On a singular system of fractional nabla difference equations with boundary conditions
© Dassios and Baleanu; licensee Springer. 2013
Received: 20 February 2013
Accepted: 18 May 2013
Published: 19 June 2013
In this article, we study a boundary value problem of a class of linear singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient matrix singular, square with an identically zero matrix pencil and non-square, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically, we study the conditions under which the boundary value problem has a unique solution, infinite solutions and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory.
Keywordsboundary conditions singular systems fractional calculus nabla operator difference equations linear discrete time system
If F is singular with a null vector X, then , so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue ; i.e., .
Regular when and .
Singular when or and .
The paper is organized as follows. In Section 2, we study the existence of solutions of the system (1) when its pencil is regular. In Section 3 we study the case of the system (1) with a singular pencil, and Section 3 contains numerical examples.
2 Regular case
In this section, we consider the case of the system (1) with a regular pencil. The class of is characterized by a uniquely defined element, known as complex Weierstrass canonical form, , see [39, 41, 44, 45], specified by the complete set of invariants of . This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over the field ℱ. In the case where is regular, we have e.d. of the following type:
e.d. of the type are called finite elementary divisors (f.e.d.), where is a finite eigenvalue of algebraic multiplicity ;
e.d. of the type are called infinite elementary divisors (i.e.d.), where q is the algebraic multiplicity of the infinite eigenvalues.
We assume that and .
Definition 2.1 Let be elements of . The direct sum of them denoted by is the .
Definition 2.2 If for the system (1) with boundary conditions (2) there exists at least one solution, the boundary value problem (1)-(2) is said to be consistent.
where is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of , and is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of .
The proof is completed. □
where is an induced matrix norm and is the discrete Mittag-Leffler function with two parameters as defined by Definition 2.3.
The proof is completed. □
by taking the sum of the above equations and using the fact that , we arrive easily at the solution (12). The proof is completed. □
- 1.The pencil has p distinct eigenvalues and all lie within the open disk
The unique solution is then given from (16). The proof is completed. □
3 Singular case
respectively. The set of minimal indices and are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of respectively. To sum up, in the case of a singular pencil, we have invariants of the following type:
finite elementary divisors of the type ;
infinite elementary divisors of the type ;
column minimal indices of the type ;
row minimal indices of the type .
where for .
where , , , and .
where , , , and . Taking into account the above expressions, we arrive easily at the subsystems (29), (30), (31), (32), and (33). The proof is completed. □
Solving the system (1) is equivalent to solving subsystems (29), (30), (31), (32) and (33). The solutions of the systems (29), (30) are given by (10) and (12) respectively; see Propositions 2.1 and 2.2.
The proof is completed. □
which means that the solution of the system (32) is unique and is the zero solution. The proof is completed. □
does not provide any non-zero equations. Hence all its solutions can be taken arbitrarily. The proof is completed. □
We can now state the following theorem.
- 2.the column minimal indices are zero, i.e.,(45)
In any other case the system has infinite solutions.
The unique solution is then given from (49). The proof is completed. □
4 Numerical examples
and thus from Theorem 2.1, and since (13) does not hold, the boundary value problem is not consistent.
for every induced matrix norm, from Theorem 3.1 the boundary value problem (1)-(2) is non-consistent.
In this article, we study the boundary value problem of a class of a singular system of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, square with an identically zero matrix pencil and non-square, we study the conditions under which the boundary value problem has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. As a further extension of this article, one can study the stability, the behavior under perturbation and possible applications in economics and engineering of singular matrix difference/differential equations of fractional order. For all this, there is already some research in progress.
We would like to express our sincere gratitude to Professor GI Kalogeropoulos for his helpful and fruitful discussions that clearly improved this article. Moreover, we are very grateful to the anonymous referees for their valuable suggestions that improved the article.
