Statement of the problem

THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS WITH CONCENTRATED AND DISTRIBUTED

DELAYS IN CONTROLS

G. KHARATISHVILI AND T. TADUMADZE

Abstract. In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal prob- lem with fixed time and free right end. It contains absolutely continu- ous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.

1. Statement of the problem. The maximum principle. LetO1 ⊂ Rⁿ, O2 ⊂ R^r be open sets and S = [−s1,0]× · · · ×[−sk,0], sk > · · · >

s1 >0. Let ann-dimensional vector-function f(t, x1, . . . , xs, u1, . . . , uv+k) with fixed t ∈ I = [0, T0] be continuously differentiable with respect to (x1, . . . , xs, u1, . . . , uν+k)∈O^s₁×O₂^ν+k. For fixed (x1, . . . , xs, u1, . . . , uν+k)∈ O₁^s×O^ν+k₂ let this function be measurable with respect to t ∈I, like the matrix-functions fxi, i = 1, . . . , s, fuj, j = 1, . . . , ν+k. For each pair of compacts K ⊂ O1, M ⊂ O2 let there exist a function m(t) = mK,M(t) summable onI such that

|f(t, x1, . . . , xs, u1, . . . , uν+k)|+ Xs i=1

|fxi|+

ν+kX

i=1

|fui| ≤m(t), (1)

∀(t, x1, . . . , xs, u1, . . . , uν+k)∈I×K^s×M^ν+k.

Let now the functions τi(t), i = 1, . . . , s, θj(t), j = 1, . . . , ν, t ∈ I, be absolutely continuous, satisfy the conditions τi(t) ≤ t, ˙τi(t) > 0, i = 1, . . . , s, θj(t)≤ t, ˙θj(t) >0, j = 1, . . . , ν, and τ = min(τ1(0), . . . , τs(0)), θ = min(θ1(0), . . . , θν(0),−sk); U1 ⊂ O2, N ⊂ O1 be convex sets and Ω1 = Ω(U1) = {u ∈ L_∞|u(t) ∈ U1, t ∈ [θ, T0], the closure of u([θ, T0]) is

1991Mathematics Subject Classification. 49K25.

Key words and phrases. Maximum principle, optimal control problem, delays in con- trols, sufficient conditions for optimality.

577

1072-947X/95/1100-0577$07.50/0 c1995 Plenum Publishing Corporation

compact and belongs to O2}. Let L_∞ be a space of essentially bounded measurable functions u: [θ, T0]→ R^r with the norm kuk = ess sup|u(t)|, t∈[θ, T0];G={ϕ∈Eϕ|ϕ(t)∈N, t∈[τ,0]},Eϕ be the space of piecewise continuous functions ϕ: [τ,0]→Rⁿ with a finite number of discontinuity points of the first kind with the norm kϕk = sup|ϕ(t)|, t ∈[τ,0], and let qⁱ(t, x0, x1), (t, x0, x1)∈I×O²₁,i= 0, . . . , l, be continuously differentiable scalar functions.

Definition 1. We shall call the elementζ= (T, x0, ϕ, u)∈I×O1×G×Ω1

admissible if the corresponding solutionx(t) =x(t;ζ) of the system

˙ x(t) =

Z

S

f€

t, x(τ1(t)), . . . , x(τs(t)), u(θ1(t)), . . . , u(θν(t)), u(t+s1), . . . . . . , u(t+sk)

dS, t∈[0, T]⊂I, u∈Ω1 (2) (here and in what followsdSstands fords1. . . dsk), with the initial condition x(t) =ϕ(t), t∈[τ,0), x(0) =x0, (3) is defined on the interval [0, T] and satisfies the condition

qⁱ(T, x0, x(T)) = 0, i= 1, . . . , l. (4) We shall denote a set of admissible elements by ∆1.

Definition 2. We call the element eζ = (T ,e ex0,ϕ,e u)e ∈∆1, Te ∈ (0, T0) optimal if there exists a number δ > 0 such that for each element ζ = (T, x0, ϕ, u)∈∆1satisfying the condition

|T−Te|+|x0−ex0|+kϕ−ϕek+ku−euk ≤δ, the inequality

q⁰(T ,e xe0,x(eTe))≤q⁰(T, x0, x(T)) (5) is fulfilled, whereex(t) =x(t;ζ),e x(t) =x(t;ζ).

The optimal problem consists in finding the optimal elements.

