K. Mansimov, T. Melikov, and T. Tadumadze
VARIATION FORMULAS OF SOLUTION FOR A CONTROLLED DELAY FUNCTIONAL-DIFFERENTIAL
EQUATION TAKING INTO ACCOUNT DELAYS PERTURBATIONS AND THE MIXED INITIAL CONDITION
Abstract. Variation formulas of solution are obtained for a nonlinear controlled delay functional-differential equation with respect to perturba- tions of initial moment, constant delays, initial vector, initial functions and control function. The effects of delay perturbations and the mixed initial condition are discovered in the variation formulas.
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2010 Mathematics Subject Classification. 34K99.
Key words and phrases. Controlled delay functional-differential equa- tion, variation formula of solution, effect of delay perturbation, effect of the mixed initial condition.
1. Introduction
In the present paper, variation formulas of solution (variation formulas) are obtained for a nonlinear controlled delay functional-differential equa- tion under perturbations of initial moment, constant delays, initial vector, initial functions and control function. The effects of delays perturbations and the mixed initial condition are discovered in the variation formulas.
The mixed initial condition means that at the initial moment, some coordi- nates of the trajectory do not coincide with the corresponding coordinates of the initial function, whereas the others coincide. The variation formula allows one to construct an approximate solution of the perturbed equation in an analytical form on the one hand, and in the theory of optimal con- trol it plays the basic role in proving the necessary conditions of optimality [1]–[11], on the other. Variation formulas for various classes of functional- differential equations without perturbation of delay are given in [2], [6], [7]
and [9]–[13]. Variation formulas for delay functional-differential equations with the continuous and discontinuous initial condition taking into account
constant delay perturbation are proved in [14] and [15], respectively. Varia- tion formulas for controlled delay functional-differential equations with the continuous initial condition taking into account constant delay perturbation are proved in [16].
2. Formulation of the Main Results
Let Rnx be the n-dimensional vector space of points x= (x1, . . . , xn)T, where T denotes transposition; suppose P ⊂ Rkp, Z ⊂Rmz and W ⊂ Rru are open sets and O= (P, Z)T ={x= (p, z)T ∈Rxn: p∈P, z∈Z}, with k+m=n.Let then-dimensional functionf(t, x, p, z, u) satisfy the following conditions: for almost allt∈I= [a, b], the functionf(t,·) :O×P×Z×W → Rnx is continuously differentiable; for any (x, p, z, u) ∈ O×P ×Z×W, the functions f(t, x, p, z, u), fx(·), fp(·), fz(·)fu(·) are measurable on I; for arbitrary compacts K ⊂ O, U ⊂ W there exists a function mK,U(·) ∈ L(I,[0,∞)),such that for anyx∈K, (p, z)T ∈K,u∈U and for almost all t∈Ithe inequality
|f(t, x, p, z, u)|+|fx(·)|+|fp(·)|+|fz(·)|+|fu(·)| ≤mK,U(t) is fulfilled.
Let 0< τ1< τ2, 0< σ1< σ2be the given numbers andEϕ=Eϕ(I1, Rkp) be the space of continuous functions ϕ : I1 → Rpk, where I1 = [bτ , b],bτ = a−max{τ2, σ2}.Further,
Φ =©
ϕ∈Eϕ: ϕ(t)∈Pª
and G=©
g∈Eg=Eg(I1, Rmz) : g(t)∈Zª are the sets of initial functions. LetEube the space of bounded measurable functions u: I →Rru and Ω ={u∈Eu : u(t)∈ W, t ∈I, clu(I)⊂W} be a set of control functions, whereu(I) ={u(t) : t∈I} and clu(I) is the closure of the setu(I).
