (0) ̸ = 0, then from formula (I 4 ) in Theorem 4, we get as  ≥   + 2, and the zero solution is unstable if | 2 | > 1 and not asymptotically stable if Similarly, with the aid of formula (II 4 ), Theorem 5, the following theorem can be proved.
The proof can be performed similarly to that of Theorem 13 with the aid of formula (II 4 ), Theorem 5.
Proof.We show that the zero solution of ( 1) is conditionally stable.For ‖ 1 ()‖,  ∈  and the zero solution is unstable.

Asymptotic Formulas
Carefully analyzing the analytical formulas for the solutions of system (1) given in Theorems 4-9, it is possible (under various assumptions for the roots of ( 6) and for the initial data) to derive asymptotic formulas for the solutions of system (1) or simplify the formulas for the exact solutions.Below, we do such investigation.Set The  1), ( 3) is where and and and formula (I 4 ), Theorem 4, for solution (),  ∈ Z ∞   +2 , can be written as ) . (39) From (39), we get ) . (40) . The statement ( 2) is an obvious consequence of (40).For  1 = 0, we get the statement (2) immediately from (I 4 ).Statement ( 3) is also a straightforward consequence of (40 Then from (40) we deduce and the statement (4) remains true.The statement (5) follows from (40) if  1 ̸ = 0. From this representation, we derive the above asymptotic formulas if  1 ̸ = 0.If  1 = 0, then, by (I 4 ),  1 () = 0 and the statement (5) remains valid.The last statement (6) is a consequence of (40) and remains valid also if  1 = 0.
The following theorem uses the vector norm, given by (20).
Theorem 24.If the assumptions of Theorem 9 hold, then the solution of the initial problem (1), (3) is where Proof.The formula 4(y 1 ), Theorem 9, can be written as if  ∈ Z ∞   +2 .The second and fourth terms are leading terms if  * 2 (0) ̸ = 0 and  ̸ = 0.If  * 2 (0) = 0, then the first and third terms are leading terms.