Advanced
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF MATRIX LYAPUNOV INTEGRO DIFFERENTIAL EQUATIONS
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF MATRIX LYAPUNOV INTEGRO DIFFERENTIAL EQUATIONS
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 807-816
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 25, 2014
  • Accepted : May 13, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
GOTETI V.R.L. SARMA
ALFRED HUGO

Abstract
The asymptotic behavior of solutions of Lyapunov type matrix Volterra integro differential equation, in which the coefficient matrices are not stable, is studied by the method of reduction. AMS Mathematics Subject Classification : 45J05, 34D05.
Keywords
1. Introduction
Integro differential equations are emerging into the main stream of research because of their wide applications in mathematical biology. Many mathematical models of biology can be represented through integro differential equations [5] . Recently the researcher [1] , [2] , [3] , [4] & [11] , studied different methods for solving the integro differential equations. The problem of stability of integro differential equations is studied by several methods like admissibility of integral operators, defining Lyapunov like functions and so on. Burton [4] studied the stability of the Volterra integro differential equation
PPT Slide
Lager Image
where X and f are n -vectors, A and C are n x n continuous matrices, by constructing a suitable Lyapunov function under various conditions on C and f . Burton also studied the stability of the Volterra integro differential equation in which A is a constant n x n matrix whose characteristic roots have negative real parts and and the uniform stability of Volterra equations [5] and [6] . Grossman and Miller [7] studied the asymptotic behavior of solutions of Volterra integro differential system of the form
PPT Slide
Lager Image
as a perturbation of the linear system
PPT Slide
Lager Image
where A and B are n x n continuous matrices.
Murty, Srinivas and Narasimham [12] studied the asymptotic behaviour of solitions of matrix integro differential equation
PPT Slide
Lager Image
where X ( t ), B ( t ), K ( t , s ) and F ( t ) are n x n continuous matrices and B ( t ) is not necessarily stable. Asymptotic behavior of solutions of Volterra equations also studied by Levin [10] .
In many of the control engineering problems, we often come across the following important matrix of Lyapunov integro differential equation.
PPT Slide
Lager Image
where A ( t ), B ( t ), K 1 ( t , s ), K 2 ( t , s ) are (n × n) matrices defined on 0 ≤ t < ∞ and 0 ≤ s ≤ t < ∞, and F(t) is an (n × n) matrix whose elements are continuous on 0 ≤ t < ∞.
This paper investigates the asymptotic behavior of the solutions of the matrix intego differential equation (1) of Volterra type in which A ( t ) and B ( t ) are not necessarily stable, by the method of reduction. This paper is organized as follows. In section 2, we obtain the solution of (1) in terms of resolvent functions which establishes variation of parameters formula. In section 3, we derive an equivalent equation of (1) which involves an arbitrary functions and by a proper choice of these functions we find new coefficient matrices A 1 ( t ) and B 1 ( t ) (corresponding to A ( t ) and B ( t ) to be stable. In section 4, we present our main results on asymptotic stability.
2. Variation of parameters formula
Theorem 2.1. The solution of the matrix linear integro differential equation
PPT Slide
Lager Image
where A ( t ), K 1 ( t , s ) are (n × n) continuous matrices for t R + and ( t , s ) ∈ R + × R + , K 1 ( t , s ) = 0 for s > t > 0 and F C [ R , R n×n ] is given by
PPT Slide
Lager Image
where R 1 ( t , s ) is the unique solution of
PPT Slide
Lager Image
with R 1 ( t , t ) = I and given by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Proof . Clearly R 1 ( t , s ) defined as above exists and satisfies (3). Let T ( t ) be the solution of (2) for t ≥ 0. Set P ( s ) = R 1 ( t , s ) T ( s ) then,
PPT Slide
Lager Image
Integrating between 0 to t gives
PPT Slide
Lager Image
Using Fubini’s theorem we get
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
Conversely suppose that T ( t ) is given as above. We will show that it satisfies (2). Consider
PPT Slide
Lager Image
From (3) and using Fubini’s theorem we get
PPT Slide
Lager Image
Since R 1 ( t , s ) is non zero continuous function for to ≤ s t < ∞ we will get
PPT Slide
Lager Image
i.e. T(t) satisfies (2).