Theorem 1 (The necessary conditions of optimality). Let ζe∈∆1

be an optimal element andt=Tebe a Lebesgue point of the function fe(t) =

Z

S

f€

t,x(τe 1(t)), . . . ,ex(τs(t)),u(θe 1(t)), . . . . . . ,u(θe ν(t)),u(te +s1), . . . ,u(te +sk)

dS.

Then there exist a nonzero vectorπ= (π0, . . . , πl),π0 ≤0, and a solution ψ(t),t∈[0,Te]of the system

ψ(t) =˙ − Xs j=1

ψ(γj(t))fexj(γj(t)) ˙γj(t), t∈[0,Te], ψ(t) = 0, t >T ,e (6) such that the following conditions are fulfilled:

1. the integral maximum principle:

Xs j=1

Z 0 τj(0)

ψ(γj(t))fexj(γj(t)) ˙γj(t)ϕ(t)dte ≥

≥ Xs j=1

Z 0 τj(0)

ψ(γj(t))fexj(γj(t)) ˙γj(t)ϕ(t)dt, ∀ϕ∈G, (7) Z e^T

0

ψ(t)nX^ν

i=1

feui(t)eu(θi(t)) + Xk i=1

Z

S

feuν+i(t, s1, . . . , sk)×

×eu(t+si)dSo dt≥

Z e^T

0

ψ(t)nX^ν

i=1

feui(t)u(θi(t)) +

+ Xk

i=1

Z

S

feuν+i(t, s1, . . . , sk)u(t+si)dSo

dt, ∀u∈Ω1; (8) 2. the conditions of transversality:

πQet=−ψ(Te)fe(Te), πQex0 =−ψ(0), πQex1 =ψ(Te). (9) Here Q= (q⁰, . . . , q^l), the tilde sign over Qmeans that the corresponding gradient is calculated at the point (T ,e ex0,ex(Te)), γj(t) is the inverse of the function τj(t),

fexj(t) = Z

S

fexj(t, s1, . . . , sk)dS, fexj(t, s1, . . . , sk) =fxj

€t,x(τe 1(t)), . . . ,ex(τs(t)),u(θe 1(t)), . . . . . . ,eu(θν(t)),u(te +s1), . . . ,eu(t+sk)

.

Remark 1. If the rank of the matrix (Qet,Qex0,Qex1) is equal to 1 +l, then ψ(t)6≡0.

Consider now the problem with a fixed time and a free right end:

˙ x(t) =

Xs i=1

Ai(t)x(τi(t)) + Z

S

f€

t, u(θ1(t)), . . .

. . . , u(θν(t)), u(t+s1), . . . , u(t+sk)

dS, (10)

x(t) =ϕ(t), t∈[τ,0), x(0) =x0, ϕ∈G, I(ζ) =

Z T0

0

hg€

t, x(τ1(t)), . . . , x(τs(t)) +

Z

S

f⁰€

t, u(θ1(t)), . . . . . . , u(θν(t)), u(t+s1), . . . , u(t+sk)

dSi dt,

where Ai(t), i = 1, . . . , s, t ∈ I, are the summable matrix-functions; the function f(t, u1, . . . , uν+k) with fixed t ∈ I is continuous with respect to (u1, . . . , uν+k) ∈ O₂^ν+k, and with fixed (u1, . . . , uν+k) ∈ O^ν+k₂ it is mea- surable with respect to t ∈ I; for each compact M ⊂ O2 there exists a summable functionm1(t) =mM(t) such that

|f(t, u1, . . . , uν+k)| ≤m1(t), ∀(t, u1, . . . , uν+k)∈I×M^ν+k; the function g(t, x1, . . . , xs) with fixed t ∈I is continuously differentiable and convex with respect to (x1, . . . , xs)∈O^ns₃ , with fixed (x1, . . . , xs)∈O^ns₃ it is measurable with respect tot∈I, and for each compactK⊂O3 there exists a summable functionm2(t) =mk(t) such that

|g(t, x1, . . . , xs)| ≤m2(t), ∀(t, x1, . . . , xs)∈I×O₃^ns;

O3 ⊂Rⁿ is a convex open set; the scalar function f⁰ satisfies some condi- tions likef; moreover, Ω2= Ω(U2), whereU3⊂O2 is an arbitrary set and x0∈Rⁿ is a given point.

Definition 3. An elementζe= (ϕ,e eu)∈∆2=G×Ω2is called optimal if for eachζ∈∆2the inequality

I(ζ)e ≤I(ζ) is fulfilled.

