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 O1 ⊂
Rn , O2 ⊂ Rr be open sets and S = [−s1 , 0] × · · · × [−sk , 0], sk > · · · >
s1 > 0. Let an n-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 ) ∈ O1s ×O2ν+k . For fixed (x1 , . . . , xs , u1 , . . . , uν+k ) ∈
O1s × O2ν+k 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 on I such that
|f (t, x1 , . . . , xs , u1 , . . . , uν+k )| +
s
X
i=1
|fxi | +
ν+k
X
i=1
s
|fui | ≤ m(t),
(1)
∀ (t, x1 , . . . , xs , u1 , . . . , uν+k ) ∈ I × K × 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, τ̇i (t) > 0, i =
1, . . . , s, θj (t) ≤ t, θ̇j (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
1991 Mathematics Subject Classification. 49K25.
Key words and phrases. Maximum principle, optimal control problem, delays in controls, sufficient conditions for optimality.
577
c 1995 Plenum Publishing Corporation
1072-947X/95/1100-0577$07.50/0 
578
G. KHARATISHVILI AND T. TADUMADZE
compact and belongs to O2 }. Let L∞ be a space of essentially bounded
measurable functions u : [θ, T0 ] → Rr 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
q i (t, x0 , x1 ), (t, x0 , x1 ) ∈ I × O12 , i = 0, . . . , l, be continuously differentiable
scalar functions.
Definition 1. We shall call the element ζ = (T, x0 , ϕ, u) ∈ I ×O1 ×G×Ω1
admissible if the corresponding solution x(t) = x(t; ζ) of the system
Z
€
f t, x(τ1 (t)), . . . , x(τs (t)), u(θ1 (t)), . . . , u(θν (t)), u(t + s1 ), . . .
ẋ(t) =
S

(2)
. . . , u(t + sk ) dS, t ∈ [0, T ] ⊂ I, u ∈ Ω1
(here and in what follows dS stands for ds1 . . . 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
q i (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 ζe = (Te, x
e0 , 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) ∈ ∆1 satisfying the condition
the inequality
|T − Te| + |x0 − x
e + ku − u
e0 | + kϕ − ϕk
ek ≤ δ,
q 0 (Te, x
e0 , x
e(Te)) ≤ q 0 (T, x0 , x(T ))
(5)
e x(t) = x(t; ζ).
is fulfilled, where x
e(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 and t = Te be a Lebesgue point of the function
Z
€
e(θ1 (t)), . . .
e(τs (t)), u
fe(t) =
f t, x
e(τ1 (t)), . . . , x
S

...,u
e(θν (t)), u
e(t + sk ) dS.
e(t + s1 ), . . . , u
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
579
Then there exist a nonzero vector π = (π0 , . . . , πl ), π0 ≤ 0, and a solution
ψ(t), t ∈ [0, Te] of the system
ψ̇(t) = −
s
X
j=1
ψ(γj (t))fexj (γj (t))γ̇j (t), t ∈ [0, Te], ψ(t) = 0, t > Te,
such that the following conditions are fulfilled:
1. the integral maximum principle:
s Z 0
X
e
ψ(γj (t))fexj (γj (t))γ̇j (t)ϕ(t)dt
≥
≥
Z Te
0
j=1 τj (0)
s
XZ 0
j=1
τj (0)
ψ(γj (t))fexj (γj (t))γ̇j (t)ϕ(t)dt, ∀ϕ ∈ G,
(6)
(7)
k Z
ν
nX
X
u(θi (t)) +
feuν+i (t, s1 , . . . , sk ) ×
ψ(t)
feui (t)e
S
i=1
i=1
Z Te
ν
nX
o
ψ(t)
u(t + si )dS dt ≥
feui (t)u(θi (t)) +
×e
0
+
k Z
X
i=1
S
i=1
o
feuν+i (t, s1 , . . . , sk )u(t + si )dS dt, ∀u ∈ Ω1 ;
(8)
2. the conditions of transversality:
e t = −ψ(Te)fe(Te), π Q
e x = ψ(Te).
e x = −ψ(0), π Q
πQ
0
1
(9)
Here Q = (q 0 , . . . , q l ), the tilde sign over Q means that the corresponding
gradient is calculated at the point (Te, x
e0 , x
e(Te)), γj (t) is the inverse of the
function τj (t),
Z
e
fexj (t, s1 , . . . , sk )dS,
fxj (t) =
S
€
e(τ1 (t)), . . . , x
e(τs (t)), u
fexj (t, s1 , . . . , sk ) = fxj t, x
e(θ1 (t)), . . .

