Abstract

Two-dimensional linear discrete systems ,  , are analyzed, where are constant integer delays, , , are constant matrices, , ,  ,  , and . Under the assumption that the system is weakly delayed, the asymptotic behavior of its solutions is studied and asymptotic formulas are derived.

1. Introduction and Preliminaries

Throughout the paper, we use the notations: is zero matrix, is the unit matrix, is zero vector, and , where and are integers, . Similarly, a set is defined.

In the paper, we investigate discrete planar systems with delayswhere are integers such that , , , are constant matrices, , ,  ,  ,and . An initial problem to (1) iswhere and . Obviously, the initial problem (1), (3) has a unique solution on .

1.1. Weakly Delayed Systems

It is well-known that the characteristic equation to (1) iswhere , and the characteristic equation to a system without delays isThe 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 with matrix is again a weakly delayed system.

In [1], the following necessary and sufficient conditions determining weakly delayed systems are also derived.

Theorem 3. System (1) is a weakly delayed system if and only ifwhere and .

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 1118. In Section 3, asymptotic formulas describing the behavior of solutions (for ) of nontrivial solutions of (1) are derived with the results formulated in Theorems 1924.

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.

1.3. Representation of Solutions
1.3.1. Jordan Forms

It is known that, for every matrix , there exists a nonsingular matrix transforming it to the corresponding Jordan matrix form . In [[1], Theorem ], formulas (8)–(11) are analyzed and it is concluded that system (1) can be a weakly delayed system only if matrix has one of the following three Jordan matrix forms (the case of the roots of (6) being complex conjugate is not compatible with (2) and (8)–(11)):if (6) has two real distinct roots , , and eitherorin the case of one double real root . Transforming (1) by , we getwhere , ,  , and . The transformed initial problem for (15) isBelow, we will use functions ,  , defined as and functions ,  , defined aswhere ,  .

1.3.2. Explicit Analytic Formulas in Case (12)

In this case, , . From conditions (8)–(11) for (15) it follows (we refer to [[1], Part 2.1.1.]) that either(I),  ,  ,or(II),  ,  .The proofs of the below results are given in [[1], Theorem ].

Theorem 4. Let (1) be a weakly delayed system and (6) have two real distinct roots , . In case (I), the solution of the initial problem (1), (3) is ,  , where (I1) if ,(I2), if ,(I3) + + if , ,(I4) + + if .

Theorem 5. Let (1) be a weakly delayed system and (6) have two real distinct roots , . In case (II), the solution of the initial problem (1), (3) is ,  , where (II1), if ,(II2) + , if ,(II3) + + + if , ,(II4) + + if .

1.3.3. Explicit Analytic Formulas in Case (13)

We have , and, from the necessary and sufficient conditions (8)–(11) for (15), it follows (see [[1], Part 2.1.3.]) that only the following three cases are possible:(a),  ,  ,(b),  ,  ,(c),  .The results formulated below are proved in [[1], Theorems and ],

Theorem 6. Let (1) be a weakly delayed system, (6) have a twofold root , and the matrix have the form (13). In case (a), the solution of the initial problem (1), (3) is ,  , where (a1), if ,(a2) + , if ,(a3) + + + if , ,(a4) + + if .

Theorem 7. Let (1) be a weakly delayed system, (6) have a twofold root , and the matrix have the form (13). In case (b), the solution of the initial problem (1), (3) is ,  , where (b1), if ,(b2) + , if ,(b3) + + + if , ,(b4) + + if .

Theorem 8. Let system (1) be a weakly delayed system, (6) have two repeated roots , and the matrix have the form (13). In case (c), the solution of the initial problem (1), (3) is ,  , where (c1), if ,(c2) + , if ,(c3) + + if , ,(c4) + if .

1.3.4. Explicit Analytic Formulas in Case (14)

We have . The necessary and sufficient conditions (8)–(11) for (15) are reduced (we refer to [[1], Part 2.1.6.]) to ,  ,  . The following result is proved in [[1], Theorem ].

