GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 6, 1995, 577-591

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 problem with fixed time and free right end. It contains absolutely continuous 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. Let O_{1} ⊂ R^{n}, O_{2} ⊂ R^{r} be open sets and S = [−s_{1}, 0] × ⋯ × [−s_{k}, 0], s_{k} > ⋯ > s_{1} > 0. Let an n-dimensional vector-function f(t, x_{1}, …, x_{s}, u_{1}, …, u_{v+k}) with fixed t ∈ I = [0, T_{0}] be continuously differentiable with respect to (x_{1}, …, x_{s}, u_{1}, …, u_{ν+k}) ∈ O_{1}^{s}×O_{2}^{ν+k}. For fixed (x_{1}, …, x_{s}, u_{1}, …, u_{ν+k}) ∈ O_{1}^{s} × O_{2}^{ν+k} let this function be measurable with respect to t ∈ I, like the matrix-functions f_{x_{i}}, i = 1, …, s, f_{u_{j}}, j = 1, …, ν + k. For each pair of compacts K ⊂ O_{1}, M ⊂ O_{2} let there exist a function m(t) = m_{K,M}(t) summable on I such that

|f(t, x_{1}, …, x_{s}, u_{1}, …, u_{ν+k})| + ∑_{i=1}^{s} |f_{x_{i}}| + ∑_{i=1}^{ν+k} |f_{u_{i}}| ≤ m(t),  (1)
∀ (t, x_{1}, …, x_{s}, u_{1}, …, u_{ν+k}) ∈ I × K^{s} × M^{ν+k}.

Let now the functions τ_{i}(t), i = 1, …, s, θ_{j}(t), j = 1, …, ν, t ∈ I, be absolutely continuous, satisfy the conditions τ_{i}(t) ≤ t, τ̇_{i}(t) > 0, i = 1, …, s, θ_{j}(t) ≤ t, θ̇_{j}(t) > 0, j = 1, …, ν, and τ = min(τ_{1}(0), …, τ_{s}(0)), θ = min(θ_{1}(0), …, θ_{ν}(0), −s_{k}); U_{1} ⊂ O_{2}, N ⊂ O_{1} be convex sets and Ω_{1} = Ω(U_{1}) = {u ∈ L_{∞}|u(t) ∈ U_{1}, t ∈ [θ, T_{0}], the closure of u([θ, T_{0}]) is

577 compact and belongs to O_{2}}. Let L_{∞} be a space of essentially bounded measurable functions u : [θ, T_{0}] → R^{r} with the norm ∥u∥ = ess sup |u(t)|, t ∈ [θ, T_{0}]; G = {φ ∈ E_{φ}|φ(t) ∈ N, t ∈ [τ, 0]}, E_{φ} be the space of piecewise continuous functions φ : [τ, 0] → R^{n} with a finite number of discontinuity points of the first kind with the norm ∥φ∥ = sup |φ(t)|, t ∈ [τ, 0], and let q^{i}(t, x_{0}, x_{1}), (t, x_{0}, x_{1}) ∈ I × O_{1}^{2}, i = 0, …, l, be continuously differentiable scalar functions.

1991 Mathematics Subject Classification. 49K25.

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

1072-947X/95/1100-0577$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Definition 1. We shall call the element ζ =(T, x_{0}, φ, u)∈I×O_{1}×G×Ω_{1} admissible if the corresponding solution x(t) = x(t; ζ) of the system

ẋ(t) = ∫_{S} f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …, u(θ_{ν}(t)), u(t + s_{1}), …

…, u(t + s_{k}))dS, t ∈ [0, T] ⊂ I, u ∈ Ω_{1}  (2)

(here and in what follows dS stands for ds_{1} … ds_{k}), with the initial condition

x(t) = φ(t), t ∈ [τ, 0), x(0) = x_{0},  (3)

is defined on the interval [0, T] and satisfies the condition

q^{i}(T, x_{0}, x(T)) = 0, i = 1, …, l.  (4)

We shall denote a set of admissible elements by Δ_{1}.

Definition 2. We call the element ζ̃ = (T̃, x̃_{0}, φ̃, ũ) ∈ Δ_{1}, T̃ ∈ (0, T_{0}) optimal if there exists a number δ > 0 such that for each element ζ = (T, x_{0}, φ, u) ∈ Δ_{1} satisfying the condition

|T − T̃| + |x_{0} − x̃_{0}| + ∥φ − φ̃∥ + ∥u − ũ∥ ≤ δ,

the inequality

q^{0}(T̃, x̃_{0}, x̃(T̃)) ≤ q^{0}(T, x_{0}, x(T))  (5)

is fulfilled, where x̃(t) = x(t; ζ̃), x(t) = x(t; ζ).

The optimal problem consists in finding the optimal elements.

Theorem 1 (The necessary conditions of optimality). Let ζ̃ ∈ Δ_{1} be an optimal element and t = T̃ be a Lebesgue point of the function

f̃(t) = ∫_{S} f(t, x̃(τ_{1}(t)), …, x̃(τ_{s}(t)), ũ(θ_{1}(t)), …
…, ũ(θ_{ν}(t)), ũ(t + s_{1}), …, ũ(t + s_{k}))dS.

THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS 579

Then there exist a nonzero vector π = (π_{0}, …, π_{l}), π_{0} ≤ 0, and a solution ψ(t), t ∈ [0, T̃] of the system

ψ̇(t) = − ∑_{j=1}^{s} ψ(γ_{j}(t))f̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t), t ∈ [0, T̃], ψ(t) = 0, t > T̃, (6)

such that the following conditions are fulfilled:

1. the integral maximum principle:

∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} ψ(γ_{j}(t))f̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)φ̃(t)dt ≥

≥ ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} ψ(γ_{j}(t))f̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)φ(t)dt, ∀φ ∈ G,  (7)

∫ _{T} ; 0 ψ(t){ ∑_{i=1}^{ν} f̃_{u_{i}}(t)ũ(θ_{i}(t)) + ∑_{i=1}^{k} ∫_{S} f̃_{u_{ν+i}}(t, s_{1}, …, s_{k}) ×

×ũ(t + s_{i})dS}dt ≥ ∫ _{T} ; 0 ψ(t){ ∑_{i=1}^{ν} f̃_{u_{i}}(t)u(θ_{i}(t)) +

+ ∑_{i=1}^{k} ∫_{S} f̃_{u_{ν+i}}(t, s_{1}, …, s_{k})u(t + s_{i})dS}dt, ∀u ∈ Ω_{1};  (8)

2. the conditions of transversality:

πQ̃_{t} = −ψ(T̃)f̃(T̃), πQ̃_{x_{0}} = −ψ(0), πQ̃_{x_{1}} = ψ(T̃).  (9)

Here Q = (q^{0}, …, q^{l}), the tilde sign over Q means that the corresponding gradient is calculated at the point (T̃, x̃_{0}, x̃(T̃)), γ_{j}(t) is the inverse of the function τ_{j}(t),

f̃_{x_{j}}(t) = ∫_{S} f̃_{x_{j}}(t, s_{1}, …, s_{k})dS,
f̃_{x_{j}}(t, s_{1}, …, s_{k}) = f_{x_{j}}(t, x̃(τ_{1}(t)), …, x̃(τ_{s}(t)), ũ(θ_{1}(t)), …
…, ũ(θ_{ν}(t)), ũ(t + s_{1}), …, ũ(t + s_{k})).

Remark 1. If the rank of the matrix (Q̃_{t}, Q̃_{x_{0}}, Q̃_{x_{1}}) is equal to 1 + l, then ψ(t) ≢ 0.

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

ẋ(t) = ∑_{i=1}^{s} A_{i}(t)x(τ_{i}(t)) + ∫_{S} f(t, u(θ_{1}(t)), …

…, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k}))dS,  (10)
x(t) = φ(t), t ∈ [τ, 0), x(0) = x_{0}, φ ∈ G,
I(ζ) = ∫_{0}^{T_{0}} [g(t, x(τ_{1}(t)), …, x(τ_{s}(t))) + ∫_{S} f^{0}(t, u(θ_{1}(t)), …
…, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k}))dS]dt,

where A_{i}(t), i = 1, …, s, t ∈ I, are the summable matrix-functions; the function f(t, u_{1}, …, u_{ν+k}) with fixed t ∈ I is continuous with respect to (u_{1}, …, u_{ν+k}) ∈ O_{2}^{ν+k}, and with fixed (u_{1}, …, u_{ν+k}) ∈ O_{2}^{ν+k} it is measurable with respect to t ∈ I; for each compact M ⊂ O_{2} there exists a summable function m_{1}(t) = m_{M}(t) such that

|f(t, u_{1}, …, u_{ν+k})| ≤ m_{1}(t), ∀(t, u_{1}, …, u_{ν+k}) ∈ I × M^{ν+k};

the function g(t, x_{1}, …, x_{s}) with fixed t ∈ I is continuously differentiable and convex with respect to (x_{1}, …, x_{s}) ∈ O_{3}^{ns}, with fixed (x_{1}, …, x_{s}) ∈ O_{3}^{ns} it is measurable with respect to t ∈ I, and for each compact K ⊂ O_{3} there exists a summable function m_{2}(t) = m_{k}(t) such that

|g(t, x_{1}, …, x_{s})| ≤ m_{2}(t), ∀(t, x_{1}, …, x_{s}) ∈ I × O_{3}^{ns};

O_{3} ⊂ R^{n} is a convex open set; the scalar function f^{0} satisfies some conditions like f; moreover, Ω_{2} = Ω(U_{2}), where U_{3} ⊂ O_{2} is an arbitrary set and x_{0} ∈ R^{n} is a given point.

Definition 3. An element ζ̃ = (φ̃, ũ) ∈ Δ_{2} = G × Ω_{2} is called optimal if for each ζ ∈ Δ_{2} the inequality

I(ζ̃) ≤ I(ζ)

is fulfilled.

