P. Dvalishvili, N. Gorgodze, and T. Tadumadze
VARIATION FORMULAS OF SOLUTION FOR NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
WITH REGARD FOR THE DELAY FUNCTION PERTURBATION AND THE CONTINUOUS
INITIAL CONDITION
Abstract. Variation formulas of solution are obtained for linear with re- spect to prehistory of the phase velocity (quasi-linear) neutral functional- differential equations with variable delays. In the variation formulas, the effect of perturbation of the delay function appearing in the phase coordi- nates is stated.
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2000 Mathematics Subject Classification: 34K38, 34K40, 34K27.
Key words and phrases: Neutral functional-differential equation, vari- ation formula of solution, effect of the delay function perturbation, contin- uous initial condition.
LetI= [a, b]be a finite interval andRnbe then-dimensional vector space of points x= (x1, . . . , xn)T, whereT is the sign of transposition. Suppose that O ⊂ Rn is an open set, and Ef is the set of functions f : I×O2 → Rn satisfying the following conditions: the function f(t,·) : O2 → Rn is continuously differentiable for almost all t ∈ I; the functions f(t, x, y), fx(t, x, y)and fy(t, x, y)are measurable on I for any(x, y)∈O2; for each f ∈Ef and compact setK⊂O, there exists a functionmf,K(t)∈L(I,R+), R+= [0,∞), such that
|f(t, x, y)|+|fx(t, x, y)|+|fy(t, x, y)| ≤mf,K(t) for all(x, y)∈K2 and almost allt∈I.
Further, let D be the set of continuous differentiable scalar functions (delay functions)τ(t), t∈I, satisfying the conditions:
τ(t)< t, τ(t)˙ >0, inf{τ(a) : τ∈D}:=τ >b −∞.
Let Φ be the set of continuously differentiable initial functions φ(t) ∈ O, t∈I1= [bτ , b].
To each elementµ= (t0, τ, φ, f)∈Λ = [a, b)×D×Φ×Ef we assign the quasi-linear neutral functional-differential equation
˙
x(t) =A(t) ˙x(σ(t)) +f(
t, x(t), x(τ(t)))
(1) with the continuous initial condition
x(t) =φ(t), t∈[τ , tb 0], (2) whereA(t)is a given continuous matrix function of dimensionn×n;σ∈D is a fixed delay function.
Definition 1. Let µ = (t0, τ, φ, f) ∈ Λ. A function x(t) = x(t;µ) ∈ O, t∈[bτ , t1],t1∈(t0, b], is said to be a solution of equation (1) with the initial condition (2), or a solution corresponding to the elementµand defined on the interval[τ , tb 1], ifx(t)satisfies condition (2) and is absolutely continuous on the interval[t0, t1]and satisfies equation (1) almost everywhere on[t0, t1].
Letµ0= (t00, τ0, φ0, f0)∈Λbe the given element andx0(t)be a solution corresponding toµ0 and defined on[τ , tb 10], witha < t00< t10< b.
Let us introduce the set of variations V =
{
δµ= (δt0, δτ, δφ, δf) : |δt0| ≤α, ∥δτ∥ ≤α, δφ=
∑k
i=1
λiδφi, δf =
∑k
i=1
λiδfi, |λi| ≤α, i= 1, k }
. Here
δt0∈R, δτ ∈D−τ0, ∥δτ∥=sup{
|δτ(t)|: t∈I} and
δφi ∈Φ−φ0, δfi∈Ef−f0, i= 1, k, are the fixed functions andα >0is a fixed number.
There exist the numbers δ1 > 0 and ε1 > 0 such that for arbitrary (ε, δµ)∈ (0, ε1]×V the elementµ0+εδµ∈ Λ and there corresponds the solutionx(t;µ0+εδµ)defined on the interval[bτ , t10+δ1]⊂I1 ( [1, Theo- rem 2]).
Due to the uniqueness, the solutionx(t;µ0) is a continuation of the so- lution x0(t) on the interval [bτ , t10+δ1]. Therefore, the solution x0(t) is assumed to be defined on the interval[bτ , t10+δ1].
Let us define the increment of the solution
x0(t) =x(t;µ0) : ∆x(t;εδµ) =x(t;µ0+εδµ)−x0(t),
∀(t, ε, δµ)∈[τ , tb 10+δ1]×(0, ε1]×V.
Theorem 1. Let the following conditions hold:
1) the function f0(t, x, y),(t, x, y)∈I×O2 is bounded;
2) there exists the limit
zlim→z0
f0(z) =f0−, z= (t, x, y)∈(a, t00]×O2,
wherez0= (t00, φ0(t00), φ0(τ0(t00))).