Theorem 2.2. The solution of the matrix linear integro differential equation
PPT Slide
Lager Image
with K 2 ( t , s ) = 0 for s > t > 0 is given by
PPT Slide
Lager Image
where R 2 ( t , s ) is the unique solution of
PPT Slide
Lager Image
where R 2 ( t , t ) = I and given by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Proof . Proof is the consequence of the theorem (2.1).
With these two results as a tool one can obtain the solution of (1) in terms of R 1 ( t , s ) and R 2 ( t , s ) which establishes the variation of parameters formula for (1).
Theorem 2.3. The solution of (1) is given by
PPT Slide
Lager Image
where R 1 ( t , s ) and R 2 ( t , s ) are stated as in previous theorems.
Proof . We refer [9] .
3. Equivalence equation
In this section we derive an equation, equivalent to (1) by defining proper choice of arbitrary functions.
Theorem 3.1. Let φ 1 ( t , s ) and φ 2 ( t , s ) are n × n matrix functions which are continuously differentiable on 0 ≤ s t < ∞ and commute with T ( t ). Then the equation (1) with T (0) = T 0 is equivalent to
PPT Slide
Lager Image
with Y ( o ) = To where A 1 ( t ) = A ( t ) − φ 1 ( t , t ), B 1 ( t ) = B ( t ) − φ 2 ( t , t ).
PPT Slide
Lager Image
Proof . Let T ( t ) be any solution of (1) with T ( o ) = To then,
PPT Slide
Lager Image
Now consider
PPT Slide
Lager Image
Integrating on both sides from 0 to t and using Fubini’s theorem, we get
PPT Slide
Lager Image
Now using
PPT Slide
Lager Image
and substituting T′ ( s ) and using Fubini’s Theorem we will get
PPT Slide
Lager Image
i.e
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
Hence T ( t ) is also a solution of (5). To prove the converse, let Y ( t ) be any solution of (3.1) existing on [0,∞). Define
PPT Slide
Lager Image
Substitute Y′ ( t ) from (3.1) we get
PPT Slide
Lager Image
Substituting L 1 ( t , s ) and L 2 ( t , s ) we get
PPT Slide
Lager Image
Now using the fact that
PPT Slide
Lager Image
and simplifying we get
PPT Slide
Lager Image
Since the solutions of the matrix Volterra integral equations are unique, then Z ( t ) ≡ 0. Therefore
PPT Slide
Lager Image
Hence Y ( t ) is a solution of (1) and the proof is complete.
Because (5) is equivalent to (1) the stability properties of (1) implies the stability properties of (5). If A ( t ) and B ( t ) are not stable in (1) we can find A 1 ( t ) and B 1 ( t ) (corresponding to A ( t ) and B ( t )) to be stable through the proper choice of ϕ 1 and ϕ 2 . If we are choosing ϕ 1 and ϕ 2 such that L 1 ( t , s ) and L 2 ( t , s ) are vanish then (5) reduced to differential equation equivalent to integro differential equation (1). Now we will present our main theorems on asymptotic stability in next section.
4. Main Results
Lemma 4.1. Let A 1 ( t ) and B 1 ( t ) are ( N × n ) continuous Matrices as defined previously and they commute with their integrals and let M and α are positive real numbers. Suppose the inequality
PPT Slide
Lager Image
holds then every solution of (5) with Y (0) = To satisfies the inequality
PPT Slide
Lager Image
Proof . Consider
PPT Slide
Lager Image
Pre-multiplying with
PPT Slide
Lager Image
and post multiplying with
PPT Slide
Lager Image
on both sides and rearranging we get
PPT Slide
Lager Image
Integrating from 0 to t on both sides we get
PPT Slide
Lager Image
Therefore,
PPT Slide
Lager Image
Taking norm on both sides and using the inequality (6) we get
PPT Slide
Lager Image
Now using Fubini’s Theorem we get
PPT Slide
Lager Image
Theorem 4.2. Let φ 1 ( e , s ) amd φ 2 ( t , s ) are continuously differentiable matrix functions such that, for 0 ≤ s t < ∞,
(i) The hypothesis of Lemma (4.1) holds.