Theorem 2. For an element ζ̃ ∈ Δ_{2} to be optimal, it suffices to fulfill the following conditions:

∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} [ψ(γ_{j}(t))A_{j}(γ_{j}(t)) − g̃_{x_{j}}(γ_{j}(t))]γ̇_{j}(t)φ̃(t)dt ≥

≥ ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} [ψ(γ_{j}(t))A_{j}(γ_{j}(t)) − g̃_{x_{j}}(γ_{j}(t))]γ̇_{j}(t)φ(t)dt, ∀φ ∈ G, (11)

∫_{0}^{T_{0}} [ψ(t)f(t, ũ(·)) − f^{0}(t, ũ(·))]dt ≥ ∫_{0}^{T_{0}} [ψ(t)f(t, u(·)) −

−f^{0}(t, u(·))]dt, ∀u ∈ Ω_{2}.  (12)

Here

f(t, u(·)) = ∫_{S} f(t, u(θ_{1}(t)), …, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k}))dS,

f^{0}(t, u(·)) is defined analogously; g̃_{x_{j}}(t) = g_{x_{j}}(t, x̃(τ_{1}(t)), …, x̃(τ_{s}(t))) and ψ(t) is the solution of the system

ψ̇(t) = ∑_{j=1}^{s} [ψ_{0}(γ_{j}(t))g̃_{x_{j}}(γ_{j}(t)) − ψ(γ_{j}(t))A_{j}(γ_{j}(t))]γ̇_{j}(t), t ∈ [0, T_{0}], (13)
ψ(t) = 0, t ≥ T_{0}, ψ_{0}(t) = 1, t ∈ [0, T_{0}], ψ_{0}(t) = 0, t > T_{0}. (14)

Remark 2. Theorems 1 and 2 are valid in the case where θ_{j}(t), t ∈ I, j = 1, …, ν, are piecewise absolutely continuous functions (with a finite number of discontinuity points) satisfying the conditions θ_{j}(t) ≤ t and θ̇_{j}(t) > 0.

2. Proof of Theorem 1.

2.1. Continuous dependence and differentiability of the solution. Let ζ̃ = (T̃, x̃_{0}, φ̃, ũ) ∈ Δ_{1}, T̃ < T_{0} be an optimal element of the problem (2)–(5) and let x̃(t) = x(t; ζ̃), t ∈ [0, T̃] be the corresponding solution of system (2).

Lemma 1. For each ε > 0 there exists a number δ = δ(ε) > 0 such that for each μ = (x_{0}, φ, u) ∈ O_{1} × G × Ω_{1} satisfying the condition

|x_{0} − x̃_{0}| + ∥φ − φ̃∥ + ∥u − ũ∥ ≤ δ,  (15)

the system

ẋ(t) = ∫_{S} f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …
…, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k}))dS,  (16)
x(t) = φ(t), t ∈ [τ, 0), x(0) = x_{0},

has the solution x(t; μ) which is defined on [0, T̃ + δ] ⊂ I. In this case if μ_{i} = (x_{0}^{i}, φ_{i}, u_{i}), i = 1, 2, satisfy the condition (15), then

|x(t; μ_{1}) − x(t; μ_{2})| ≤ ε, t ∈ [0, T̃ + δ].

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

ẋ(t) = f̃(t, x(τ_{1}(t)), …, x(τ_{s}(t))) + δf(t, x(τ_{1}(t)), …, x(τ_{s}(t))),
x(t) = φ(t), t ∈ [τ, 0), x(0) = x_{0},

where

f̃(t, x_{1}, …, x_{s}) = ∫_{S} f(t, x_{1}, …, x_{s}, ũ(θ_{1}(t)), …

…, ũ(θ_{ν}(t)), ũ(t + s_{1}), …, ũ(t + s_{k}))dS,
δf(t, x_{1}, …, x_{s}) = ∫_{S} [f(t, x_{1}, …, x_{s}, u(θ_{1}(t)), …

…, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k})) − f(t, x_{1}, …, x_{s}, ũ(θ_{1}(t)), …

…, ũ(θ_{ν}(t)), ũ(t + s_{1}), …, ũ(t + s_{k}))]dS.

Let M_{0} ⊂ O_{2} be the closure of the set ũ([θ, T_{0}]). Then there exists a number δ > 0 such that the functions u ∈ V_{δ} = {u ∈ Ω_{1}| ∥u − ũ∥ ≤ δ} for almost all t ∈ [θ, T_{0}] take values from the compact M_{1} ⊂ O_{2} containing some neighborhood of the compact M_{0}.

Let K_{0} ⊂ O_{1} be a compact containing some neighborhood of the set {x̃(t)|t ∈ [τ, T̃]}.

