A Mathematical Journal
Vol. 7, No 2, (21-38). August 2005.

Optimization of differential inclusions of
Bolza type with state constraints and

duality

E. N. Mahmudov
Istanbul University, Faculty of Engineering,

Dep. of Industrial Engineering, Avcilar, İstanbul, Turkey
elimhan22@yahoo.com

G.Çiçek
Istanbul University, Faculty of Science,

Dep. of Mathematics, Vezneciler, İstanbul, Turkey
gulseren cicek@hotmail.com

ABSTRACT
Sufficient conditions for optimization are obtained and duality theorems are

also derived for the problems under consideration on the basis of the apparatus
of locally conjugate mappings and the subdifferential calculus.

RESUMEN

Se obtienen condiciones suficientes para optimización y se derivan teoremas
de dualidad para los problemas, bajo consideraciones en las bases de los aparatos
de funciones conjugadas localmente y del cálculo subdiferencial.

Key words and phrases: Multi-valued, conjugate mappings, subdifferential,
sufficient conditions, conjugate function, duality

Math. Subj. Class.: Primary: 49K20, 35K99, 49K24, Secondary: 54C60



22 E. N. Mahmudov and G. Çiçek
7, 2(2005)

1 Introduction

The present article is devoted to an investigation of the problem of Bolza type
differential inclusions with state constraints:

I(x(.), t1) = ϕ(x(t1), t1) +
∫ t1

t0

g(x(t), t)dt → inf, (1)
ẋ(t) ∈ a(x(t), t), t ∈ [to, t1], (2)

x(t0) = x0,x(t1) ∈ M, (3)
x(t) ∈ F(t), t ∈ [t0, t1], (4)

where a is a bounded non-autonomous convex multi-valued mapping [1],
a(., t) : R

n → 2Rn , the target set M ⊂ Rn is a convex set of final states, g is a
convex function, g,ϕ : R

n+1 → R1 and F : [t0, t1] → 2R
n

is a convex-valued function.
The initial moment of the time t0 is fixed, and the last moment t1 is generally free.
An admissible solution x(t) of the differential inclusion (2) with boundary conditions
(3) is an absolutely continuous function (x(t) ∈ F(t) for all t ∈ [t0, t1]).

In our optimization problem we use the apparatus of locally conjugate mapping
and we observe that relationship between locally conjugate mapping and conjugate
function is useful for detailed investigations.

In Section 2., using locally conjugate mapping[1], we formulate sufficient condi-
tions of optimality. In addition we show that conjugate variable has jumps,which
are typical for control systems with state constraints and among sufficient conditions
there appears a condition of jumps(see [3]), where the number of jump points may be
countable.

In Section 3., we prove the theorem of duality for convex problems, and we show
that conjugate differential inclusion play the role of extremal relation for a direct and
dual problem. For construction of the dual problem, the convex continuous problem
is interchanged with the discrete approximation problem and results from [8] are used.

Former investigations[1,13-15,16] have made an intensive development of the the-
ory of extremal problems described by multivalued mappings with discrete time and
with lumped parameters. Many problems in economic dynamics, as well as classical
problems on optimal control, differential games, and so on, can be reduced to such
investigations.
The papers [11-12] are a survey of optimality conditions for optimal control problems
involving differential inclusions and so-called differential-difference inclusions. The
papers[11,17-18,23] establish necessary conditions for optimal control problems with
state constraints, formulated in terms of differential inclusions.
Definition 1.1: 1) h(x̄,x) is called the upper convex approximation(UCA) of a
function g(x) at a point x ∈ domg = {x : |g(x)| < +∞} [1] if:

i) h(x̄,x) ≥ F(x̄,x) = sup
τ (.)

lim sup
λ↓0

1
λ
(g(x + λx̄ + τ(λ)) −g(x))

λ−1τ(λ) → 0, λ ↓ 0 for all x̄ �= 0.



7, 2(2005)
Optimization of differential inclusions ... 23

ii) h(x̄,x) is a convex closed (lower semicontinuous) positive homogeneous func-
tion of x̄.

2) The set ∂h(0,x) = {x∗ ∈ Rn : h(x̄,x) ≥< x̄,x∗ >,x̄ ∈ Rn}, is called a subdif-
ferential of the function g at the point x and is denoted by ∂g(x), there symbol < .,. >
denotes scalar product. It is known that when g(x) is convex, the given definition co-
incides with the usual definition of the subdifferential.(see [1])

3) The mapping a∗(y∗; z) = {x∗ : (−x∗,y∗) ∈ K∗a(z)} is called a locally conjugate
mapping (LCM) to the convex mapping a at the point z.
Theorem 1.1: Let a : R

n → 2Rn be convex-valued closed bounded continuous map-
ping such that the function Wa(x,y∗) = inf

y∈a(x)
< y,y∗ > is continuous differentiable

on x. Let us suppose that the vector z̄1 = (x̄1, ȳ1) satisfies the inequality

< x̄1,
∂Wa(x0,y∗)

∂x
> − < ȳ1,y∗ >< 0.