e(θν (t)), u
e(t + sk ) .
...,u
e(t + s1 ), . . . , u
et , Q
ex , Q
e x ) is equal to 1 + l, then
Remark 1. If the rank of the matrix (Q
0
1
ψ(t) 6≡ 0.
Consider now the problem with a fixed time and a free right end:
Z
s
X
€
Ai (t)x(τi (t)) +
ẋ(t) =
f t, u(θ1 (t)), . . .
i=1
S
580
G. KHARATISHVILI AND T. TADUMADZE

. . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) dS,
(10)
x(t) = ϕ(t), t ∈ [τ, 0), x(0) = x0 , ϕ ∈ G,
Z
Z T0 h
€

€
f 0 t, u(θ1 (t)), . . .
g t, x(τ1 (t)), . . . , x(τs (t)) +
I(ζ) =
S
0
 i
. . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) dS 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 ) ∈ O2ν+k , and with fixed (u1 , . . . , uν+k ) ∈ O2ν+k it is measurable with respect to t ∈ I; for each compact M ⊂ O2 there exists a
summable function m1 (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 ) ∈ O3ns , with fixed (x1 , . . . , xs ) ∈ O3ns
it is measurable with respect to t ∈ I, and for each compact K ⊂ O3 there
exists a summable function m2 (t) = mk (t) such that
|g(t, x1 , . . . , xs )| ≤ m2 (t), ∀(t, x1 , . . . , xs ) ∈ I × O3ns ;
O3 ⊂ Rn is a convex open set; the scalar function f 0 satisfies some conditions like f ; moreover, Ω2 = Ω(U2 ), where U3 ⊂ O2 is an arbitrary set and
x0 ∈ Rn is a given point.
e) ∈ ∆2 = G × Ω2 is called optimal if
Definition 3. An element ζe = (ϕ,
e u
for each ζ ∈ ∆2 the inequality
e ≤ I(ζ)
I(ζ)
is fulfilled.
Theorem 2. For an element ζe ∈ ∆2 to be optimal, it suffices to fulfill
the following conditions:
s Z 0
X
ƒ
‚
≥
ψ(γj (t))Aj (γj (t)) − gexj (γj (t)) γ̇j (t)ϕ(t)dt
e
≥
j=1 τj (0)
s
XZ 0
j=1
Z
τj (0)
T0
0
‚
‚
ƒ
ψ(γj (t))Aj (γj (t)) − gexj (γj (t)) γ̇j (t)ϕ(t)dt, ∀ϕ ∈ G, (11)
Z T0
ƒ
‚
e(·)) − f 0 (t, u
e(·)) dt ≥
ψ(t)f (t, u(·)) −
ψ(t)f (t, u
0
ƒ
−f 0 (t, u(·)) dt, ∀u ∈ Ω2 .
(12)
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
Here
f (t, u(·)) =
Z
S
581