We shall estimate δf(t, x_{1}, …, x_{s}) for each (x_{1}, …, x_{s}) ∈ K_{0}^{s}, u ∈ V_{δ}. It is not difficult to notice that

|δf| ≤ ∫_{S} \{ ∫_{0}^{1} |\frac{d}{dξ} f(t, x_{1}, …, x_{s}, ũ(θ_{1}(t)) + ξδu(θ_{1}(t)), …

…, ũ(θ_{ν}(t)) + ξδu(θ_{ν}(t)), ũ(t + s_{1}) + ξδu(t + s_{1}), …

…, ũ(t + s_{k}) + ξδu(t + s_{k}))|dξ}dS,

where δu(t) = u(t) − ũ(t).

It is clear that ũ + ξδu ∈ V_{δ} and hence, ũ(t) + ξδu(t) ∈ M_{1} for almost all t ∈ [θ, T_{0}].

Taking (1) into consideration, we obtain

|δf|≤∫_{S}{∫_{0}^{1}[∑_{j=1}^{ν}|f_{u_{i}}(t, x_{1}, …, x_{s}, ũ(θ_{1}(t))+ξδu(θ_{1}(t)), …

…, ũ(θ_{ν}(t)) + ξδu(θ_{ν}(t)), ũ(t + s_{1}) + ξδu(t + ξ_{1}), …

…, ũ(t + s_{k}) + ξδu(t + ξ_{k}))||δu(θ_{j}(t))| +

+ ∑_{j=1}^{k} |f_{u_{ν+j}}(t, x_{1}, …, x_{s}, ũ(θ_{1}(t)) + ξδu(θ_{1}(t)), …, ũ(θ_{ν}(t)) +

+ξδu(θ_{ν}(t)), ũ(t + s_{1}) + ξδu(t + s_{1}), …, ũ(t + s_{k}) + ξδu(t + s_{k}))| ×

×|δu(t + s_{j})|]dξ}dS ≤ (s_{1} ⋯ s_{k})m_{k_{0},M_{0}}(t)∥u − ũ∥.

Thus for each (x_{1}, …, x_{s}) ∈ K_{0}^{s}, u ∈ V_{δ}, t′, t′′ ∈ I we have

| ∫_{t′}^{t′′} δf(t, x_{1}, …, x_{s})dt| ≤ c_{0}δ.  (17)

Now it is not difficult to see that

∫_{I} {|δf(t, x_{1}, …, x_{s})| + ∑_{j=1}^{s} |δf_{x_{j}} |}dt ≤ c_{1} = 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 V_{1} = {δμ = (δx_{0}, δφ, δu)|δx_{0} ∈ O_{1} − x̃_{0}, δφ ∈ G − φ̃, δu ∈ Ω − ũ, |δx_{0}| ≤ c = const, ∥δφ∥ ≤ c, ∥δu∥ ≤ c}.

Lemma 2. There exist numbers ε_{0} > 0, δ_{0} > 0 such that for each ε ∈ [0, ε_{0}] and δμ ∈ V_{1} the solution x(t; εδμ) = x(t; μ̃ + εδμ), μ̃ = (x̃_{0}, φ̃, ũ), is defined on [0, T̃ + δ_{0}], and

lim_{ε→0} x(t; εδμ) = x(t; μ̃) is uniform over (t, δμ) ∈ [0, T̃ + δ_{0}] × V_{1}.

Moreover

x(t; εδμ) = x̃(t) + εδx(t; δμ) + R(t; εδμ), t ∈ [0, T̃ + δ_{0}],  (19)

where x̃(t) = x(t; μ̃),

δx(t; δμ) = Y (0; t)δx_{0} + ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} Y (γ_{j}(ξ); t)f̃_{x_{j}}(γ_{j}(ξ)) ×

×γ̇_{j}(ξ)δφ(ξ)dξ + ∫_{0}^{t} Y (ξ; t){ ∑_{j=1}^{ν} f̃_{u_{j}}(ξ)δu(θ_{j}(ξ)) +

+ ∑_{j=1}^{k} ∫_{S} f̃_{u_{ν+j}}(ξ, s_{1}, …, s_{k})δu(ξ + s_{j})dS}dξ,  (20)
lim_{ε→0} ε^{−1}R(t; εδμ) = 0 is uniform over (t, δμ) ∈ [0, T̃ + δ_{0}] × V_{1}. (21)

Here Y (ξ; t), ξ, t ∈ I is a matrix function satisfying both the matrix equation

\frac{∂Y (ξ; t)}{∂ξ} + ∑_{j=1}^{s} Y (γ_{j}(ξ); t)f̃_{x_{j}}(γ_{j}(ξ))γ̇_{j}(ξ) = 0, 0 ≤ ξ ≤ t,

and the condition

Y (ξ; t) = \begin{cases}E, ξ = t, \\ 0, ξ > t. \end{cases}

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

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

\frac{d}{dt}Δx(t) = ∫_{S} f(t, x(τ_{1}(t), …, x(τ_{s}(t), ũ(θ_{1}(t)) +

+εδu(θ_{1}(t)), …, ũ(θ_{ν}(t)) + εδu(θ_{ν}(t)), ũ(t + s_{1}) +

+εδu(t + s_{1}), …, ũ(t + s_{k}) + εδu(t + s_{k}))dS − f̃(t),  (22)

and the initial condition

Δx(t) = εδφ(t), t ∈ [τ, 0), Δx(0) = εδx_{0}.