Theorem 2. For an element ζe∈∆2 to be optimal, it suffices to fulfill the following conditions:

Xs j=1

Z 0 τj(0)

‚ψ(γj(t))Aj(γj(t))−egxj(γj(t))ƒ

˙

γj(t)ϕ(t)dte ≥

≥ Xs j=1

Z 0 τj(0)

‚ψ(γj(t))Aj(γj(t))−egxj(γj(t))ƒ

˙

γj(t)ϕ(t)dt, ∀ϕ∈G, (11) Z T0

0

‚ψ(t)f(t,eu(·))−f⁰(t,u(e ·))ƒ dt≥

Z T0

0

‚ψ(t)f(t, u(·))−

−f⁰(t, u(·))ƒ

dt, ∀u∈Ω2. (12)

Here f(t, u(·)) =

Z

S

f€

t, u(θ1(t)), . . . , u(θν(t)), u(t+s1), . . . , u(t+sk) dS, f⁰(t, u(·))is defined analogously; egxj(t) =gxj

€t,ex(τ1(t)), . . . ,ex(τs(t)) and ψ(t)is the solution of the system

ψ(t) =˙ Xs j=1

‚ψ0(γj(t))egxj(γj(t))−ψ(γj(t))Aj(γj(t))ƒ

˙

γj(t), t∈[0, T0],(13) ψ(t) = 0, t≥T0, ψ0(t) = 1, t∈[0, T0], ψ0(t) = 0, t > T0. (14) Remark 2. Theorems 1 and 2 are valid in the case whereθj(t),t∈I,j= 1, . . . , ν, are piecewise absolutely continuous functions (with a finite number of discontinuity points) satisfying the conditions θj(t)≤tand ˙θj(t)>0.

2. Proof of Theorem 1.

2.1. Continuous dependence and differentiability of the solution. Let ζe = (T ,e xe0,ϕ,e eu) ∈ ∆1, T < Te 0 be an optimal element of the problem (2)–(5) and let x(t) =e x(t;eζ), t ∈ [0,Te] be the corresponding solution of system (2).

Lemma 1. For eachε >0 there exists a numberδ=δ(ε)>0 such that for each µ= (x0, ϕ, u)∈O1×G×Ω1 satisfying the condition

|x0−ex0|+kϕ−ϕek+ku−euk ≤δ, (15) the system

˙ x(t) =

Z

S

f€

t, x(τ1(t)), . . . , x(τs(t)), u(θ1(t)), . . . . . . , u(θν(t)), u(t+s1), . . . , u(t+sk)

dS, (16)

x(t) =ϕ(t), t∈[τ,0), x(0) =x0,

has the solution x(t;µ) which is defined on [0,Te+δ] ⊂I. In this case if µi= (xⁱ₀, ϕi, ui),i= 1,2, satisfy the condition (15), then

|x(t;µ1)−x(t;µ2)| ≤ε, t∈[0,Te+δ].

Proof. We rewrite the system (16) in the form

˙

x(t) =fe€

t, x(τ1(t)), . . . , x(τs(t)) +δf€

t, x(τ1(t)), . . . , x(τs(t)) , x(t) =ϕ(t), t∈[τ,0), x(0) =x0,

where

fe(t, x1, . . . , xs) = Z

S

f€

t, x1, . . . , xs,u(θe 1(t)), . . .

. . . ,eu(θν(t)),u(te +s1), . . . ,eu(t+sk) dS, δf(t, x1, . . . , xs) =

Z

S

hf€

t, x1, . . . , xs, u(θ1(t)), . . . . . . , u(θν(t)), u(t+s1), . . . , u(t+sk)

−f€

t, x1, . . . , xs,u(θe 1(t)), . . . . . . ,eu(θν(t)),eu(t+s1), . . . ,eu(t+sk)i

dS.

Let M0 ⊂ O2 be the closure of the set u([θ, Te 0]). Then there exists a number δ > 0 such that the functions u∈ Vδ = {u ∈ Ω1| ku−uek ≤ δ} for almost allt∈[θ, T0] take values from the compactM1⊂O2containing some neighborhood of the compactM0.

Let K0 ⊂ O1 be a compact containing some neighborhood of the set {ex(t)|t∈[τ,Te]}.

