LYAPUNOV FUNCTIONALS CONSTRUCTION FOR STOCHASTIC DIFFERENCE SECOND-KIND VOLTERRA EQUATIONS WITH doc

FOR STOCHASTIC DIFFERENCE SECOND-KINDVOLTERRA EQUATIONS WITH CONTINUOUS TIME LEONID SHAIKHET Received 4 August 2003 The general method of Lyapunov functionals construction which was deve

FOR STOCHASTIC DIFFERENCE SECOND-KIND

VOLTERRA EQUATIONS WITH CONTINUOUS TIME

LEONID SHAIKHET

Received 4 August 2003

The general method of Lyapunov functionals construction which was developed duringthe last decade for stability investigation of stochastic differential equations with afteref-fect and stochastic difference equations is considered It is shown that after some mod-ification of the basic Lyapunov-type theorem, this method can be successfully used alsofor stochastic difference Volterra equations with continuous time usable in mathematicalmodels The theoretical results are illustrated by numerical calculations

1 Stability theorem

Construction of Lyapunov functionals is usually used for investigation of stability ofhereditary systems which are described by functional differential equations or Volterraequations and have numerous applications [3,4,8,21] The general method of Lyapunovfunctionals construction for stability investigation of hereditary systems was proposedand developed (see [2,5,6,7,9,10,11,12,13,17,18,19]) for both stochastic differentialequations with aftereffect and stochastic difference equations Here it is shown that aftersome modification of the basic Lyapunov-type stability theorem, this method can also beused for stochastic difference Volterra equations with continuous time, which are popularenough in researches [1,14,15,16,20]

Let {Ω,F, P} be a probability space, {F t, t ≥ t0}a nondecreasing family of sub-

σ-algebras ofF, that is,Ft1⊂Ft2 for t1< t2, and H a space of Ft-measurable functions

Copyright©2004 Hindawi Publishing Corporation

Advances in Di fference Equations 2004:1 (2004) 67–91

2000 Mathematics Subject Classification: 39A11, 37H10

URL: http://dx.doi.org/10.1155/S1687183904308022

with the initial condition

A solution of problem (1.2), (1.3) is anFt-measurable processx(t) = x(t;t0,φ), which

is equal to the initial functionφ(t) from (1.3) fort ≤ t0and with probability 1 defined by(1.2) fort > t0

Definition 1.1 A function x(t) from H is called

(i) uniformly mean square bounded if x 2< ∞;

(ii) asymptotically mean square trivial if

Remark 1.2 It is easy to see that if the function x(t) is uniformly mean square summable,

then it is uniformly mean square bounded and asymptotically mean square quasitrivial

Remark 1.3 It is evident that condition (1.7) follows from (1.6), but the inverse statement

is not true The corresponding function is considered inExample 5.1

Together with (1.2) we will consider the auxiliary difference equation

Definition 1.4 The trivial solution of (1.10) is called

(i) mean square stable if, for any
> 0 and t0≥0, there exists aδ = δ(
,t0)> 0 such

Then the solution of ( 1.2 ), ( 1.3 ) is uniformly mean square summable.

Proof Rewrite condition (1.14) in the form

We show that the right-hand side of inequality (1.18) is bounded Really, using (1.14),(1.15), fort ≥ t0, we have

From (1.2), (1.3), and (1.5), fort ∈[t0,t0+h0], we obtain

From here and (1.8), it follows that the solution of (1.2), (1.3) is uniformly mean square

Remark 1.6 Replace condition (1.12) inTheorem 1.5by the condition

c1(1 + 2A)  φ 2and condition (1.7) follows It means that the trivial solution of (1.10) isasymptotically mean square quasistable

FromTheorem 1.5andRemark 1.6, it follows that an investigation of the solution of(1.2) can be reduced to the construction of appropriate Lyapunov functionals Below,some formal procedure of Lyapunov functionals construction for (1.2) is described.

Remark 1.8 Suppose that in (1.2)h0= h > 0, h j = jh, j =1, 2, Putting t = t0+sh, y(s) = x(t0+sh), and ξ(s) = η(t0+sh), one can reduce (1.2) to the form

y(s + 1) = ξ(s + 1) + G

s, y(s), y(s −1),y(s −2), 

, s > −1,

Below, the equation of type (1.31) is considered

2 Formal procedure of Lyapunov functionals construction

The proposed procedure of Lyapunov functionals construction consists of the followingfour steps

Step 1 Represent the functional F at the right-hand side of (1.2) in the form

which satisfies the conditions ofTheorem 1.5

Step 3 Consider Lyapunov functional V(t) for (1.2) in the formV(t) = V1(t) + V2(t),

where the main component is

Calculate E∆V1(t) and, in a reasonable way, estimate it.