We can rewrite the system (22) in the form

\frac{d}{dt}Δx(t) = ∑_{j=1}^{s} f̃_{x_{j}}(t)Δx(τ_{j}(t)) + ε ∑_{j=1}^{ν} f̃_{u_{j}}(t)δu(θ_{j}(t)) +

+ε ∑_{j=1}^{k} ∫_{S} f_{u_{ν+j}}(t, s_{1}, …, s_{k})δu(t + s_{j})dS + Γ(t; εδμ),  (23)

where

Γ(t; εδμ) = ∫_{S} [f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), ũ(θ_{1}(t)) + εδu(θ_{1}(t)), …

…, ũ(θ_{ν}(t)) + εδu(θ_{ν}(t)), ũ(t + s_{1}) + εδu(t + s_{1}), …, ũ(t + s_{k})) +

+εδu(t + s_{k})) − f̃(t) − ∑_{j=1}^{s} f̃_{x_{j}}(t)Δx(τ_{j}(t)) − ε ∑_{j=1}^{ν} f_{u_{j}}(t) ×

×δu(θ_{j}(t)) − ε ∑_{j=1}^{k} f̃_{u_{ν+j}}(t, s_{1}, …, s_{k})δu(t + s_{j})]dS.

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

Δx(t) = ε{Y (0; t)δx_{0} + ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} Y (γ_{j}(ξ); t)f̃_{x_{j}}(γ_{j}(ξ)) ×

×γ̇_{j}(ξ)δφ(ξ)dξ + ∫_{0}^{t} Y (ξ; t)[ ∑_{j=1}^{ν} f̃_{u_{j}}(ξ)δu(θ_{j}(ξ)) +

+ ∑_{j=1}^{k} ∫_{S} f̃_{u_{ν+j}}(ξ; s_{1}, …, s_{k})δu(ξ + s_{j})dS]dξ} +

THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS 585

+ ∫_{0}^{t} Y (ξ; t)Γ(ξ)dξ = εδx(t; εδμ) + R(t; εδμ).

Now we shall estimate R. We have

|R(t; εδμ)| ≤ ∥Y ∥ ∫_{0}^{_{T}+δ_{0}} |Γ(t; εδμ)|dt ≤

≤ ∥Y ∥ ∫_{0}^{_{T}+δ_{0}} \{ ∫_{S} [ ∫_{0}^{1} |\frac{d}{dξ} f(t, x̃(τ_{1}(t)) + ξΔx(τ_{1}(t)), …

…, x̃(τ_{s}(t)) + ξΔx(τ_{s}(t)), ũ(θ_{1}(t)) + εξδu(θ_{1}(t)), … …, ũ(θ_{ν}(t)) + εξδu(θ_{ν}(t)), ũ(t + s_{1}) + εξδu(t + s_{1}), …

…, ũ(t + s_{k}) + εξδu(t + s_{k})) − ∑_{j=1}^{s} f̃_{x_{j}}(t)Δx(τ_{j}(t)) −

−ε( ∑_{j=1}^{ν} f̃_{u_{j}}(t)δu(θ_{j}(t)) + ∑_{j=1}^{k} f̃_{u_{ν+j}}(t, s_{1}, …, s_{k})δu(t + s_{j}))|dξ]dS}dt,

where

Δx(t) = x(t) − x̃(t), ∥Y ∥ = max_{ξ,t∈[0,_{T}+δ_{0}]} |Y (ξ; t)|.

Differentiating the integrand with respect to ξ and grouping corresponding terms we get

|R(t; εδμ)| ≤ εσ_{1}(εδμ) + ∥Δx∥σ_{2}(εδμ)

with

∥Δx∥ = max_{0≤t≤_{T}+δ_{0}} |Δx(t)|,

σ_{1}(εδμ) = c∥Y ∥ ∫_{0}^{_{T}+δ_{0}} { ∫_{S} [ ∫_{0}^{1} [ ∑_{j=1}^{ν} |f_{u_{j}}(t, x̃(τ_{1}(t)) + ξΔx(τ_{1}(t)), …

…, x̃(τ_{s}(t)) + ξΔx(τ_{s}(t)), ũ(θ_{1}(t)) + εξδu(θ_{1}(t)), …, ũ(θ_{ν}(t)) + +εξδu(θ_{ν}(t)), ũ(t + s_{1}) + εξδu(t + s_{1}), …, ũ(t + s_{k}) + εξδu(t + s_{k})) −

−f̃_{u_{j}}(t)| + ∑_{j=1}^{k} |f̃_{u_{ν+j}}(t, x̃(τ_{1}(t)) + ξΔx(τ_{1}(t)), …, x̃(τ_{s}(t)) +