To any element
µ= (t0, τ, σ, p0, ϕ, g, u)∈Λ = (a, b)×(τ1, τ2)×(σ1, σ2)×P×Φ×G×Ω, we assign the controlled delay functional-differential equation
˙
x(t) = ( ˙p(t),z(t))˙ T =f¡
t, x(t), p(t−τ), z(t−σ), u(t)¢
(2.1) with a mixed initial condition
x(t) = (ϕ(t), g(t))T, t∈[bτ , t0), x(t0) = (p0, g(t0))T. (2.2) The condition (2.2) is said to be a mixed initial condition; it consists of two parts: the first part isp(t) =ϕ(t),t∈[bτ , t0),p(t0) =p0,the discontinuous part, since generallyp(t0)6=ϕ(t0); the second part isz(t) =g(t),t∈[bτ , t0], the continuous part, since alwaysz(t0) =g(t0).
Definition 2.1. Let µ = (t0, τ, σ, p0, ϕ, g, u) ∈ Λ. A function x(t) = x(t;µ) ∈ O, t ∈ [bτ , t1], t1 ∈ (t0, b), is called a solution of equation (2.1) with the initial condition (2.2) or a solution corresponding to the element µand defined on the interval [bτ , t1] if it satisfies the condition (2.2) and is
absolutely continuous on the interval [t0, t1] and satisfies the equation (2.1) almost everywhere on [t0, t1].
Letµ0= (t00, τ0, σ0, p00, ϕ0, g0, u0)∈Λ be a fixed element. In the space Eµ=R1t0×R1τ×R1σ×Rkp×Eϕ×Eg×Euwe introduce the set of variations
V =
½
δµ= (δt0, δτ, δσ, δp0, δϕ, δg, δu)∈Eµ−µ0: |δt0| ≤α,
|δτ| ≤α, |δσ| ≤α, |δp0| ≤α, δϕ= Xν
i=1
λiδϕi,
δg= Xν
i=1
λiδgi, δu= Xν
i=1
λiδui, |λi| ≤α, i= 1, ν
¾ ,
whereδϕi ∈Eϕ−ϕ0, δgi ∈Eg−g0, δui∈Eu−u0, i= 1, ν, are the fixed functions;α >0 is a fixed number.
Letx0(t) = (p0(t), z0(t))T be the solution corresponding to the element µ0 and defined on the interval [bτ , t10], with t10 < b. There exist numbers δ1 > 0 and ε1 > 0 such that for arbitrary (ε, δµ) ∈ [0, ε1]×V we have µ0+εδµ ∈Λ. In addition, to this element there corresponds the solution x(t;µ0+εδµ) defined on the interval [bτ , t10+δ1]⊂I1 (see Theorem 5.3 in [17, p. 111]).
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 solutionx0(t) =x(t;µ0):
∆x(t;εδµ) =x(t;µ0+εδµ)−x0(t), (t, ε, δµ)∈[bτ , t10+δ1]×[0, ε1]×V.
Theorem 2.1. Let the following conditions hold:
2.1. t00+τ0< t10;
2.2. the functions ϕ0(t), g0(t), t ∈ I1, are absolutely continuous and
˙
ϕ0(t),g˙0(t)are bounded; there exist compact setsK0⊂OandU0⊂ W containing neighborhoods of sets(ϕ0(I1), g0(I1))T∪x0([t00, t10]) and clu0(I), respectively, such that the function f(t, x, p, z, u), (t, x)∈I×K0,(p, z)T ∈K0,u∈U0, is bounded;
2.3. there exist the limits
t→tlim00−g˙0(t) = ˙g−0,
w→wlim0
f(w, u0(t)) =f0−, w∈(t00−τ0, t00]×O×P×Z,
(w1,w2)→(wlim01,w02)
£f(w1, u0(t))−f(w2, u0(t))¤
=f01−,
w1, w2∈(t00, t00+τ0]×O×P×Z,
where
w= (t, x, p, z), w0=¡