We shall estimateδf(t, x1, . . . , xs) for each (x1, . . . , xs)∈K₀^s,u∈Vδ. It is not difficult to notice that

|δf| ≤ Z

S

n Z ¹

0

ŒŒ

Œ d dξf€

t, x1, . . . , xs,eu(θ1(t)) +ξδu(θ1(t)), . . . . . . ,u(θe ν(t)) +ξδu(θν(t)),eu(t+s1) +ξδu(t+s1), . . .

. . . ,u(te +sk) +ξδu(t+sk)ŒŒŒdξo dS, whereδu(t) =u(t)−eu(t).

It is clear thateu+ξδu∈Vδ and hence,u(t) +e ξδu(t)∈M1for almost all t∈[θ, T0].

Taking (1) into consideration, we obtain

|δf|≤

Z

S

nZ ¹

0

hX^ν

j=1

ŒŒfui

€t, x1, . . . , xs,eu(θ1(t))+ξδu(θ1(t)), . . .

. . . ,eu(θν(t)) +ξδu(θν(t)),u(te +s1) +ξδu(t+ξ1), . . . . . . ,eu(t+sk) +ξδu(t+ξk)ŒŒ|δu(θj(t))|+ +

Xk j=1

ŒŒfuν+j

€t, x1, . . . , xs,u(θe 1(t)) +ξδu(θ1(t)), . . . ,eu(θν(t)) +

+ξδu(θν(t)),eu(t+s1) +ξδu(t+s1), . . . ,eu(t+sk) +ξδu(t+sk)ŒŒ×

×|δu(t+sj)|i dξo

dS≤(s1· · ·sk)mk0,M0(t)ku−euk. Thus for each (x1, . . . , xs)∈K₀^s,u∈Vδ,t⁰, t⁰⁰∈Iwe have

ŒŒ

Œ Z t⁰⁰

t0

δf(t, x1, . . . , xs)dtŒŒŒ≤c0δ. (17)

Now it is not difficult to see that Z

I

nŒŒδf(t, x₁, . . . , xs)ŒŒ+ Xs j=1

ŒŒδf_xjŒŒo

dt≤c1=const. (18)

The inequalities (17) and (18) allow us to conclude that Lemma 1 implies Theorem 2.8 (see [1], p.52).

Now let us introduce the set V1 =ˆ

δµ = (δx0, δϕ, δu)|δx0 ∈ O1−ex0, δϕ∈G−ϕ,e δu∈Ω−u,e |δx0| ≤c=const,kδϕk ≤c,kδuk ≤c‰

.

Lemma 2. There exist numbers ε0 > 0, δ0 > 0 such that for each ε ∈ [0, ε0] and δµ ∈ V1 the solution x(t;εδµ) = x(t;µe + εδµ), e

µ= (ex0,ϕ,e u), is defined one [0,Te+δ0], and

εlim→0x(t;εδµ) =x(t;µ)e is uniform over (t, δµ)∈[0,Te+δ0]×V1. Moreover

x(t;εδµ) =ex(t) +εδx(t;δµ) +R(t;εδµ), t∈[0,Te+δ0], (19) wherex(t) =e x(t;µ),e

δx(t;δµ) =Y(0;t)δx0+ Xs j=1

Z 0 τj(0)

Y(γj(ξ);t)fexj(γj(ξ))×

×γ˙j(ξ)δϕ(ξ)dξ+ Z t

0

Y(ξ;t)nX^ν

j=1

feuj(ξ)δu(θj(ξ)) +

+ Xk j=1

Z

S

feuν+j(ξ, s1, . . . , sk)δu(ξ+sj)dSo

dξ, (20)

εlim→0ε⁻¹R(t;εδµ) = 0 is uniform over (t, δµ)∈[0,Te+δ0]×V1. (21) HereY(ξ;t),ξ, t∈Iis a matrix function satisfying both the matrix equation

∂Y(ξ;t)

∂ξ +

Xs j=1

Y(γj(ξ);t)fexj(γj(ξ)) ˙γj(ξ) = 0, 0≤ξ≤t, and the condition

Y(ξ;t) =

(E, ξ=t, 0, ξ > t.

Proof. The first part of the Lemma is a simple corollary of Lemma 1.