Theorem 9. Let (1) be a weakly delayed system, (6) have a double root , and the matrix have the form (14). Then, the solution of the initial problem (1), (3) is ,  , where , , if , + , if , + + + if , , + + if , if

2. Conditional Stability

In [[1], Theorem ], it is explained that the space of solutions of a weakly delayed system (1), depending initially on parameters (i.e., on the initial data (3)) is reduced (as ) to a space of solutions depending either on or even only on 2 parameters. This is also visible from an analysis of formulas describing the behavior of solutions for given in Theorems 49. 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 , 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; the zero solution can be conditionally stable in spite of the fact that it is unstable in terms of the traditional definition of stability.

Define a norm of a matrix ,  , as and, for vectors , a vector normFor a discrete vector-function , we define 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 and , there exists such that , , implies for all , uniformly stable if may be chosen independently of , and unstable if it is not stable;(b)asymptotically stable if it is stable and ;(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 , , 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).

Theorem 11. If the assumptions of Theorem 4 hold, , , and , then the zero solution of (1) is conditionally asymptotically stable.

Proof. In this case, and . From formula in Theorem 4, for , we get Now, it is easy to see that that is, the zero solution is conditionally asymptotically stable. If , then from formula in Theorem 4, we get as , and the zero solution is unstable if and not asymptotically stable if .

Similarly, with the aid of formula , Theorem 5, the following theorem can be proved.

Theorem 12. If the assumptions of Theorem 5 hold, , , and , then the zero solution of (1) is conditionally asymptotically stable.

Theorem 13. If the assumptions of Theorem 4 hold, , , and , then the zero solution of (1) is conditionally stable.

Proof. We utilize formula in Theorem 4 again. Since and , for , we get We setThen if . As in the proof of Theorem 11, we can show that if , the zero solution is unstable.

Theorem 14. If the assumptions of Theorem 5 hold, , , and , then the zero solution of (1) is conditionally stable.

The proof can be performed similarly to that of Theorem 13 with the aid of formula , Theorem 5.

Theorem 15. If the assumptions of Theorem 6 hold, and , then the zero solution of (1) is conditionally stable.

Proof. We have and For , we get by Theorem 6, formula , Let and be given by (27). Then, the last equality implies if . So conditional stability is proved. If , then and formula yields Then, the zero solution is unstable since, obviously,

The following two theorems can be proved using a scheme similar to that of the proof of Theorem 15 and with the aid of formula , Theorem 7, and formula , Theorem 8, respectively.

Theorem 16. If the assumptions of Theorem 7 hold, and , then the zero solution of (1) is conditionally stable.

Theorem 17. If the assumptions of Theorem 8 hold, and , then the zero solution of (1) is conditionally stable.

Theorem 18. If the assumptions of Theorem 9 hold, and , then the zero solution of (1) is conditionally stable.

Proof. We show that the zero solution of (1) is conditionally stable. For , , we get (by , Theorem 9)and, for , , we get (by , Theorem 9)From (33), (34), it is easy to see that the zero solution of (1) is conditionally stable. If , then (by , Theorem 9) and the zero solution is unstable.

3. Asymptotic Formulas

Carefully analyzing the analytical formulas for the solutions of system (1) given in Theorems 49, 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 symbol ~ used below stands for what is called the asymptotic equivalence. By definition, two nonzero real functions and defined for are asymptotically equivalent if .

Theorem 19. If the assumptions of Theorem 4 hold, then the solution of the initial problem (1), (3) is where , , and (1) if , and + ,(2), if and ,(3) if , and + ,(4) if and and ,(5) if and ,(6) + if and .

Proof. Let . Then and formula , Theorem 4, for solution , , can be written asFrom (39), we getThen (40), if , implies the statement . The statement ( is an obvious consequence of (40). For , we get the statement ( immediately from . Statement ( is also a straightforward consequence of (40). Let . Then from (40) we deduce if and the statement is valid. If , then implies and the statement ( remains true. The statement follows from (40) if . From this representation, we derive the above asymptotic formulas if . If , then, by (, and the statement ( remains valid. The last statement is a consequence of (40) and remains valid also if .

Theorem 20. If the assumptions of Theorem 5 hold, then the solution of the initial problem (1), (3) is where , , and (1)<