+ξΔx(τ_{s}(t)), ũ(θ_{1}(t)) + εξδu(θ_{1}(t)), …, ũ(θ_{ν}(t)) + εξδu(θ_{ν}(t)), ũ(t + s_{1}) + +εξδu(t + s_{1}), …, ũ(t + s_{k}) + εξδu(t + s_{k})) − f̃_{u_{ν+j}}(t)|]dξ]dS}dt,

σ_{2}(εδμ) = ∥Y ∥ ∫_{0}^{_{T}+δ_{0}} { ∫_{S} [ ∫_{0}^{1} ∑_{j=1}^{s} |f_{x_{j}}(t, x̃(τ_{1}(t)) +

+ξΔx(τ_{1}(t)), …, x̃(τ_{s}(t)) + ξΔx(τ_{s}(t)), ũ(θ_{1}(t)) + εξδu(θ_{1}(t)), … …, ũ(θ_{ν}(t)) + εξδu(θ_{ν}(t)), ũ(t + s_{1}) + εξδu(t + s_{1}), …

…, ũ(t + s_{k}) + εξδu(t + s_{k})) − f̃_{x_{j}}(t)|dξ]dS}dt.

It is not difficult to see that the convergence lim_{ε→0} σ_{i}(εδμ) = 0, i = 1, 2, is uniform for δμ ∈ V_{1}.

Now,

∥Δx∥ ≤ ε∥δx(δμ)∥ + ∥R(εδμ)∥ ≤
≤ ε(∥δx(δμ)∥ + σ_{1}(εδμ)) + σ_{2}(εδμ)∥Δx∥.

Here

∥δx(δμ)∥ = max_{0≤t≤_{T}+δ_{0}} |δx(t; δμ)|, ∥R(εδμ)∥ = max_{0≤t≤_{T}+δ_{0}} |R(t; εδμ)|.

Hence

∥Δx∥ ≤ \frac{ε(∥δx(δμ)∥ + σ_{1}(εδμ))}{1 − σ_{2}(εδμ)} = εα(εδμ),

where the value α(εδμ) is bounded when ε → 0 uniformly for δμ ∈ V_{1}.

Thus

∥R(εδμ)∥/ε ≤ σ_{1}(εδμ) + α(εδμ)σ_{2}(εδμ),

whence follows (21).

Let D_{0} be a set of points ζ = (T, x_{0}, φ, u) ∈ I × O_{1} × G × Ω_{1} defined by the following condition: the system

ẋ(t) = ∫_{S} f(t, x(τ_{1}(t)), …, x(τ_{s}(t)), u(θ_{1}(t)), …
…, u(θ_{ν}(t)), u(t + s_{1}), …, u(t + s_{k}))dS,
x(t) = φ(t), t ∈ [τ, 0), x(0) = x_{0},

has the solution x(t) = x(t; ζ) which is defined on [0, T].

Thus the mapping

L : D_{0} → R^{1+n}  (24)

is given on D_{0} by the formula L(ζ) = (T, x_{0}, x(T)). It follows from Lemma 1 that the set D_{0} is open in the topology induced from the space E_{ζ} = R^{1} × R^{n} × Eφ × L_{∞}.

Lemma 3. The mapping (24) is differentiable at the point ζ̃ = (T̃, x̃_{0}, φ̃, ũ). Namely,

\begin{matrix}dL_{ζ}(δζ) = (δT, δx_{0}, δx_{1}), \\ where δx_{1} = f̃(T̃)δT + δx(T̃; δμ), δζ = (δT, δμ) ∈ E_{ζ} − ζ̃. \end{matrix} (25)

Proof. Let V = {δζ = (δT, δμ)| |δT | ≤ c, δμ ∈ V_{1}}. With small enough ε > 0 we have εδT ≤ δ_{0} for each δζ ∈ V. Therefore, using the representation (19) and the fact that t = T̃ is the Lebesgue point of the function f̃(t), we have

x(T̃ + εδT; εδμ) = x̃(T̃ + εδT) + εδx(T̃ + εδT; δμ) +

+R(T̃ + εδT; εδμ) = x̃(T̃) + ∫ _{T}+εδT ; _{T} f̃(t)dt + εδx(T̃; δμ) +

+ε(δx(T̃ + εδT; δμ) − δx(T̃; δμ)) + R(T̃ + εδT; δμ) =

= x̃(T̃) + εδx_{1} + o_{1}(εδζ),

where lim_{ε→0} o_{1}(εδζ)/ε = 0 uniformly for δζ ∈ V.

Therefore

L(ζ̃+ εδζ) − L(ζ̃) = (T̃ + εδT, x̃_{0} + εδx_{0}, x(T̃ + εδT)) −

−(T̃, x̃_{0}, x̃(T̃)) = (δT, δx_{0}, δx_{1}) · ε + o(εδζ) = dL_{ζ}(δζ)ε + o(εδζ),

where o(εδζ) = (0, 0, o_{1}(εδζ)).

2.2. Deduction of the maximum principle. Consider the linear topological space E_{z} = R^{1} × E_{ζ} of the points z = (σ, ζ).