It is easily seen that the function ∆x(t) = x(t)−ex(t), where x(t) = x(t;εδµ), satisfies the system

d

dt∆x(t) = Z

S

f€

t, x(τ1(t), . . . , x(τs(t),u(θe 1(t)) + +εδu(θ1(t)), . . . ,u(θe ν(t)) +εδu(θν(t)),eu(t+s1) + +εδu(t+s1), . . . ,eu(t+sk) +εδu(t+sk)

dS−fe(t), (22) and the initial condition

∆x(t) =εδϕ(t), t∈[τ,0), ∆x(0) =εδx0. We can rewrite the system (22) in the form

d

dt∆x(t) = Xs j=1

fexj(t)∆x(τj(t)) +ε Xν j=1

feuj(t)δu(θj(t)) +

+ε Xk j=1

Z

S

fuν+j(t, s1, . . . , sk)δu(t+sj)dS+ Γ(t;εδµ), (23) where

Γ(t;εδµ) = Z

S

hf€

t, x(τ1(t)), . . . , x(τs(t)),u(θe 1(t)) +εδu(θ1(t)), . . . . . . ,u(θe ν(t)) +εδu(θν(t)),u(te +s1) +εδu(t+s1), . . . ,u(te +sk)) +

+εδu(t+sk)

−fe(t)− Xs j=1

fexj(t)∆x(τj(t))−ε Xν j=1

fuj(t)×

×δu(θj(t))−ε Xk j=1

feuν+j(t, s1, . . . , sk)δu(t+sj)i dS.

By means of the Cauchy formula the solution of the system (23) can be represented in the form

∆x(t) =εn

Y(0;t)δx0+ Xs j=1

Z 0 τj(0)

Y(γj(ξ);t)fexj(γj(ξ))×

×γ˙j(ξ)δϕ(ξ)dξ+ Z t

0

Y(ξ;t)hX^ν

j=1

feuj(ξ)δu(θj(ξ)) +

+ Xk j=1

Z

S

feuν+j(ξ;s1, . . . , sk)δu(ξ+sj)dSi dξo

+

+ Z t

0

Y(ξ;t)Γ(ξ)dξ=εδx(t;εδµ) +R(t;εδµ).

Now we shall estimateR. We have

|R(t;εδµ)| ≤ kYk Z e^T+δ0

0 |Γ(t;εδµ)|dt≤

≤ kYkZ e^T+δ0

0

n Z

S

h Z ¹

0

ŒŒ

Œd dξf€

t,ex(τ1(t)) +ξ∆x(τ1(t)), . . . . . . ,x(τe s(t)) +ξ∆x(τs(t)),u(θe 1(t)) +εξδu(θ1(t)), . . . . . . ,u(θe ν(t)) +εξδu(θν(t)),eu(t+s1) +εξδu(t+s1), . . .

. . . ,u(te +sk) +εξδu(t+sk)

− Xs j=1

fexj(t)∆x(τj(t))−

−εX^ν

j=1

feuj(t)δu(θj(t)) + Xk j=1

feuν+j(t, s1, . . . , sk)δu(t+sj)‘ŒŒŒdξi dSo

dt,

where

∆x(t) =x(t)−ex(t), kYk= max

ξ,t∈[0,^Te^+δ0]|Y(ξ;t)|.

Differentiating the integrand with respect toξand grouping correspond- ing terms we get

|R(t;εδµ)| ≤εσ1(εδµ) +k∆xkσ2(εδµ) with

k∆xk= max

0≤t≤^T+δe ⁰|∆x(t)|, σ1(εδµ) =ckYkZ e^T+δ0

0

n Z

S

h Z ¹

0

hX^ν

j=1

ŒŒ

Œfuj

€t,x(τe 1(t)) +ξ∆x(τ1(t)), . . .

. . . ,x(τe s(t)) +ξ∆x(τs(t)),eu(θ1(t)) +εξδu(θ1(t)), . . . ,eu(θν(t)) + +εξδu(θν(t)),u(te +s1) +εξδu(t+s1), . . . ,u(te +sk) +εξδu(t+sk))−

−feuj(t)ŒŒŒ+ Xk j=1

ŒŒ

Œ efuν+j(t,x(τe 1(t)) +ξ∆x(τ1(t)), . . . ,x(τe s(t)) + +ξ∆x(τs(t)),u(θe 1(t)) +εξδu(θ1(t)), . . . ,u(θe ν(t)) +εξδu(θν(t)),eu(t+s1) +

+εξδu(t+s1), . . . ,eu(t+sk) +εξδu(t+sk))−feuν+j(t)ŒŒŒi dξi

dSo dt,

σ2(εδµ) =kYkZ e^T+δ0

0

n Z

S

h Z ¹

0

Xs j=1

ŒŒ

Œfxj

€t,x(τe 1(t)) +

+ξ∆x(τ1(t)), . . . ,ex(τs(t)) +ξ∆x(τs(t)),u(θe 1(t)) +εξδu(θ1(t)), . . . . . . ,u(θe ν(t)) +εξδu(θν(t)),eu(t+s1) +εξδu(t+s1), . . .