- Baleanu D, Diethelm K, Scalas E: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore; 2012.MATHGoogle Scholar
- Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 1995, 68(1):46–53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
- Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge; 2000.Google Scholar
- Kaczorek T 411. In Selected Problems of Fractional Systems Theory. Springer, Berlin; 2011.View ArticleGoogle Scholar
- Malinowska AB, Torres DFM: Introduction to the Fractional Calculus of Variations. Imperial College Press, London; 2012. ISBN:978–1-84816–966–1/hbkMATHView ArticleGoogle Scholar
- Metzler R, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103(16):7180–7186. 10.1063/1.470346View ArticleGoogle Scholar
- Podlubny I Mathematics in Science and Engineering. In Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Klamka J: Controllability and minimum energy control problem of fractional discrete-time systems. In New Trends in Nanotechnology and Fractional Calculus. Springer, New York; 2010:503–509.View ArticleGoogle Scholar
- Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 2010, 109(3):973–1033. 10.1007/s10440-008-9356-6MATHMathSciNetView ArticleGoogle Scholar
- Ahrendt, K, Castle, L, Holm, M, Yochman, K: Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. (2011)MATHGoogle Scholar
- Atici FM, Eloe PW: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 2011, 41(2):353–370. 10.1216/RMJ-2011-41-2-353MATHMathSciNetView ArticleGoogle Scholar
- Atici FM, Eloe PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 2009, 137(3):981–989.MATHMathSciNetView ArticleGoogle Scholar
- Baleanu D, Mustafa OG, Agarwal RP: Asymptotically linear solutions for some linear fractional differential equations. Abstr. Appl. Anal. 2010., 2010: Article ID 865139Google Scholar
- Baleanu D, Mustafa OG: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 2010, 59(5):1835–1841. 10.1016/j.camwa.2009.08.028MATHMathSciNetView ArticleGoogle Scholar
- Baleanu D, Babakhani A: Employing of some basic theory for class of fractional differential equations. Adv. Differ. Equ. 2011., 2011: Article ID 296353Google Scholar
- Bastos NRO, Ferreira RAC, Torres DFM: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 2011, 29(2):417–437.MATHMathSciNetView ArticleGoogle Scholar
- Bastos NRO, Ferreira RAC, Torres DFM: Discrete-time fractional variational problems. Signal Process. 2011, 91(3):513–524. 10.1016/j.sigpro.2010.05.001MATHView ArticleGoogle Scholar
- Bastos NRO, Mozyrska D, Torres DFM: Fractional derivatives and integrals on time scales via the inverse generalized Laplace transform. Int. J. Math. Comput. 2011, 11: 1–9.MathSciNetGoogle Scholar
- Debbouche A: Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems. Adv. Differ. Equ. 2011., 2011: Article ID 5Google Scholar
- Debbouche A, Baleanu D: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 2011, 62(3):1442–1450. 10.1016/j.camwa.2011.03.075MATHMathSciNetView ArticleGoogle Scholar
- Debbouche A, Baleanu D, Agarwal RP: Nonlocal nonlinear integrodifferential equations of fractional orders. Adv. Differ. Equ. 2012., 2012: Article ID 78Google Scholar
- Ferreira RAC, Torres DFM: Fractional h -difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 2011, 5(1):110–121. 10.2298/AADM110131002FMATHMathSciNetView ArticleGoogle Scholar
- Hein J, McCarthy Z, Gaswick N, McKain B, Speer K: Laplace transforms for the nabla-difference operator. Panam. Math. J. 2011, 21(3):79–97.MATHMathSciNetGoogle Scholar
- Kaczorek T: Positive stable realizations of fractional continuous-time linear systems. Int. J. Appl. Math. Comput. Sci. 2011, 21(4):697–702.MATHMathSciNetView ArticleGoogle Scholar