Then the following statements are true for a point z0 = (x0,y0),y0 ∈ a(x0,y∗):
i)The cone

Ka(z0) =
{
z̄ :< x̄,

∂Wa(x0,y∗)
∂x

> − < ȳ,y∗ >< 0
}

is the smooth local tent, which is the cone of tangent directions to gfa(graph of a) at
the point z0.

ii) LCM a∗ corresponding to the cone Ka(z0) may be given by the formula

a∗(y∗; z0) =
{
∂Wa(x0,y∗)

∂x

}

Proof. If Sa(x,y∗), y∗ ∈ Rn, is the support function to a(x) , then by the theory
of convex analysis it is known that y ∈ a(x) if and only if < y,y∗ >≤ Sa(x,y∗) for
all y∗ ∈ Rn. Since Sa(x,y∗) = −Wa(x,−y∗), the preceding inequality means that
< y,y∗ >≥ Wa(x,y∗). Thus a(x) is given by

a(x) = {y : Wa(x,y∗)− < y,y∗ >≤ 0}, y∗ ∈ Rn. (5)

Suppose

fy∗ (z) = Wa(x,y
∗)− < y,y∗ >, (6)

then by the Lemma 3.1[1,p.225], fy∗ (z) is continuous on y∗ and is continuous dif-
ferentiable on z. By the Theorem 2.2[1,p.211], UCA(upper convex approximation)
hy∗ (z̄,z) of the function fy∗ (z) is

hy∗ (z̄,z) =< z̄,
∂Wa(x,y∗)

∂x
×{−y∗} > . (7)



24 E. N. Mahmudov and G. Çiçek
7, 2(2005)

Furthermore fy∗ (z0) = 0 and fy∗ (z) has an UCA hy∗ (z̄,z0), which is continuous on
z̄ and by the condition on the vector z̄1, we have hy∗ (z̄1,z0) < 0. Then applying
Theorem 3.3[1,p.234] by (7) we see that i) of the theorem follows. Since in this case

−con∂fy∗ (z0) = con
{
−∂Wa(x0,y

∗)
∂x

,y∗
}
,

then by the same Theorem 3.3[1,p.234] the equality

a∗(y∗; z0) =
{
∂Wa(x0,y∗)

∂x

}

holds. This, in turn, implies that ii) is correct. The Theorem is proved.

Let O+(gfa) be the recession cone[2] to a convex function a in the space
Z = X ×Y , i.e.

O+(gfa) = {z̄ : z + λz̄ ∈ gfa,λ ≥ 0,∀z ∈ gfa}. (8)
For such convex function a, let us define

Ωa(x
∗,y∗) = inf{− < x,x∗ > + < y,y∗ >: (x,y) ∈ gfa}. (9)

It is evident that

Ωa(x
∗,y∗) = inf

x
{− < x,x∗ > +Wa(x,y∗)}. (10)

Definition 1.2: The function

a∗(y∗) = {x∗ : (−x∗,y∗) ∈ (O+gfa)∗}
is called conjugate function to a convex function a. It is clear that if mapping a is
superlinear[5], i.e. gfa is a cone, then this definition coincides with the definition of
B.H.Pshenichnyi [1].

Conjugate function can be used in different problems connected with duality the-
orems.
Definition 1.3: Multivalued mapping a is called quasisuperlinear if its graph is in
the form of

gfa = M + K,

where M is a convex compactum, K is a closed convex cone.
Lemma 1.1: For a convex mapping a we have

domΩa = {(−x∗,y∗) : Ωa(x∗,y∗) > −∞}⊆ (O+gfa)∗.
If a is a quasisuperlinear mapping then

domΩa = K
∗.