. . . ,u(te +sk) +εξδu(t+sk)

−fexj(t)ŒŒŒdξi dSo

dt.

It is not difficult to see that the convergence limε→0σi(εδµ) = 0,i= 1,2, is uniform forδµ∈V1.

Now,

k∆xk ≤εkδx(δµ)k+kR(εδµ)k ≤

≤ε€

kδx(δµ)k+σ1(εδµ)

+σ2(εδµ)k∆xk. Here

kδx(δµ)k= max

0≤t≤^T+δe ⁰|δx(t;δµ)|, kR(εδµ)k= max

0≤t≤^Te^+δ0|R(t;εδµ)|. Hence

k∆xk ≤ ε€

kδx(δµ)k+σ1(εδµ)

1−σ2(εδµ) =εα(εδµ),

where the valueα(εδµ) is bounded whenε→0 uniformly for δµ∈V1. Thus

kR(εδµ)k/ε≤σ1(εδµ) +α(εδµ)σ2(εδµ), whence follows (21).

LetD0 be a set of pointsζ= (T, x0, ϕ, u)∈I×O1×G×Ω1 defined by the following condition: the system

˙ x(t) =

Z

S

f€

t, x(τ1(t)), . . . , x(τs(t)), u(θ1(t)), . . . . . . , u(θν(t)), u(t+s1), . . . , u(t+sk)

dS, x(t) =ϕ(t), t∈[τ,0), x(0) =x0, has the solutionx(t) =x(t;ζ) which is defined on [0, T].

Thus the mapping

L:D0→R¹⁺ⁿ (24)

is given onD0by the formulaL(ζ) = (T, x0, x(T)). It follows from Lemma 1 that the set D0 is open in the topology induced from the space Eζ = R¹×Rⁿ×Eϕ×L_∞.

Lemma 3. The mapping (24) is differentiable at the point ζe = (T ,e xe0,ϕ,e eu). Namely,

dLe^ζ(δζ) = (δT, δx0, δx1),

where δx1=fe(Te)δT+δx(T;e δµ), δζ= (δT, δµ)∈Eζ−ζ.e (25) Proof. Let V ={δζ = (δT, δµ)| |δT| ≤c, δµ ∈ V1}. With small enough ε >0 we haveεδT ≤δ0for eachδζ∈V. Therefore, using the representation (19) and the fact thatt=Teis the Lebesgue point of the function fe(t), we have

x(Te+εδT;εδµ) =x(e Te+εδT) +εδx(Te+εδT;δµ) + +R(Te+εδT;εδµ) =x(e T) +e

Z e^T+εδT e

T

f(t)dte +εδx(Te;δµ) + +ε€

δx(Te+εδT;δµ)−δx(Te;δµ)

+R(Te+εδT;δµ) =

=x(eTe) +εδx1+o1(εδζ), where limε→0o1(εδζ)/ε= 0 uniformly forδζ∈V.

Therefore

L(eζ+εδζ)−L(ζ) =e € eT+εδT,xe0+εδx0, x(Te+εδT))−

−(T ,e ex0,x(e Te)

= (δT, δx0, δx1)·ε+o(εδζ) =dLe^ζ(δζ)ε+o(εδζ), whereo(εδζ) =€

0,0, o1(εδζ) .

2.2. Deduction of the maximum principle. Consider the linear topological spaceEz=R¹×Eζ of the pointsz= (σ, ζ).

LetDbe a subset of this space, D=ˆ

z|z= (σ, ζ)|σ∈R¹, ζ ∈D0

‰.

The setD is open (the more so, finitely open) (see [2,3]). Define onD the mappingP :D→R^1+lby the formula

P(z) =Q(L(ζ)) + (σ,0, . . . ,0) =€

q^o(L(ζ)), . . . , q^l(L(ζ))

+ (σ,0, . . . ,0).