- Kaczorek T: Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils. Int. J. Appl. Math. Comput. Sci. 2013, 23(1):29–33.MATHMathSciNetView ArticleGoogle Scholar
- Klamka J: Controllability of dynamical systems. Mat. Stosow. 2008, 50(9):57–75.MathSciNetMATHGoogle Scholar
- Klamka J, Wyrwał J: Controllability of second-order infinite-dimensional systems. Syst. Control Lett. 2008, 57(5):386–391. 10.1016/j.sysconle.2007.10.002MATHView ArticleMathSciNetGoogle Scholar
- Dai L Lecture Notes in Control and Information Sciences. Singular Control Systems 1988. (edited by M Thoma and A Wyner)Google Scholar
- Dassios IK: On non homogeneous linear generalized linear discrete time systems. Circuits Syst. Signal Process. 2012, 31(5):1699–1712. 10.1007/s00034-012-9400-7MATHMathSciNetView ArticleGoogle Scholar
- Dassios IK, Kalogeropoulos G: On a non-homogeneous singular linear discrete time system with a singular matrix pencil. Circuits Syst. Signal Process. 2013. doi:10.1007/s00034–012–9541–8Google Scholar
- Dassios I: On solutions and algebraic duality of generalized linear discrete time systems. Discrete Math. Appl. 2012, 22(5–6):665–682.MATHMathSciNetView ArticleGoogle Scholar
- Dassios I: On stability and state feedback stabilization of singular linear matrix difference equations. Adv. Differ. Equ. 2012., 2012: Article ID 75Google Scholar
- Dassios I: On robust stability of autonomous singular linear matrix difference equations. Appl. Math. Comput. 2012, 218(12):6912–6920. 10.1016/j.amc.2011.12.067MATHMathSciNetView ArticleGoogle Scholar
- Dassios I: On a boundary value problem of a class of generalized linear discrete time systems. Adv. Differ. Equ. 2011., 2011: Article ID 51Google Scholar
- Grispos E, Giotopoulos S, Kalogeropoulos G: On generalised linear discrete-time regular delay systems. J. Inst. Math. Comput. Sci., Math. Ser. 2000, 13(2):179–187.MATHMathSciNetGoogle Scholar
- Grispos E, Kalogeropoulos G, Mitrouli M: On generalised linear discrete-time singular delay systems. In Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications (HERCMA 2001). Edited by: Lipitakis EA. LEA, Athens; 2002:484–486. 2 volumesGoogle Scholar
- Grispos E, Kalogeropoulos G, Stratis I: On generalised linear discrete-time singular delay systems. J. Math. Anal. Appl. 2000, 245(2):430–446. 10.1006/jmaa.2000.6761MATHMathSciNetView ArticleGoogle Scholar
- Grispos E: Singular generalised autonomous linear differential systems. Bull. Greek Math. Soc. 1992, 34: 25–43.MATHMathSciNetGoogle Scholar
- Kalogeropoulos, GI: Matrix pencils and linear systems. PhD thesis, City University, London (1985)Google Scholar
- Kalogeropoulos G, Stratis IG: On generalized linear regular delay systems. J. Math. Anal. Appl. 1999, 237(2):505–514. 10.1006/jmaa.1999.6458MATHMathSciNetView ArticleGoogle Scholar
- Karcanias N, Kalogeropoulos G: Geometric theory and feedback invariants of generalized linear systems: a matrix pencil approach. Circuits Syst. Signal Process. 1989, 8(3):375–397. 10.1007/BF01598421MATHMathSciNetView ArticleGoogle Scholar
- Rugh WJ: Linear System Theory. Prentice Hall International, London; 1996.MATHGoogle Scholar
- Sandefur JT: Discrete Dynamical Systems. Academic Press, San Diego; 1990.MATHGoogle Scholar
- Gantmacher RF: The Theory of Matrices. Vols. I, II. Chelsea, New York; 1959.MATHGoogle Scholar
- Mitrouli M, Kalogeropoulos G: A compound matrix algorithm for the computation of the Smith form of a polynomial matrix. Numer. Algorithms 1994, 7(2–4):145–159.MATHMathSciNetView ArticleGoogle Scholar
- Nagai A: Discrete Mittag-Leffler function and its applications. Publ. Res. Inst. Math. Sci. 2003, 1302: 1–20.MathSciNetGoogle Scholar
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