Let D be a subset of this space,

D = {z|z = (σ, ζ)|σ ∈ R^{1}, ζ ∈ D_{0}}.

The set D is open (the more so, finitely open) (see [2,3]). Define on D the mapping P : D → R^{1+l} by 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 ζ̃ = (0, T̃, x̃_{0}, φ̃, ũ) has the form

dP_{z}(δz) = σ̃ + \frac{∂Q̃}{∂t}δT + \frac{∂Q̃}{∂x_{0}} δx_{0} + \frac{∂Q̃}{∂x_{1}} δx_{1},

σ̃ = (δσ, 0, …, 0), δz ∈ E_{z} − z̃.

Using the expressions (20) and (25), we obtain

dP_{z}(δz) = σ̃ + (\frac{∂Q̃}{∂t} + \frac{∂Q̃}{∂x_{1}} f̃(T̃))δT + (\frac{∂Q̃}{∂x_{0}} + \frac{∂Q̃}{∂x_{1}} Y (0; T̃))δx_{0} +

+ ∑_{j=1}^{s} \frac{∂Q̃}{∂x_{1}} ∫_{τ_{j}(0)}^{0} Y (γ_{j}(t); T̃)f̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)δφ(t)dt +

+\frac{∂Q̃}{∂x_{1}} ∫ _{T} ; 0 Y (t; T̃)[ ∑_{j=1}^{ν} f̃_{u_{j}}(t)δu(Q_{j}(t)) +

+ ∑_{j=1}^{k} ∫_{S} f̃_{u_{ν+j}}(t; s_{1}, …, s_{k})δu(t + s_{j})dS]dt.

In the space E_{z} let us define a filter by the following elements:

W_{z} = (R_{+}^{1} ∩ V_{0}) × V_{T} × V_{x_{0}} × (G ∩ V_{φ}) × (Ω_{1} ∩ V_{u}),

where V_{0}, V_{T}, V_{x_{0}}, V_{φ}, V_{u} are arbitrary convex neighborhoods of the points

0 ∈ R^{1}, T̃ ∈ (0, T_{0}), x̃_{0} ∈ O_{1}, φ̃ ∈ G, ũ ∈ Ω_{1}, correspondingly, R_{+}^{1} = [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 mapping P is continuous on Φ (see [2,3]).

The optimality of the element ζ̃ is 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 vector π = (π_{0}, …, π_{l}) and an element

W̃_{z} = (R_{+}^{1} ∩ Ṽ_{0}) × Ṽ_{T} × Ṽ_{x_{0}} × (G ∩ Ṽ_{φ}) × (Ω_{1} ∩ Ṽ_{u})

of the filter Φ such that

πdP_{z}(δz) ≤ 0, ∀ δz ∈ K(W̃_{z} − z̃),  (26)

where K(W) is a cone stretched on the set W. It is easily seen that the condition δz ∈ K(W̃_{z} − z̃) is equivalent to the condition δσ ∈ R_{+}^{1}, δT ∈

R^{1} − T̃, δx_{0} ∈ R^{n} − x̃_{0}, δφ ∈ K(Ṽ_{φ}− φ̃) ⊃ G − φ̃, δu ∈ K(Ṽ_{u}− ũ) ⊃ Ω_{1} − ũ.

Using the expression for the differential dP_{z}(δz), assuming that δu = 0, δφ = 0, and taking into consideration that δσ ∈ R_{+}^{1}, δT, δx_{0} may take arbitrary values, we obtain for π_{0} ≤ 0

\begin{cases}π(\frac{∂_{Q}}{∂t} + \frac{∂_{Q}}{∂x_{1}} f̃(T̃)) = 0, \\ π(\frac{∂_{Q}}{∂x_{0}} + \frac{∂_{Q}}{∂x_{1}} Y (0; T̃)) = 0. \end{cases}  (27)

Assuming that in (26) δt_{1} = 0, δx_{0} = 0, δu = 0, δσ = 0 we obtain

∑_{j=1}^{s} π \frac{∂Q̃}{∂x_{1}} ∫_{τ_{j}(0)}^{0} Y (γ_{j}(t); T̃)f̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)δφ(t)dt ≤ 0, δφ ∈ G − φ̃. (28)

If we assume δt_{1} = 0, δx_{0} = δφ = 0, δσ = 0 we shall get

π \frac{∂Q̃}{∂x_{1}} ∫ _{T} ; 0 Y (t; T̃)[ ∑_{j=1}^{ν} f̃_{u_{j}}(t)δu(θ_{j}(t)) +

+ ∫_{S} ∑_{j=1}^{k} f̃_{u_{ν+j}}(t; s_{1}, …, s_{k})δu(t + s_{j})dS]dt ≤ 0.  (29)

Let us introduce the notation

ψ(t) = π \frac{∂Q̃}{∂x_{1}} Y (t; T̃), t ∈ [0, T̃].  (30)

It is clear that ψ(t) satisfies the system (6) and the condition

ψ(T̃) = π \frac{∂Q̃}{∂x_{1}} .  (31)