(ii) | φ 1 ( t , s ) + φ 2 ( t , s )| ≤ Loe −γ(ts) .
(iii)
PPT Slide
Lager Image
where L 0 , γ (> α ), α 0 are positive real numbers.
(iv)F ( t ) ≡ 0.
If α 0 > 0, then every solution T ( t ) of (1) tends to zero exponentially as t → ∞.
Proof . In order to show every solution of (1) tends to zero exponentially, it is enough to show every solution of (5) tends to zero exponentially as t → ∞.
From the previous lemma (4.1) and condition (ii) and (iv) implies
PPT Slide
Lager Image
then,
PPT Slide
Lager Image
Now applying Grownwall - Belllman inequality we get
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
Since α −Mαo > 0, the theorem follows.
Remark 4.1. From the Theorem (4.2) the solution of (1) is exponentially asymptotically stable if F ( t ) ≡ 0.
Remark 4.2. If F ( t ) ≠ 0 in the Theorem (4.2), then the solutions of (1) tends to zero as t → ∞.
Remark 4.3. It is possible to select the matrices φ 1 ( t , s ) and φ 2 ( t , s ) satisfying the conditions (i) and (ii) of the Theorem (4.1).
BIO
Dr. Goteti V.R.L. Sarma, after obtained his Ph.D in Applied Mathematics from Andhra University, India in 2003, worked in Eritrea Institute of Technology, Eritrea from 2003 to 2012. Later he joined in University of Dodoma Tanzania in 2013. His research interests includes Dynamical Systems, Mathematical Modelling and Ordinary Differential Equations.
Department of Mathematics,University of Dodoma P.O. Box: 259, Dodoma, Tanzania.
e-mail:gvrlsarma@rediffmail.com
Alfred Hugo, obtained MSc in Mathematical Modelling from University of Dar Es Salaam, 2011. Before that he employed by the University of Dodoma since 2009 to date. His research interests include Mathematical Modelling, Dynamical Systems and Partial Differential equations.
Department of Mathematics,University of Dodoma P.O. Box: 259, Dodoma, Tanzania.
e-mail:alfredhugo@ymail.com
References
Arikoglu A. , Ozkol I. (2008) Solutions of Integral and Integro-Differential Equation Systems by Using Differential Transform Method Computers Mathematics with Applications 56 2411 - 2417    DOI : 10.1016/j.camwa.2008.05.017
Biazar J. (2003) Solution of systems of integro-differential equation by Adomian decomposition method Appl. Math. Comput. 168 1232 - 1238    DOI : 10.1016/j.amc.2004.10.015
Biazar J. , Ghazvini H. , Eslami M. (2009) Hes Homotopy perturbation method for systems of integro-differential Equations Chaos, Solitions and Fractals 39 1253 - 1258    DOI : 10.1016/j.chaos.2007.06.001
Burton T.A. (1980) An integro differential equations Proc. Amer. Math. Soc. 79 393 - 399
Burton T.A. (1980) Uniform stability for Volterra equations J. Diff Equs. 36 40 - 53    DOI : 10.1016/0022-0396(80)90074-1
Burton T.A. (1979) Stability theory for Volterra equations J. Diff Equs. 32 101 - 118    DOI : 10.1016/0022-0396(79)90054-8
Grossman S.I. , Miller R.K. (1970) Perturbation theory for Volterra Integro Differential system J. diff. Equs. 8 451 - 474    DOI : 10.1016/0022-0396(70)90018-5
Hoppensteadt F.C. 1982 Mathematical methods of population biology Cambridge Univ. Press Cambridge
Lakshmikantham V. , Deo S.G. Method of variation of parameters for dynamic systems Gordon and breach scientific Publishers
Levin J.J. (1963) The asymptotic behaviour of the solutions of a Volterra equation Proc. Amer. Math. Soc. 14 534 - 541    DOI : 10.1090/S0002-9939-1963-0152852-8
Maleknejad K. , Kajani M. Tavassoli (2004) Solving Linear integro-differential equation system by Galerkin methods with Hybrid functions 159 603 - 612
Murty K.N. , Srinivas M.A.S. , Narasimham V.A. (1987) Asymptotic behaviour of solutions of matrix integro differential equations Tamkang J. Math. 18 1 - 8