Step 4 In order to satisfy the conditions ofTheorem 1.5, the additional componentV2(t)

is chosen by some standard way

3 Linear Volterra equations with constant coefficients

We demonstrate the formal procedure of Lyapunov functionals construction describedabove for stability investigation of the scalar equation

wherer ≥0 is a given integer,a j are known constants, and the processη(t) is uniformly

mean square summable

3.1 The first way of Lyapunov functionals construction Following the procedure,

represent (Step 1) equation (3.1) in the form (2.1) withF3(t) =0,

and suppose that the solutionD of this equation is a positive semidefinite symmetric

ma-trix of dimensionk + 1 with the elements d i jsuch that the conditiond k+1,k+1 > 0 holds In

this case the functionv(t) = Y (t)DY(t) is a Lyapunov function for (3.4), that is, it isfies the conditions ofTheorem 1.5, in particular, condition (1.14) withγ(t) =0 Really,using (3.4), we have

sat-∆v(t) = Y (t + 1)DY(t + 1) − Y (t)Dy(t)

= Y (t)[A DA − D]Y(t) = − Y (t)UY(t) = − y2(t). (3.6)

FollowingStep 3of the procedure, we will construct a Lyapunov functionalV(t) for

(3.1) in the formV(t) = V1(t) + V2(t), where

V1(t) = X (t)DX(t), X(t) =x(t − k), ,x(t −1),x(t)

Rewrite now (3.1) using representation (3.2) as

X(t + 1) = AX(t) + B(t), B(t) =0, ,0,b(t)

x(t) k

and using (3.2), (3.10), and (3.17), we obtain

Choose now (Step 4) the functionalV2(t) in the form

sum-Using (3.23), (3.21), (3.17), and (3.10), transformq0in the following way:

Note that condition (3.30) can also be represented in the form

3.2 The second way of Lyapunov functionals construction We get another stability

condition Equation (3.1) can be represented (Step 1) in the form (2.1) withF2(t) =0,

x(t + 1) = η(t + 1) + βx(t) + ∆F3(t). (3.35)

In this case the auxiliary equation (2.3) (Step 2) is y(t + 1) = βy(t) The function v(t) = y2(t) is a Lyapunov function for this equation if | β | < 1 We will construct the

Lyapunov functional V(t) (Step 3) for (3.1) in the form V(t) = V1(t) + V2(t), where

V1(t) =(x(t) − F3(t))2 Calculating E∆V1(t), by virtue of representation (3.33), we have

then there exists a so bigµ > 0 that condition β2+ 2α | β −1|+µ −1(α + | β |)< 1 holds also,

and, therefore, the solution of (3.1) is uniformly mean square summable

It is easy to see that condition (3.44) can be written also in the form

4 Particular cases

Here, particular cases of condition (3.31) for different k ≥0 are considered

4.1 Casek =0 Equation (3.5) gives the solutiond11=(1− a2)−1, which is a positiveone if| a0| < 1 From (3.17), it follows thatβ0= | a0| Condition (3.31) takes the form

So, under condition (4.1), the solution of (3.1) is uniformly mean square summable

4.2 Casek =1 The matrix equation (3.5) is equivalent to the system of equations

with the solution

with the solution

If the matrixD with the elements defined by (4.8) is a positive semidefinite one with

d33> 0, then under the condition

For numerical investigation of the solution of (5.1), we determine one of the possibletrajectories of the processη(t), t ≥ t0, in the following way On the interval [t0+nh0,t0+(n + 1)h0),n =0, 1, , put

The graph of the functionη(t) for t0=0,h0=1 is shown onFigure 5.2.

The functionη(t) constructed above satisfies the following conditions:

Therefore, limt →∞ η(t) does not exist So, the function η(t) is an asymptotically

quasitriv-ial function (satisfies condition (1.7)) but not an asymptotically trivquasitriv-ial one (does notsatisfy condition (1.6))

The trajectory of (5.1) with the initial functionφ(θ) =cos 2θ −1 is shown in the point

A(1.1, −0.9), which belongs to the summability region, onFigure 5.3withη(t) ≡0 and

on Figure 5.4withη(t) described above The trajectory of (5.1) with the initial tionφ(θ) =0.05cos2θ is shown in the point B( −0.5,0.6), which does not belong to the

func-summability region, onFigure 5.5withη(t) ≡0 and onFigure 5.6withη(t) described

above The pointsA and B are shown onFigure 5.1

Example 5.2 Consider the difference equation

−2 10 20 30 40 50 60 70 t

−2

−1

1 2

−2 10 20 30 40 t

−4

−2

2 4

−1

0.4 b

As it is shown onFigure 5.7(and naturally it can be shown analytically), forb ≥0condition (5.15) coincides with condition (5.16) and, fora ≥0,b ≥0, conditions (5.15),(5.16), (5.17), and (5.18) give the same region of uniformly mean square summability,

Note also that the region of uniformly mean square summabilityQ k of the solution

of (5.14), obtained by condition (3.31), expands ifk increases, that is, Q0⊂ Q1⊂ Q2⊂

Q3⊂ Q4 So, to get a greater region of uniformly mean square summability, one can usecondition (3.31) fork =5,k =6, and so forth But it is clear that each regionQ kcan beobtained by the condition| b | < 1 only.

To obtain a condition of another type for uniformly mean square summability of thesolution of (5.14), transform the sum from (5.14) fort > 0 in the following way:

The corresponding matrixD is defined by (4.3) witha0= a + b, a1= b(1 − a), and it

is a positive semidefinite one if and only if

b(1 − a)

< 1, | a + b | < 1 − b(1 − a). (5.24)

OnFigure 5.8the graph onFigure 5.7is shown together with the region of uniformlymean square summability obtained by condition (5.24) (the yellow curve)

The trajectory of (5.14) withr =1 and the initial functionalφ(θ) =0.8cosθ is shown

in the pointA(1.2, −1.8), which belongs to the summability region, onFigure 5.9with

η(t) ≡0 and onFigure 5.10withη(t) described above The trajectory of (5.14) withr =1and the initial functionalφ(θ) =0.1cosθ is shown in the point B(1.33, −1.8), which does

not belong to the summability region, onFigure 5.11withη(t) ≡0 and onFigure 5.12withη(t) described above The points A and B are shown onFigure 5.8

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