t00, x00, ϕ0(t00−τ0), g0(t00−σ0)¢ , x00= (p00, g0(t00))T,
w01=¡
t00+τ0, x0(t00+τ0), p00, z0(t00+τ0−σ0)¢ , w02=¡
t00+τ0, x0(t00+τ0), ϕ0(t00), z0(t00+τ0−σ0)¢ . Then there exist numbersε2∈(0, ε1]andδ2∈(0, δ1] such that
∆x(t;εδµ) =εδx(t;δµ) +o(t;εδµ) (2.3) for arbitrary
(t, ε, δµ)∈[t10−δ2, t10+δ2]×[0, ε2]ש
δµ∈V :δt0≤0, δτ≤0, δσ≤0ª , where
δx(t;δµ) =n
Y(t00;t)£
(Θk×1,g˙−0)T −f0−¤
−Y(t00+τ0;t)f01−o δt0−
−Y(t00+τ0;t)f01−δτ+β(t;εδµ), (2.4) β(t;εδµ) =Y(t00;t)(δp0, δg(t00))T−
−
½Zt
t00
Y(ξ;t)fp[ξ] ˙p0(ξ−τ0)dξ
¾ δτ−
−
½Zt
t00
Y(ξ;t)fz[ξ] ˙z0(ξ−σ0)dξ
¾ δσ+
+
t00
Z
t00−τ0
Y(ξ+τ0;t)fp[ξ+τ0]δϕ(ξ)dξ+
+
t00
Z
t00−σ0
Y(ξ+σ0;t)fz[ξ+σ0]δg(ξ)dξ+
+ Zt
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ; (2.5)
ε→0lim
o(t;εδµ)
ε = 0
uniformly for
(t, δµ)∈[t10−δ2, t10+δ2]ש
δµ∈V : δt0≤0, δτ≤0, δσ≤0ª
;
Θk×1is thek×1zero matrix,Y(s;t)is then×nmatrix function satisfying on the interval[t00, t] the equation
Yξ(ξ;t) =−Y(ξ;t)fx[ξ]−³
Y(ξ+τ0;t)fp[ξ+τ0], Y(ξ+σ0;t)fz[ξ+σ0]´ ,
and the condition
Y(ξ;t) = (
Hn×n for ξ=t, Θn×n forξ > t.
Here,Hn×n is then×nidentity matrix,
fx[ξ] =fx
³
ξ, x0(ξ), p0(ξ−τ0), z0(ξ−σ0), u0(ξ)
´
, p˙0(ξ−τ0) = ˙p0(s)|s=ξ−τ0,
under p˙0(s)is assumed derivative of the functionp0(s) on the set[bτ , t00)∪ (t00, t10+δ2].
Some comments. The function δx(t;δµ) is called the variation of the solutionx0(t) on the interval [t10−δ2, t10+δ2] and the expression (2.4) is called the variation formula.
c 1) Theorem 2.1 corresponds to the case where the variations at the pointst00, τ0, σ0 are performed simultaneously on the left.
c 2) The addend
−
½
Y(t00+τ0;t)f01−+ Zt
t00
Y(ξ;t)fp[ξ] ˙p0(ξ−τ0)dξ
¾ δτ−
−
½Zt
t00
Y(ξ;t)fz[ξ] ˙z0(ξ−σ0)dξ
¾ δσ
is the effect of perturbations of the delaysτ0 andσ0 (see (2.4) and (2.5)).
c 3) The expression
Y(t00;t)(δp0, δg(t00))T+ +
½
Y(t00;t)£
(Θk×1,g˙−0)T −f0−¤
−Y(t00+τ0;t)f01−
¾ δt0
is the effect of the mixed initial condition (2.2) under perturbations of initial momentt00, initial vectorp00and functiong0(t).
c 4) The expression
t00
Z
t00−τ0
Y(ξ+τ0;t)fp[ξ+τ0]δϕ(ξ)dξ+
+
t00
Z
t00−σ0
Y(ξ+σ0;t)fz[ξ+σ0]δg(ξ)dξ+ Zt
t00
Y(ξ;t)fu[ξ]δu(ξ)dξ
in the formula (2.5) is the effect of perturbations of the initial func- tionsϕ0(t),g0(t) and the control function u0(t).
c 5) The variation formula allows one to obtain an approximate solution of the perturbed functional-differential equation
˙
x(t) =f³
t, x(t), p(t−τ0−εδτ), z(t−σ0−εδσ), u0(t) +εδu(t)´ with the perturbed initial condition
x(t) =¡
ϕ0(t) +εδϕ(t), g0(t) +εδg(t)¢T
, t∈[bτ , t00+εδt0), x(t00+εδt0) =¡
p00+εδp0, g0(t00) +εδg(t00)¢T .
