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.
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
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
as a perturbation of the linear system
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
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.
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
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
where
R
_{1}
(
t
,
s
) is the unique solution of
with
R
_{1}
(
t
,
t
) =
I
and given by
where
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,
Integrating between 0 to
t
gives
Using Fubini’s theorem we get
Therefore
Conversely suppose that
T
(
t
) is given as above. We will show that it satisfies (2). Consider
From (3) and using Fubini’s theorem we get
Since
R
_{1}
(
t
,
s
) is non zero continuous function for to ≤
s
≤
t
< ∞ we will get
i.e. T(t) satisfies (2).
Theorem 2.2.
The solution of the matrix linear integro differential equation
with K
_{2}
(
t
,
s
) = 0
for s
>
t
> 0
is given by
where R
_{2}
(
t
,
s
)
is the unique solution of
where R
_{2}
(
t
,
t
) =
I and given by
where
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
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
with Y
(
o
) =
T_{o}
where A
_{1}
(
t
) =
A
(
t
) −
φ
_{1}
(
t
,
t
),
B
_{1}
(
t
) =
B
(
t
) −
φ
_{2}
(
t
,
t
).
Proof
. Let
T
(
t
) be any solution of (1) with
T
(
o
) =
T_{o}
then,
Now consider
Integrating on both sides from 0 to
t
and using Fubini’s theorem, we get
Now using
and substituting
T′
(
s
) and using Fubini’s Theorem we will get
i.e
Therefore
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
Substitute
Y′
(
t
) from (3.1) we get
Substituting
L
_{1}
(
t
,
s
) and
L
_{2}
(
t
,
s
) we get
Now using the fact that
and simplifying we get
Since the solutions of the matrix Volterra integral equations are unique, then
Z
(
t
) ≡ 0. Therefore
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
holds then every solution of (5) with Y
(0) =
T_{o} satisfies the inequality
Proof
. Consider
Premultiplying with
and post multiplying with
on both sides and rearranging we get
Integrating from 0 to
t
on both sides we get
Therefore,
Taking norm on both sides and using the inequality (6) we get
Now using Fubini’s Theorem we get
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(t−s)}
.
(iii)
where
L
_{0}
,
γ
(>
α
),
α
_{0}
are positive real numbers.
(iv)F
(
t
) ≡ 0.
If α
−
Mα
_{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
then,
Now applying Grownwall  Belllman inequality we get
Therefore
Since
α
−Mα_{o}
> 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.
email: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.
email:alfredhugo@ymail.com
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