€
f t, u(θ1 (t)), . . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) dS,
€

e(τ1 (t)), . . . , x
e(τs (t)) and
f (t, u(·)) is defined analogously; gexj (t) = gxj t, x
ψ(t) is the solution of the system
0
ψ̇(t) =
s
X
‚
j=1
ƒ
ψ0 (γj (t))e
gxj (γj (t)) − ψ(γj (t))Aj (γj (t)) γ̇j (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 where θj (t), t ∈ I, j =
1, . . . , ν, are piecewise absolutely continuous functions (with a finite number
of discontinuity points) satisfying the conditions θj (t) ≤ t and θ̇j (t) > 0.
2. Proof of Theorem 1.
2.1. Continuous dependence and differentiability of the solution. Let
e u
e) ∈ ∆1 , Te < T0 be an optimal element of the problem
ζe = (Te, x
e0 , ϕ,
e t ∈ [0, Te] be the corresponding solution of
(2)–(5) and let x
e(t) = x(t; ζ),
system (2).
Lemma 1. For each ε > 0 there exists a number δ = δ(ε) > 0 such that
for each µ = (x0 , ϕ, u) ∈ O1 × G × Ω1 satisfying the condition
ek ≤ δ,
|x0 − x
e0 | + kϕ − ϕk
e + ku − u
the system
Z
€
f t, x(τ1 (t)), . . . , x(τs (t)), u(θ1 (t)), . . .
S

. . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) dS,
x(t) = ϕ(t), t ∈ [τ, 0), x(0) = x0 ,
ẋ(t) =
(15)
(16)
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
€

€

ẋ(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
€
e(θ1 (t)), . . .
f t, x1 , . . . , xs , u
582
G. KHARATISHVILI AND T. TADUMADZE

...,u
e(t + s1 ), . . . , u
e(θν (t)), u
e(t + sk ) dS,
Z h
€
f t, x1 , . . . , xs , u(θ1 (t)), . . .
δf (t, x1 , . . . , xs ) =
S