In fact, for a sufficiently smallε∈(0, ε2] from (2.3) it follows that x(t;µ0+εδµ)≈x0(t) +εδx(t;δµ).
Theorem 2.2. Let the conditions 2.1 and 2.2 of Theorem 2.1 hold.
Moreover, there exist the limits
t→tlim00+g˙0(t) = ˙g0+,
w→wlim0
f(w, u0(t)) =f0+, w∈[t00, t10)×O×P×Z,
(w1,w2)→(wlim01,w02)
£f(w1, u0(t))−f0(w2, u0(t))¤
=f01+,
w1, w2∈[t00+τ0, t10)×O×P×Z.
Then there exist numbersε2∈(0, ε1]andδ2∈(0, δ1]such that for arbitrary (t, ε, δµ)∈[t10−δ2, t10+δ2]×[0, ε2]× {δµ∈V : δt0≥0, δτ ≥0, δσ≥0}
the formula(2.3)holds, where δx(t;δµ) =n
Y(t00;t)£
(Θk×1,g˙0+)T−f0+¤
−Y(t00+τ0;t)f01+o δt0−
−Y(t00+τ0;t)f01+δτ+β(t;εδµ).
Theorem 2.2 corresponds to the case where the variations at the points t00,τ0, σ0 are performed simultaneously on the right.
Theorem 2.3. Let the conditions of Theorems2.1 and2.2 hold. More- over,
(Θk×1,g˙0−)T−f0−= (Θk×1,g˙0+)T −f0+=:fb0, f01− =f01+ =:fb01. Then there exist numbersε2∈(0, ε1]andδ2∈(0, δ1]such that for arbitrary (t, ε, δµ)∈[t10−δ2, t10+δ2]×[0, ε2]×V the formula(2.3)holds, where
δx(t;δµ) = n
Y(t00;t)fb0−Y(t00+τ0;t)fb01
o δt0−
−Y(t00+τ0;t)fb01δτ+β(t;εδµ).
Theorem 2.3 corresponds to the case where at the points t00,τ0, σ0 the two-sided variations are simultaneously performed. Theorems 2.1–2.3 are proved by the method given in [10]. Ift00+τ0> t10,then Theorems 2.1–
2.3 are also valid. In this case the number δ2 is so small that t00+τ0 >
t10+δ2,therefore in the variation formulas we have Y(t00+τ0;t) = Θn×n, t ∈[t10−δ2, t10+δ2]. If t00+τ0 =t10, then Theorem 2.1 is valid on the interval [t10, t10+δ2] and Theorem 2.2 is valid on the interval [t10−δ2, t10].
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(Received 09.01.2012) Author’s address:
K. Mansimov
1. Baku State University, 23 Z. Khalilov St., Baku-Az 1148, Azerbaijan.
2. Institute of Cybernetics of NAS Azerbaijan, 9 F. Agaev St., Baku-Az 1141, Azerbaijan.
E-mail: [email protected]; mansimov [email protected] T. Melikov
Azerbaijan Technological University, 103, Sh. I. Khatai Ave., Ganja-Az 2011, Azerbaijan
E-mail: [email protected] T. Tadumadze
I. Javakhishvili Tbilisi State University, Department of Mathematics &
I.Vekua Institute of Applied Mathematics, 2 University St., Tbilisi 0186, Georgia.
E-mail: [email protected]; [email protected]