Then there exist the numbersε2∈(0, ε1)andδ2∈(0, δ1)such that
∆x(t;εδµ) =εδx(t;δµ) +o(t;εδµ) (3) for arbitrary(t, ε, δµ)∈[t00, t10+δ2]×(0, ε2]×V−, whereV−={δµ∈V : δt0≤0} and
δx(t;δµ) =Y(t00−;t) [
˙
φ0(t00)−A(t00) ˙φ0(σ(t00))−f0− ]
δt0+
+β(t;δµ), (4)
β(t;δµ) = Ψ(t00;t)δφ(t00)+
+
t00
∫
τ0(t00)
Y(γ0(s);t)f0y[γ0(s)] ˙γ0(s)δφ(s)ds+
+
t00
∫
σ(t00)
Y(ϱ(s);t)A(ϱ(s)) ˙ϱ(s) ˙δφ(s)ds+
+
∫t
t00
Y(s;t)f0y[s] ˙x0(τ0(s))δτ(s)ds+
+
∫t
t00
Y(s;t)δf[s]ds, (5)
εlim→0
o(t;εδµ)
ε = 0 uniformly for (t, δµ)∈[t00, t10+δ2]×V−, Y(s;t)andΨ(s;t)are then×n-matrix functions satisfying the system
{
Ψs(s;t) =−Y(s;t)f0x[t]−Y(γ0(s);t)f0y[γ0(s)] ˙γ0(s), Y(s;t) = Ψ(s;t) +Y(ϱ(s);t)A(ϱ(s)) ˙ϱ(s), s∈[t00, t], and the condition
Ψ(s;t) =Y(s;t) = {
H, s=t, Θ, s > t;
f0y[s] =f0y
(s, x0(s), x0(τ0(s)))
, δf[s] =δf(
s, x0(s), x0(τ0(s)))
; γ0(s)is the function, inverse toτ0(t), ϱ(s)is the function, inverse toσ(t), H is the identity matrix and Θis the zero matrix.
Some comments. The function δx(t;δµ) is called the variation of the solutionx0(t), t∈[t00, t10+δ2], and the expression (4) is called the variation formula.
The addend
∫t
t00
Y(s;t)f0y[s] ˙x0(τ0(s))δτ(s)ds
in formula (5) is the effect of perturbation of the delay functionτ0(t).
The expression
Y(t00−;t) [
˙
φ0(t00)−A(t00) ˙φ0(σ(t00))−f0− ]
δt0
is the effect of the continuous initial condition (2) and perturbation of the initial momentt00.
The expression
Ψ(t00;t)δφ(t00) +
t00
∫
τ0(t00)
Y(γ0(s);t)f0y[γ0(s)] ˙γ0(s)δφ(s)ds+
+
t00
∫
σ(t00)
Y(ϱ(s);t)A(ϱ(s)) ˙ϱ(s) ˙δφ(s)ds+
∫t
t00
Y(s;t)δf[s]ds
in formula (5) is the effect of perturbations both of the initial functionφ0(t) and of the functionf0(t, x, y).
Variation formulas of solutions for various classes of neutral functional- differential equations without perturbation of delay function can be found in [2–4]. The variation formula of solution plays the basic role in proving the necessary conditions of optimality and under sensitivity analysis of mathe- matical models [5–8]. Finally, it should be noted that the variation formula allows one to get an approximate solution of the perturbed equation
˙
x(t) =A(t) ˙x(σ(t))+
+f0
(t, x(t), x(τ0(t) +εδτ(t))) +εδf(
t, x(t), x(τ0(t) +εδτ(t))) with the perturbed initial condition
x(t) =φ0(t) +εδφ(t), t∈[bτ , t00+εδt0].
In fact, for a sufficiently smallε∈(0, ε2]it follows from (3) that x(t;µ0+εδµ) =x0(t) +εδx(t;δµ).
Theorem 2. Let the following conditions hold:
1) the function f0(t, x, y),(t, x, y)∈I×O2 is bounded;
2) there exists the limit
zlim→z0
f0(z) =f0+, z∈[t00, b)×O2.
Then for eachbt0 ∈(t00, t10) there exist the numbers ε2 ∈(0, ε1) andδ2 ∈ (0, δ1) such that for arbitrary(t, ε, δµ)∈[bt0, t10+δ2]×(0, ε2]×V+, where V+={δµ∈V : δt0≥0}, formula(3) holds, where
δx(t;δµ) =Y(t00+;t)( ˙φ(t00)−A(t00) ˙φ0(σ(t00))−f0+)δt0+β(t;δµ).
The following assertion is a corollary to Theorems 1 and 2.
Theorem 3. Let the assumptions of Theorems1 and2 be fulfilled. More- over, f0− = f0+ := fb0 and t00 ̸∈ {σ(t10), σ2(t10)), . . .}. Then there exist the numbers ε2∈(0, ε1) andδ2∈(0, δ1)such that for arbitrary (t, ε, δµ)∈ [t10−δ2, t10+δ2]×(0, ε2]×V formula (3)holds, where
δx(t;δµ) =Y(t00;t)( ˙φ(t00)−A(t00) ˙φ0(σ(t00))−fb0)δt0+β(t;δµ).
All assumptions of Theorem 3 are satisfied if the function f0(t, x, y) is continuous and bounded. Clearly, in this case
fb0=f0(
t00, φ0(t00), φ0(τ0(t00))) . Acknowledgement
The work was supported by the Shota Rustaveli National Science Foun- dation (Grant No. 31/23).
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(Received 06.07.2014)
Authors’ addresses:
P. Dvalishvili
Department of Computer Sciences, Iv. Javakhishvili Tbilisi State Uni- versity, 13 University St., Tbilisi 0186, Georgia.
E-mail: [email protected] N. Gorgodze
Department of Mathematics, A. Tsereteli Kutaisi University, 59 King Tamari St., Kutaisi 4600, Georgia.
E-mail: nika−[email protected] T. Tadumadze
1. Department of Mathematics, Iv. Javakhishvili Tbilisi State University, 13 University St., Tbilisi 0186, Georgia.
2. I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, 2 University St., Tbilisi 0186, Georgia.
E-mail: [email protected]