€
. . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) − f t, x1 , . . . , xs , u
e(θ1 (t)), . . .
i
e(θν (t)), u
...,u
e(t + s1 ), . . . , u
e(t + sk ) dS.
Let M0 ⊂ O2 be the closure of the set u
e([θ, T0 ]). Then there exists a
number δ > 0 such that the functions u ∈ Vδ = {u ∈ Ω1 | ku − u
ek ≤ δ}
for almost all t ∈ [θ, T0 ] take values from the compact M1 ⊂ O2 containing
some neighborhood of the compact M0 .
Let K0 ⊂ O1 be a compact containing some neighborhood of the set
{e
x(t)|t ∈ [τ, Te]}.
We shall estimate δf (t, x1 , . . . , xs ) for each (x1 , . . . , xs ) ∈ K0s , u ∈ Vδ . It
is not difficult to notice that
Z nZ 1Œ
Œd €
e(θ1 (t)) + ξδu(θ1 (t)), . . .
|δf | ≤
Πf t, x1 , . . . , xs , u
dξ
0
S
e(t + s1 ) + ξδu(t + s1 ), . . .
...,u
e(θν (t)) + ξδu(θν (t)), u
ŒŒ o
e(t + sk ) + ξδu(t + sk ) Œdξ dS,
...,u
where δu(t) = u(t) − u
e(t).
e(t) + ξδu(t) ∈ M1 for almost all
It is clear that u
e + ξδu ∈ Vδ and hence, u
t ∈ [θ, T0 ].
Taking (1) into consideration, we obtain
Z nZ 1 hX
ν
Œ €
Œfui t, x1 , . . . , xs , u
|δf | ≤
e(θ1 (t))+ξδu(θ1 (t)), . . .
S
+
0
j=1
e(t + s1 ) + ξδu(t + ξ1 ), . . .
e(θν (t)) + ξδu(θν (t)), u
...,u
Œ
...,u
e(t + sk ) + ξδu(t + ξk ) Œ|δu(θj (t))| +
k
X
Œ
€
Œfuν+j t, x1 , . . . , xs , u
e(θ1 (t)) + ξδu(θ1 (t)), . . . , u
e(θν (t)) +
j=1
Œ
e(t + s1 ) + ξδu(t + s1 ), . . . , u
e(t + sk ) + ξδu(t + sk ) Œ ×
+ξδu(θν (t)), u
i o
×|δu(t + sj )| dξ dS ≤ (s1 · · · sk )mk0 ,M0 (t)ku − u
ek.
Thus for each (x1 , . . . , xs ) ∈ K0s , u ∈ Vδ , t0 , t00 ∈ I we have
ΠZ t00
Œ
Œ
Œ
δf (t, x1 , . . . , xs )dtŒ ≤ c0 δ.
Œ
t0
(17)
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
Now it is not difficult to see that
Z n
s
Œo
Œ
Œ
ΠX
Œδfxj Œ dt ≤ c1 = const.
Œδf (t, x1 , . . . , xs )Œ +
I
583
(18)
j=1
The inequalities (17) and (18) allow us to conclude that Lemma 1 implies
Theorem 2.8 (see [1], p.52).
ˆ
Now let us introduce the set V1 = δµ = (δx0 , δϕ, δu)|δx0 ‰∈ O1 − x
e0 ,
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; µ
e + εδµ),
µ
e = (e
x0 , ϕ,
e u
e), is defined on [0, Te + δ0 ], and
lim x(t; εδµ) = x(t; µ
e) is uniform over (t, δµ) ∈ [0, Te + δ0 ] × V1 .
ε→0
Moreover
e(t) + εδx(t; δµ) + R(t; εδµ), t ∈ [0, Te + δ0 ],
x(t; εδµ) = x
where x
e),
e(t) = x(t; µ
δx(t; δµ) = Y (0; t)δx0 +
s Z
X
j=1
×γ̇j (ξ)δϕ(ξ)dξ +
+
k Z
X
j=1
S
Z
t
0
τj (0)
Y (ξ; t)
0
(19)
Y (γj (ξ); t)fexj (γj (ξ)) ×
ν
nX
j=1
feuj (ξ)δu(θj (ξ)) +
o
feuν+j (ξ, s1 , . . . , sk )δu(ξ + sj )dS dξ,
(20)
lim ε−1 R(t; εδµ) = 0 is uniform over (t, δµ) ∈ [0, Te + δ0 ] × V1 . (21)
ε→0
Here Y (ξ; t), ξ, t ∈ I is a matrix function satisfying both the matrix equation
s
∂Y (ξ; t) X
+
Y (γj (ξ); t)fexj (γj (ξ))γ̇j (ξ) = 0, 0 ≤ ξ ≤ t,
∂ξ
j=1
and the condition
Y (ξ; t) =
(
E, ξ = t,
0, ξ > t.
584
G. KHARATISHVILI AND T. TADUMADZE
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) − x
e(t), where x(t) =
x(t; εδµ), satisfies the system
Z
€
d
e(θ1 (t)) +
f t, x(τ1 (t), . . . , x(τs (t), u
∆x(t) =
dt
S
+εδu(θ1 (t)), . . . , u
e(θν (t)) + εδu(θν (t)), u
e(t + s1 ) +

+εδu(t + s1 ), . . . , u
e(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
ν
s
X
X
d
fexj (t)∆x(τj (t)) + ε
feuj (t)δu(θj (t)) +
∆x(t) =
dt
j=1
j=1
+ε
k Z
X
j=1
where
Γ(t; εδµ) =
fuν+j (t, s1 , . . . , sk )δu(t + sj )dS + Γ(t; εδµ),
(23)
S
Z h
€
f t, x(τ1 (t)), . . . , x(τs (t)), u
e(θ1 (t)) + εδu(θ1 (t)), . . .
S
e(t + sk )) +
e(t + s1 ) + εδu(t + s1 ), . . . , u
...,u
e(θν (t)) + εδu(θν (t)), u
s
ν
X
X

