ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF MATRIX LYAPUNOV INTEGRO DIFFERENTIAL EQUATIONS

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.

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 nxn 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 nxn 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 nxn 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),K1(t, s),K2(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 A1(t) and B1(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),K1(t, s) are (n × n) continuous matrices for t ∈ R+ and (t, s) ∈ R+ × R+,K1(t, s) = 0 for s > t > 0 and F ∈ C[R,Rn×n] is given by

where R1(t, s) is the unique solution of

with R1(t, t) = I and given by

where

Proof. Clearly R1(t, s) defined as above exists and satisfies (3). Let T(t) be the solution of (2) for t ≥ 0. Set P(s) = R1(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 R1(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 K2(t, s) = 0 for s > t > 0 is given by

where R2(t, s) is the unique solution of

where R2(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 R1(t, s) and R2(t, s) which establishes the variation of parameters formula for (1).

Theorem 2.3. The solution of (1) is given by

where R1(t, s) and R2(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 φ1(t, s) and φ2(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) = T0 is equivalent to

with Y (o) = To where A1(t) = A(t) − φ1(t, t),B1(t) = B(t) − φ2(t, t).

Proof. Let T(t) be any solution of (1) with T(o) = To 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 L1(t, s) and L2(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 A1(t) and B1(t) (corresponding to A(t) and B(t)) to be stable through the proper choice of ϕ1 and ϕ2. If we are choosing ϕ1 and ϕ2 such that L1(t, s) and L2(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 A1(t) and B1(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) = To satisfies the inequality

Proof. Consider

Pre-multiplying 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 φ1(e,s) amd φ2(t,s) are continuously differentiable matrix functions such that, for 0 ≤ s ≤ t < ∞,

(i) The hypothesis of Lemma (4.1) holds.

(ii) |φ1(t, s) + φ2(t, s)| ≤ Loe−γ(t−s).

(iii) where L0, γ(> α), α0 are positive real numbers.

(iv)F(t) ≡ 0.

If α −Mα0 > 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 −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 φ1(t, s) and φ2(t, s) satisfying the conditions (i) and (ii) of the Theorem (4.1).