The differential of the mapping P at the point ζe= (0,T ,e ex0,ϕ,e eu) has the form

dPe^z(δz) =σe+∂Qe

∂tδT+ ∂Qe

∂x0

δx0+ ∂Qe

∂x1

δx1, e

σ= (δσ,0, . . . ,0), δz∈Ez−z.e

Using the expressions (20) and (25), we obtain dPe^z(δz) =eσ+∂Qe

∂t + ∂Qe

∂x1

fe(Te)‘

δT+∂Qe

∂x0

+ ∂Qe

∂x1

Y(0;Te)‘ δx0+ +

Xs j=1

∂Qe

∂x1

Z 0 τj(0)

Y(γj(t);Te)fexj(γj(t)) ˙γj(t)δϕ(t)dt+

+∂Qe

∂x1

Z e^T

0

Y(t;Te)hX^ν

j=1

feuj(t)δu(Qj(t)) +

+ Xk j=1

Z

S

feuν+j(t;s1, . . . , sk)δu(t+sj)dSi dt.

In the spaceEz let us define a filter by the following elements:

We^z= (R¹₊∩V0)×V^Te×Ve^x0×(G∩V^ϕe)×(Ω1∩Ve^u),

whereV0,V^Te,Ve^x0,V^ϕe,Ve^u are arbitrary convex neighborhoods of the points 0∈R¹,Te∈(0, T0),ex0∈O1,ϕe∈G,eu∈Ω1, correspondingly,R¹₊= [0,∞).

The filter Φ is convex (the more so, quasi-convex, (see [2,3]). It is evident that [Φ] = Φ, where [Φ] is the filter whose elements are the convex hulls of the elements of the filter Φ. It follows from Lemma 1 that the mappingP is continuous on Φ (see [2,3]).

The optimality of the elementζeis equivalent to its extremality and the latter implies criticality of the mapping P on Φ, which is proved by the well-known method (see [2,3]).

Thus all the premises for the necessary criticality condition are fulfilled [2,3].

Therefore there exist a nonzero vectorπ= (π0, . . . , πl) and an element Wfe^z= (R¹₊∩Ve0)×Ve^Te×Vee^x0×(G∩Ve^ϕe)×(Ω1∩Vee^u)

of the filter Φ such that

πdPe^z(δz)≤0, ∀δz∈K(fWe^z−ez), (26) where K(W) is a cone stretched on the set W. It is easily seen that the condition δz ∈ K(Wfe^z−ez) is equivalent to the condition δσ ∈ R₊¹, δT ∈ R¹−Te,δx0∈Rⁿ−xf0,δϕ∈K(Ve^ϕe−ϕ)e ⊃G−ϕ,e δu∈K(Vee^u−eu)⊃Ω1−u.e Using the expression for the differential dPe^z(δz), assuming thatδu= 0, δϕ = 0, and taking into consideration that δσ ∈ R¹₊, δT, δx0 may take

arbitrary values, we obtain forπ0≤0



 π

∂^Qe

∂t + ^∂Qe

∂x1fe(Te)‘

= 0, π

∂^Qe

∂x0 + ^∂Qe

∂x1Y(0;Te)‘

= 0. (27)

Assuming that in (26)δt1= 0, δx0= 0, δu= 0,δσ= 0 we obtain Xs

j=1

π∂Qe

∂x1

Z 0 τj(0)

Y(γj(t);T)e fexj(γj(t)) ˙γj(t)δϕ(t)dt≤0, δϕ∈G−ϕ.e (28) If we assumeδt1= 0,δx0=δϕ= 0,δσ= 0 we shall get

π∂Qe

∂x1

Z e^T

0

Y(t;Te)hX^ν

j=1

feuj(t)δu(θj(t)) +

+ Z

S

Xk j=1

feuν+j(t;s1, . . . , sk)δu(t+sj)dSi

dt≤0. (29)

Let us introduce the notation ψ(t) =π∂Qe

∂x1

Y(t;Te), t∈[0,T].e (30) It is clear thatψ(t) satisfies the system (6) and the condition

ψ(Te) =π∂Qe

∂x1

. (31)

We can now rewrite the condition (27) in the form (π^∂Qe

∂t =−ψ(Te)fe(T),e π^∂Qe

∂x0 =−ψ(0). (32)

Equations (31) and (32) give the conditions of transversality (9). Taking into account (30), we can rewrite inequalities (28), (29) in the form (7), (8).

Thus Theorem 1 is completely proved.