+εδu(t + sk ) − fe(t) −
fexj (t)∆x(τj (t)) − ε
fuj (t) ×
j=1
×δu(θj (t)) − ε
k
X
j=1
j=1
i
feuν+j (t, s1 , . . . , sk )δu(t + sj ) dS.
By means of the Cauchy formula the solution of the system (23) can be
represented in the form
s Z 0
n
X
Y (γj (ξ); t)fexj (γj (ξ)) ×
∆x(t) = ε Y (0; t)δx0 +
j=1
×γ̇j (ξ)δϕ(ξ)dξ +
+
k Z
X
j=1
S
Z
0
τj (0)
t
Y (ξ; t)
ν
hX
j=1
feuj (ξ)δu(θj (ξ)) +
i o
feuν+j (ξ; s1 , . . . , sk )δu(ξ + sj )dS dξ +
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
+
Z
585
t
Y (ξ; t)Γ(ξ)dξ = εδx(t; εδµ) + R(t; εδµ).
0
Now we shall estimate R. We have
|R(t; εδµ)| ≤ kY k
Z Te+δ0 n Z h Z
Z Te+δ0
0
|Γ(t; εδµ)|dt ≤
Œd €
Œ
e(τ1 (t)) + ξ∆x(τ1 (t)), . . .
Πf t, x
dξ
S
0
0
e(τs (t)) + ξ∆x(τs (t)), u
...,x
e(θ1 (t)) + εξδu(θ1 (t)), . . .
e(t + s1 ) + εξδu(t + s1 ), . . .
...,u
e(θν (t)) + εξδu(θν (t)), u
≤ kY k
1
s
 X
...,u
e(t + sk ) + εξδu(t + sk ) −
fexj (t)∆x(τj (t)) −
j=1
−ε
ν
X
j=1
where
feuj (t)δu(θj (t)) +
k
X
j=1
‘Œ i o
Œ
feuν+j (t, s1 , . . . , sk )δu(t + sj ) Œdξ dS dt,
∆x(t) = x(t) − x
e(t),
kY k =
max
e+δ0 ]
ξ,t∈[0,T
|Y (ξ; t)|.
Differentiating the integrand with respect to ξ and grouping corresponding terms we get
|R(t; εδµ)| ≤ εσ1 (εδµ) + k∆xkσ2 (εδµ)
with
k∆xk =
σ1 (εδµ) = ckY k
Z Te+δ0 n Z h Z
0
S
max
e+δ0
0≤t≤T
1
0
|∆x(t)|,
ν Œ
hX
Œ €
e(τ1 (t)) + ξ∆x(τ1 (t)), . . .
Œfuj t, x
j=1
e(θ1 (t)) + εξδu(θ1 (t)), . . . , u
...,x
e(τs (t)) + ξ∆x(τs (t)), u
e(θν (t)) +
+εξδu(θν (t)), u
e(t + s1 ) + εξδu(t + s1 ), . . . , u
e(t + sk ) + εξδu(t + sk )) −
k Œ
Œ
Œ XŒe
−feuj (t)Œ +
e(τ1 (t)) + ξ∆x(τ1 (t)), . . . , x
e(τs (t)) +
Œfuν+j (t, x
j=1
+ξ∆x(τs (t)), u
e(θ1 (t)) + εξδu(θ1 (t)), . . . , u
e(θν (t)) + εξδu(θν (t)), u
e(t + s1 ) +
Œi i o
Œ
+εξδu(t + s1 ), . . . , u
e(t + sk ) + εξδu(t + sk )) − feuν+j (t)Œ dξ dS dt,
586
G. KHARATISHVILI AND T. TADUMADZE
σ2 (εδµ) = kY k
Z Te+δ0 n Z h Z
0
S
1
s Œ
X
Œ €
e(τ1 (t)) +
Œfxj t, x
0 j=1
e(θ1 (t)) + εξδu(θ1 (t)), . . .
+ξ∆x(τ1 (t)), . . . , x
e(τs (t)) + ξ∆x(τs (t)), u
...,u
e(θν (t)) + εξδu(θν (t)), u
e(t + s1 ) + εξδu(t + s1 ), . . .
Πi o

Œ
...,u
e(t + sk ) + εξδu(t + sk ) − fexj (t)Œdξ dS dt.
It is not difficult to see that the convergence limε→0 σi (εδµ) = 0, i = 1, 2,
is uniform for δµ ∈ V1 .
Now,
k∆xk ≤ εkδx(δµ)k + kR(εδµ)k ≤
€

