Conditional Stability and Asymptotic Behavior of Solutions of Weakly Delayed Linear Discrete Systems in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06569958pt" id="M1" height="15.3664pt" version="1.1" viewBox="-0.0657574 -15.3007 19.5274 15.3664" width="19.5274pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.012,0,0,-0.012,12.819,-7.578)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg>

Diblík, Josef; Halfarová, Hana; Šafařík, Jan

doi:https://doi.org/10.1155/2017/6028078

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction and Preliminaries Conclusions Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 6028078 | https://doi.org/10.1155/2017/6028078

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

Josef Diblík,¹Hana Halfarová,¹and Jan Šafařík^1,2

Academic Editor: Garyfalos Papashinopoulos

Received27 Feb 2017

Accepted28 Mar 2017

Published11 Jun 2017

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 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.

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 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 , 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 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 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) if , and + ,(2) if and ,(3) if , , and + ,(4) if , , and ,(5) if and ,(6) if and .

Proof. Let . Then, and formula of Theorem 5 for the solution , , can be written asFrom (45), we get Now, to prove statements – of the theorem, we can proceed in much the same way as in the proof of Theorem 19.

Theorem 21. If the assumptions of Theorem 6 hold, then the solution of the initial problem (1), (3) is where , , and(1) if , and ,(2) if and .

Proof. Formula , Theorem 6, can be written as If and , then the last addition term is the leading term and formula ( is proved. If , we get formula .

Theorem 22. If the assumptions of Theorem 7 hold, then the solution of the initial problem (1), (3) is where , , and (1) if , , and ,(2) if and .

Proof. Formula , Theorem 7, can be written as Further, we can proceed as in the proof of Theorem (34).

The following theorem uses the vector norm, given by (20).

Theorem 23. If the assumptions of Theorem 8 hold and , then the solution of the initial problem (1), (3) is where (1) if , ,(2) + , if , .

Proof. Formula , Theorem 8, can be written as if . If , then the second sum is a leading term in this formula. If , the first two terms are leading.

Theorem 24. If the assumptions of Theorem 9 hold, then the solution of the initial problem (1), (3) is where if and (1) + if , and ,(2) if and .

Proof. The formula , Theorem 9, can be written as if . The second and fourth terms are leading terms if and . If , then the first and third terms are leading terms.

4. Example

To illustrate the results derived, consider the following example. The computations below were partly implemented by WolframAlpha software (http://www.wolframalpha.com/). Let system (1) be reduced to and let initial problem (3) beIn this case, , , , The eigenvalues of are , . System (55) is weakly delayed by Theorem 3. Indeed, (8) holds since (9) holds since (10) holds since and (11) holds since A nonsingular matrix transforming to the corresponding Jordan form is Then, The transformed initial problem (56) is The transformed system (55) with the initial problem iswhere or where the initial data are The function is defined as All assumptions of Theorem 4 are true. Therefore, the solution of the initial problem (55), (56) is , , where () = if ,() = if ,() + + = + () + + = + + + if .

The asymptotic behavior of the solutions of systems satisfying the assumptions of Theorem 4 is covered by Theorem 19. In the considered case, the assumptions of formula in Theorem 19 are satisfied. Then, that is, The last formula can be deduced from as it is done (in the general case) in the proof of Theorem 19.

Finally, we select initial data for system (55) to derive a result on conditional asymptotic stability by Theorem 11. Let, for , the initial data be given by the equation where is an arbitrary constant. Then, the initial data for (65), deduced from (16), are Since , , and , all assumptions of Theorem 11 are true. Therefore, the zero solution of system (55) is conditionally stable if the initial data satisfy the equation .

In the proof of Theorem 11 this statement is derived from formula in Theorem 4. In the considered particular case we have Then since, by the above formula,

5. Conclusions

The paper deals with conditional stability and asymptotic representation of solutions to linear weakly delayed discrete systems (1) if the Jordan form of the matrix equals (12), (13), or (14), that is, , , or . For these cases, conditional stability is investigated and formulas are deduced expressing the asymptotic behavior of solutions as . For further results related to weakly delayed systems, representations of solutions of discrete systems, their stability, and asymptotic behavior, we refer to [1–11] and to the references therein. It is an open question whether similar results can be derived in the case of system (1) consisting of three equations.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author has been supported by the Project no. LO1408, AdMaS UP-Advanced Materials, Structures and Technologies (supported by Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme I). The third author has been supported by the Grant FEKT-S-17-4225 of Faculty of Electrical Engineering and Communication, BUT.

References

J. Diblík and H. Halfarová, “General explicit solution of planar weakly delayed linear discrete systems and pasting its solutions,” Abstract and Applied Analysis, vol. 2014, Article ID 627295, 37 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, CRC Press, Boca Raton, Fla, USA, 2000.
S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2005.
View at: MathSciNet
L. Berezansky and S. Pinelas, “Oscillation properties for a scalar linear difference equation of mixed type,” Mathematica Bohemica, vol. 141, no. 2, pp. 169–182, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
L. Berezansky, M. Migda, and E. Schmeidel, “Some stability conditions for scalar Volterra difference equations,” Opuscula Mathematica, vol. 36, no. 4, pp. 459–470, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
J. Diblík and H. Halfarová, “Explicit general solution of planar linear discrete systems with constant coefficients and weak delays,” Advances in Difference Equations, vol. 2013, article 50, pp. 1–29, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
J. Diblík and D. Y. Khusainov, “Representation of solutions of discrete delayed system $x (k + 1) = A x (k) + B x (k - m) + f (k)$ with commutative matrices,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 63–76, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
J. Diblík, D. Y. Khusainov, and Z. Šmarda, “Construction of the general solution of planar linear discrete systems with constant coefficients and weak delay,” Advances in Difference Equations, vol. 2009, Article ID 784935, 18 pages, 2009.
View at: Publisher Site | Google Scholar
J. Diblík and B. Morávková, “Representation of the solutions of linear discrete systems with constant coefficients and two delays,” Abstract and Applied Analysis, vol. 2014, Article ID 320476, 19 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
M. Migda, M. Růžičková, and E. Schmeidel, “Boundedness and stability of discrete Volterra equations,” Advances in Difference Equations, vol. 2015, article 47, pp. 1–11, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
J. Morchało and M. Migda, “Boundedness of solutions of difference systems with delays,” Computers & Mathematics with Applications, vol. 64, no. 7, pp. 2233–2240, 2012.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2017 Josef Diblík 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1079

Downloads

548

Citations