We can now rewrite the condition (27) in the form

\begin{cases}π \frac{∂_{Q}}{∂t} = −ψ(T̃)f̃(T̃), \\ π \frac{∂_{Q}}{∂x_{0}} = −ψ(0). \end{cases}  (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 ζ̃ = (φ̃, ũ) ∈ Δ_{2} the conditions (11),

(12) be fulfilled. Introducing the notation Δx(t) = x(t; ζ) − x(x; ζ̃), ζ ∈ Δ_{2} and taking into account (10), (13), and (14), we obtain

0 = ψ(T_{0})Δx(t_{0}) − ψ(0)Δx(0) = ∫_{0}^{T_{0}} \frac{d}{dt}(ψ(t)Δx(t))dt =

= ∫_{0}^{T_{0}} [ψ̇(t)Δx(t) + ψ(t)\frac{d}{dt}Δx(t)]dt =

590 G. KHARATISHVILI AND T. TADUMADZE

=∫_{0}^{T_{0}}{∑_{j=1}^{s}ψ_{0}(γ_{j}(t))g̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)Δx(t)−∑_{j=1}^{s}ψ(γ_{j}(t))A_{j}(γ_{j}(t))γ̇_{j}(t)Δx(t)+

+ψ(t) ∑_{j=1}^{s} A_{j}(t)Δx(τ_{j}(t)) + ψ(t)[f(t, u(·)) − f(t, ũ(·))]}dt.

Further, performing elementary transformations, in view of (14) we get

0 = ψ(T_{0})Δx(t_{0}) − ψ(0)Δx(0) = ∑_{j=1}^{s} ∫_{0}^{τ_{j}(T_{0})} g̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)Δx(t)dt −

− ∑_{j=1}^{s} ∫_{0}^{τ_{j}(T_{0})} ψ(γ_{j}(t))A_{j}(γ_{j}(t))γ̇_{j}(t)Δx(t)dt +

+ ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{τ_{j}(T_{0})} ψ(γ_{j}(t)) × A_{j}(γ_{j}(t))γ̇_{j}(t)Δx(t)dt +

+ ∫_{0}^{T_{0}} ψ(t)[f(t, u(·)) − f(t, ũ(·))]dt = ∑_{j=1}^{s} ∫_{γ_{j}(0)}^{T_{0}} g̃_{x_{j}}(t)Δx(τ_{j}(t))dt +

+ ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} ψ(γ_{j}(t))A_{j}(γ_{j}(t))γ̇_{j}(t)Δx(t)dt + ∑_{j=1}^{s} ∫_{0}^{γ_{j}(0)} g̃_{x_{j}}(t)Δx(τ_{j}(t))dt −

− ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} g̃_{x_{j}}(γ_{j}(t))γ̇_{j}(t)Δx(t)dt + ∫_{0}^{T_{0}} ψ(t)[f(t, u(·)) − f(t, ũ(·))]dt =

= ∑_{j=1}^{s} ∫_{0}^{T_{0}} g̃_{x_{j}}(t)Δx(τ_{j}(t))dt + ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} [ψ(γ_{j}(t))A_{j}(γ_{j}(t)) −

−g̃_{x_{j}}(γ_{j}(t))]γ̇_{j}(t)Δx(t)dt + ∫_{0}^{T_{0}} ψ(t)[f(t, u(·)) − f(t, ũ(·))]dt. Further,

I(ζ̃) − I(ζ) = ∫_{0}^{T_{0}} [g(t, x̃(τ_{1}(t)), …, x̃(τ_{s}(t))) − g(t, x(τ_{1}(t)), …

…, x(τ_{s}(t))) + f^{0}(t, ũ(·)) − f^{0}(t, u(·))]dt + ψ(T_{0})Δx(T_{0}) − −ψ(0)Δx(0) = ∫_{0}^{T_{0}} [g(t, x̃(τ_{1}(t)), …, x̃(τ_{s}(t))) − g(t, x(τ_{1}(t)), …

…, x(τ_{s}(t))) + ∑_{j=1}^{s} g̃_{x_{j}}(t)Δx(τ_{j}(t))]dt + ∑_{j=1}^{s} ∫_{τ_{j}(0)}^{0} [ − g̃_{x_{j}}(γ_{j}(t)) +

+ψ(γ_{j}(t))A_{j}(γ_{j}(t))]γ̇_{j}(t)(φ(t) − φ̃(t))dt + ∫_{0}^{T_{0}} { − f^{0}(t, u(·)) +

+ψ(t)f(t, u(·)) − [ − f^{0}(t, ũ(·)) + ψ(t)f(t, ũ(·))]}dt.  (33)

Due to the convexity of the function g, by virtue of the inequalities (11), (12), it follows from (33) that I(ζ̃) − I(ζ) ≤ 0 for ζ ∈ Δ_{2}.

References

1. G. L. Kharatishvili, Z. A. Machaidze, N. I. Markozashvili, and T. A. Tadumadze, Abstract variational theory and its applications to optimal 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