≤ ε kδx(δµ)k + σ1 (εδµ) + σ2 (εδµ)k∆xk.
Here
kδx(δµ)k =
Hence
max
e+δ0
0≤t≤T
|δx(t; δµ)|, kR(εδµ)k =
max
e+δ0
0≤t≤T
|R(t; εδµ)|.
€

ε kδx(δµ)k + σ1 (εδµ)
= εα(εδµ),
k∆xk ≤
1 − σ2 (εδµ)
where the value α(εδµ) is bounded when ε → 0 uniformly for δµ ∈ V1 .
Thus
kR(εδµ)k/ε ≤ σ1 (εδµ) + α(εδµ)σ2 (εδµ),
whence follows (21).
Let D0 be a set of points ζ = (T, x0 , ϕ, u) ∈ I × O1 × G × Ω1 defined by
the following condition: the system
Z
€
ẋ(t) =
f t, x(τ1 (t)), . . . , x(τs (t)), u(θ1 (t)), . . .
S

. . . , u(θν (t)), u(t + s1 ), . . . , u(t + sk ) dS,
x(t) = ϕ(t), t ∈ [τ, 0), x(0) = x0 ,
has the solution x(t) = x(t; ζ) which is defined on [0, T ].
Thus the mapping
L : D0 → R1+n
(24)
is given on D0 by the formula L(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∞ .
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
587
Lemma 3. The mapping (24) is differentiable at the point ζe =
e
(T , x
e0 , ϕ,
e). Namely,
e u
(δζ) = (δT, δx0 , δx1 ),
dLe
ζ
e
where δx1 = fe(Te)δT + δx(Te; δµ), δζ = (δT, δµ) ∈ Eζ − ζ.
(25)
Proof. Let V = {δζ = (δT, δµ)| |δT | ≤ c, δµ ∈ V1 }. With small enough
ε > 0 we have εδT ≤ δ0 for each δζ ∈ V . Therefore, using the representation
(19) and the fact that t = Te is the Lebesgue point of the function fe(t), we
have
x(Te + εδT ; εδµ) = x
e(Te + εδT ) + εδx(Te + εδT ; δµ) +
Z Te+εδT
e
e
fe(t)dt + εδx(Te; δµ) +
+R(T + εδT ; εδµ) = x
e(T ) +
e
T

€
+ε δx(Te + εδT ; δµ) − δx(Te; δµ) + R(Te + εδT ; δµ) =
=x
e(Te) + εδx1 + o1 (εδζ),
where limε→0 o1 (εδζ)/ε = 0 uniformly for δζ ∈ V .
Therefore
€
e = Te + εδT, x
e0 + εδx0 , x(Te + εδT )) −
L(ζe + εδζ) − L(ζ)

−(Te, x
e0 , x
e(Te) = (δT, δx0 , δx1 ) · ε + o(εδζ) = dLe
(δζ)ε + o(εδζ),
ζ
€

where o(εδζ) = 0, 0, o1 (εδζ) .
2.2. Deduction of the maximum principle. Consider the linear topological
space Ez = R1 × Eζ of the points z = (σ, ζ).
Let D be a subset of this space,
ˆ
‰
D = z|z = (σ, ζ)|σ ∈ R1 , ζ ∈ D0 .
The set D is open (the more so, finitely open) (see [2,3]). Define on D
the mapping P : D → R1+l by the formula
€