3. Proof of Theorem 2. Let for ζe = (ϕ,e u)e ∈ ∆2 the conditions (11), (12) be fulfilled. Introducing the notation ∆x(t) =x(t;ζ)−x(x;eζ),ζ∈∆2

and taking into account (10), (13), and (14), we obtain 0 =ψ(T0)∆x(t0)−ψ(0)∆x(0) =

Z T0

0

d

dt(ψ(t)∆x(t))dt=

= Z T0

0

‚ψ(t)∆x(t) +˙ ψ(t)d dt∆x(t)ƒ

dt=

= Z T0

0

nX^s

j=1

ψ0(γj(t))egxj(γj(t)) ˙γj(t)∆x(t)− Xs j=1

ψ(γj(t))Aj(γj(t)) ˙γj(t)∆x(t)+

+ψ(t) Xs j=1

Aj(t)∆x(τj(t)) +ψ(t)‚

f(t, u(·))−f(t,u(e ·))ƒo dt.

Further, performing elementary transformations, in view of (14) we get 0 =ψ(T0)∆x(t0)−ψ(0)∆x(0) =

Xs j=1

Z τj(T0)

0 egxj(γj(t)) ˙γj(t)∆x(t)dt−

− Xs j=1

Z τj(T0) 0

ψ(γj(t))Aj(γj(t)) ˙γj(t)∆x(t)dt+

+ Xs j=1

Z τj(T0) τj(0)

ψ(γj(t))×Aj(γj(t)) ˙γj(t)∆x(t)dt+

+ Z T0

0

ψ(t)‚

f(t, u(·))−f(t,eu(·))ƒ dt=

Xs j=1

Z T0

γj(0)egxj(t)∆x(τj(t))dt+ +

Xs j=1

Z 0 τj(0)

ψ(γj(t))Aj(γj(t)) ˙γj(t)∆x(t)dt+ Xs j=1

Z γj(0)

0 egxj(t)∆x(τj(t))dt−

− Xs j=1

Z 0

τj(0)gexj(γj(t)) ˙γj(t)∆x(t)dt+ Z T0

0

ψ(t)‚

f(t, u(·))−f(t,u(e ·))ƒ dt=

= Xs j=1

Z T0

0 egxj(t)∆x(τj(t))dt+ Xs j=1

Z 0 τj(0)

h

ψ(γj(t))Aj(γj(t))−

−egxj(γj(t))i

˙

γj(t)∆x(t)dt+ Z T0

0

ψ(t)‚

f(t, u(·))−f(t,eu(·))ƒ dt.

Further,

I(ζ)e −I(ζ) = Z T0

0

hg€

t,ex(τ1(t)), . . . ,x(τe s(t))

−g€

t, x(τ1(t)), . . . . . . , x(τs(t))

+f⁰(t,u(e·))−f⁰(t, u(·))i

dt+ψ(T0)∆x(T0)−

−ψ(0)∆x(0) = Z T0

0

h g€

t,x(τe 1(t)), . . . ,ex(τs(t))

−g€

t, x(τ1(t)), . . . . . . , x(τs(t))

+ Xs j=1

e

gxj(t)∆x(τj(t))i dt+

Xs j=1

Z 0 τj(0)

h

−egxj(γj(t)) +

+ψ(γj(t))Aj(γj(t))i

˙

γj(t)(ϕ(t)−ϕ(t))dte + Z T0

0

n

−f⁰(t, u(·)) + +ψ(t)f(t, u(·))−‚

−f⁰(t,u(e ·)) +ψ(t)f(t,u(e·))ƒo

dt. (33)

Due to the convexity of the functiong, by virtue of the inequalities (11), (12), it follows from (33) thatI(eζ)−I(ζ)≤0 for ζ∈∆2.

References

1. G. L. Kharatishvili, Z. A. Machaidze, N. I. Markozashvili, and T. A. Tadumadze, Abstract variational theory and its applications to opti- mal problems with delays. (Russian)Metsniereba, Tbilisi, 1973.

2. R. V. Gamkrelidze and G. L. Kharatishvili, Extremal problems in linear topological spaces I. Math. Systems Theory 1 (1969), No. 3, 229–

256.

3. R. V. Gamkrelidze and G. L. Kharatishvili, Extremal problems in linear topological spaces. (Russian)Izv. Akad. Nauk SSSR, Ser. Math. 33 (1969), 781–839.

(Received 13.07.1994) Authors’ addresses:

G. Kharatishvili Institute of Cybernetics Georgian Academy of Sciences 5, S. Euli St., Tbilisi 380086 Republic of Georgia

T. Tadumadze

I. Vekua Institute of Applied Mathematics Tbilisi State University

2, University St., Tbilisi 380043 Republic of Georgia