P (z) = Q(L(ζ)) + (σ, 0, . . . , 0) = q o (L(ζ)), . . . , q l (L(ζ)) + (σ, 0, . . . , 0).
e0 , ϕ,
e u
e) has the
The differential of the mapping P at the point ζe = (0, Te, x
form
e
e
e
∂Q
∂Q
∂Q
+
=
δT
+
δx
δx1 ,
dPe
(δz)
σ
e
+
0
z
∂t
∂x0
∂x1
σ
e = (δσ, 0, . . . , 0), δz ∈ Ez − ze.
588
G. KHARATISHVILI AND T. TADUMADZE
Using the expressions (20) and (25), we obtain
dPe
(δz) = σ
e+
z
 ∂Q
e
+
‘
‘
 ∂Q
e
e
e
∂Q
∂Q
fe(Te) δT +
+
Y (0; Te) δx0 +
∂x1
∂x0
∂x1
∂t
Z
s
X
e
∂Q 0
+
Y (γj (t); Te)fexj (γj (t))γ̇j (t)δϕ(t)dt +
∂x
1
τ
(0)
j
j=1
+
+
e
∂Q
∂x1
k Z
X
j=1
Z Te
0
S
Y (t; Te)
ν
hX
j=1
feuj (t)δu(Qj (t)) +
i
feuν+j (t; s1 , . . . , sk )δu(t + sj )dS dt.
In the space Ez let us define a filter by the following elements:
1
∩ V0 ) × VTe × Ve
We
= (R+
× (G ∩ Vϕe) × (Ω1 ∩ Ve
),
x0
z
u
where V0 , VTe, Ve
, Vϕe, Ve
are arbitrary convex neighborhoods of the points
u
x0
1 e
0 ∈ R , T ∈ (0, T0 ), x
e0 ∈ O1 , ϕ
e ∈ G, u
e ∈ 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 mapping P
is continuous on Φ (see [2,3]).
The optimality of the element ζe 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 = (π0 , . . . , πl ) and an element
f = (R1 ∩ Ve0 ) × Ve × Ve × (G ∩ Ve ) × (Ω1 ∩ Ve )
W
+
ϕ
e
e
e
e
z
x0
e
u
T
of the filter Φ such that
f − ze),
(δz) ≤ 0, ∀ δz ∈ K(W
πdPe
e
z
z
(26)
where K(W ) is a cone stretched on the set W . It is easily seen that the
f − ze) is equivalent to the condition δσ ∈ R1 , δT ∈
condition δz ∈ K(W
+
z
e
e
e.
e ⊃ G − ϕ,
−
u
)
⊃
Ω1 − u
R1 − Te, δx0 ∈ Rn − x
f0 , δϕ ∈ K(Veϕe − ϕ)
e δu ∈ K(Vee
u
Using the expression for the differential dPe
(δz), assuming that δu = 0,
z
1
, δT , δx0 may take
δϕ = 0, and taking into consideration that δσ ∈ R+
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
arbitrary values, we obtain for π0 ≤ 0
 
‘
e + ∂Q
e fe(Te) = 0,
π ∂ Q
∂x1
 ∂t
‘
e + ∂Q
e Y (0; Te) = 0.
π ∂ Q
∂x0
589
(27)
∂x1
Assuming that in (26) δt1 = 0, δx0 = 0, δu = 0, δσ = 0 we obtain
s
X
e Z 0
∂Q
e (28)
Y (γj (t); Te)fexj (γj (t))γ̇j (t)δϕ(t)dt ≤ 0, δϕ ∈ G − ϕ.
π
∂x1 τj (0)
j=1
If we assume δt1 = 0, δx0 = δϕ = 0, δσ = 0 we shall get
e
∂Q
π
∂x1
+
Z X
k
S j=1
Z Te
0
Y (t; Te)
ν
hX
j=1
feuj (t)δu(θj (t)) +
i
feuν+j (t; s1 , . . . , sk )δu(t + sj )dS dt ≤ 0.
(29)
Let us introduce the notation
ψ(t) = π
e
∂Q
Y (t; Te), t ∈ [0, Te].
∂x1
(30)
It is clear that ψ(t) satisfies the system (6) and the condition
ψ(Te) = π
e
∂Q
.
∂x1
We can now rewrite the condition (27) in the form
(
e = −ψ(Te)fe(Te),
π ∂∂tQ
e = −ψ(0).
π ∂Q
(31)
(32)
∂x0
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),
e ζ ∈ ∆2
(12) be fulfilled. Introducing the notation ∆x(t) = x(t; ζ) − x(x; ζ),
and taking into account (10), (13), and (14), we obtain
Z T0
d
0 = ψ(T0 )∆x(t0 ) − ψ(0)∆x(0) =
(ψ(t)∆x(t))dt =
dt
0
Z T0
ƒ
‚
d
ψ̇(t)∆x(t) + ψ(t) ∆x(t) dt =
=
dt
0
590
=
G. KHARATISHVILI AND T. TADUMADZE
Z
0
s
T0nX
j=1
s
X
ψ0 (γj (t))e
gxj (γj (t))γ̇j (t)∆x(t)− ψ(γj (t))Aj (γj (t))γ̇j (t)∆x(t)+
j=1
+ψ(t)
s
X
j=1
ƒo
‚
e(·)) dt.
Aj (t)∆x(τj (t)) + ψ(t) f (t, u(·)) − f (t, u
Further, performing elementary transformations, in view of (14) we get
0 = ψ(T0 )∆x(t0 ) − ψ(0)∆x(0) =
−
+
s Z
X
+
T0
0
+
s Z
X
j=1
gexj (γj (t))γ̇j (t)∆x(t)dt −
0
τj (0)
ψ(γj (t)) × Aj (γj (t))γ̇j (t)∆x(t)dt +
s
X
‚
ƒ
e(·)) dt =
ψ(t) f (t, u(·)) − f (t, u
j=1
0
ψ(γj (t))Aj (γj (t))γ̇j (t)∆x(t)dt +
j=1 τj (0)
s Z 0
X
−
j=1
τj (T0 )
ψ(γj (t))Aj (γj (t))γ̇j (t)∆x(t)dt +
j=1 0
s
X Z τj (T0 )
j=1
Z
τj (T0 )
s Z
X
τj (0)
=
T0
γj (0)
s Z
X
j=1
gexj (γj (t))γ̇j (t)∆x(t)dt +
s Z T0
X
j=1
Z
0
gexj (t)∆x(τj (t))dt +
Z
i
−e
gxj (γj (t)) γ̇j (t)∆x(t)dt +
0
Further,
Z
Z
T0
0
T0
γj (0)
gexj (t)∆x(τj (t))dt −
0
ƒ
‚
ψ(t) f (t, u(·)) − f (t, u
e(·)) dt =
s Z 0
X
j=1
gexj (t)∆x(τj (t))dt +
τj (0)
h
ψ(γj (t))Aj (γj (t)) −
‚
ƒ
ψ(t) f (t, u(·)) − f (t, u
e(·)) dt.
h €

€
g t, x
e(τ1 (t)), . . . , x
e(τs (t)) − g t, x(τ1 (t)), . . .
0
i

. . . , x(τs (t)) + f 0 (t, u
e(·)) − f 0 (t, u(·)) dt + ψ(T0 )∆x(T0 ) −
Z T0 h
€
€

−ψ(0)∆x(0) =
g t, x
e(τ1 (t)), . . . , x
e(τs (t)) − g t, x(τ1 (t)), . . .
e − I(ζ) =
I(ζ)
T0
0

. . . , x(τs (t)) +
s
X
j=1
s Z
i
X
gexj (t)∆x(τj (t)) dt +
j=1
0
τj (0)
h
− gexj (γj (t)) +
THE MAXIMUM PRINCIPLE IN OPTIMAL CONTROL PROBLEMS
Z
i
+ψ(γj (t))Aj (γj (t)) γ̇j (t)(ϕ(t) − ϕ(t))dt
e
+
T0
591
n
− f 0 (t, u(·)) +
‚
ƒo
+ψ(t)f (t, u(·)) − − f 0 (t, u
e(·)) + ψ(t)f (t, u
(33)
e(·)) dt.
0
Due to the convexity of the function g, by virtue of the inequalities (11),
e − I(ζ) ≤ 0 for ζ ∈ ∆2 .
(12), it follows from (33) that I(ζ)
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