7, 2(2005)
Optimization of differential inclusions ... 25

Proof. Let us assume the contrary: let (−x∗0,y∗0 ) ∈ domΩa, but (−x∗0,y∗0 ) �∈
(O+gfa)∗ . It means that there exists a pair (x̄0, ȳ0) ∈ O+gfa, for which

− < x∗0, x̄0 > + < y∗0, ȳ0 >< 0.
By the definition of O+gfa, we have

(x,y) + λ(x̄0, ȳ0) ∈ gfa, (x,y) ∈ gfa, λ > 0.
Then

− < x + λx̄0,x∗0 > + < y + λȳ0,y∗0 >= − < x∗0,x > + < y∗0,y > +
+λ{− < x̄0,x∗0 > + < ȳ0,y∗0 >}→−∞ for λ → +∞,

which contradicts the fact that (−x∗0,y∗0 ) ∈ domΩa. This proves the first statement
of the lemma. Furthermore, when a is a quasisuperlinear mapping, applying Result
9.1.2[2] and Lemma3.6.1[1],we get

(O+gfa)∗ = [O+(M + K)]∗ = (O+M)∗ ∩ (O+K)∗ = Rn ∩K∗ = K∗.
On the other hand

domΩa = dom(ΩM + ΩK ) = domΩM ∩domΩK = domΩK = K∗.
Hence

domΩa = K
∗.

Lemma is proved.

The following example shows that the inverse inclusion generally is not true. In
fact, let a : X → 2Y (X,Y one-dimensional axises) is given as:

a(x) = {y : y ≥ x2} , gfa = {(x,y) : y ≥ x2}.
Check that O+gfa = {0}×Y +, where Y + is the positive y-axis. Therefore (O+gfa)∗ =
{(−x∗,y∗) : x∗ ∈ X,y∗ ∈ Y +}. Then it is clear that (−x∗0,y∗0 ) ∈ (O+gfa)∗,
x∗0 = 1,y

∗
0 = 0, but (−x∗0,y∗0 ) �∈ domΩa.

Lemma 1.2: Let a be a quasisuperlinear mapping and Wa(.,y∗) be proper closed
function. Then the relation

sup
x∗∈a∗(y∗)

{< x,x∗ > +ΩM (x∗,y∗)} = inf
y∈a(x)

< y,y∗ >

holds.
Proof. From Lemma 1.1, we have

domΩa = (O
+gfa)∗ = K∗.



26 E. N. Mahmudov and G. Çiçek
7, 2(2005)

Therefore with regard to Theorem 4.1.III[1] we find the relation

sup
x∗
{Ωa(x∗,y∗)+ < x,x∗ >} =

sup
x∗
{< x,x∗ > +ΩM (x∗,y∗) : x∗ ∈ a∗(y∗)} = Wa(x,y∗).

Remark 1.2.1: If M = {0}, then ΩM = 0 and so the result of the above lemma
coincides with the result of the Theorem 4.5.III[1,p.129].
Lemma 1.3: Let a be a convex mapping. Then the point x0 is a solution of the
problem

inf
x
{− < x,x∗ > +Wa(x,y∗)}, x∗,y∗ ∈ Rn

if and only if

x∗ ∈ a∗(y∗,z0), y0 ∈ a(x0,y∗).
Proof. By the Theorem 2.1.IV[1], x0 is a minimum point of the convex function

− < x,x∗ > +Wa(x,y∗)
if and only if

0 ∈ ∂x[− < x0,x∗ > +Wa(x0,y∗)],
i.e.

x∗ ∈ ∂xWa(x0,y∗).
And, therefore by the definition of Ωa it is evident that y0 ∈ a(x0,y∗). Then by the
Theorem 2.1.III[1], we find the required result.
Theorem 1.2: Let a be a convex-valued closed bounded continuous mapping, satis-
fying the Lipschitz condition, and let the function Waz (x̄,y

∗) be closed, where

az (x̄) = {ȳ : (x̄, ȳ) ∈ Ka(z)}.

Then for arbitrary y ∈ a(x,y∗),z = (x,y) ∈ gfa, the function Waz (.,y∗) is an UCA
for Wa(.,y∗) and, besides,

a∗(y∗; z) = ∂xWa(x,y
∗).

Proof. If z̄ = (x̄, ȳ) ∈ Ka(z),z = (x,y),y ∈ a(x), then by the definition of the cone of
tangent directions, there is a function τ(λ), λ−1τ(λ) → 0, λ ↓ 0 (τ(λ) ∈ Z = X ×Y )
such that z + λz̄ + τ(λ) ∈ gfa for a sufficiently small λ ≥ 0. That means

y + λȳ + τy(λ) ∈ a(x + λx̄ + τx(λ)),τ = (τx,τy),τx(λ) ∈ X,τy(λ) ∈ Y.



7, 2(2005)
Optimization of differential inclusions ... 27

Since a satisfies the Lipschitz condition, Wa(x,y∗) also satisfies the same condition
by Lemma 3.2.V[1,p.226]. For such functions we have

F(x̄,x) = lim sup
λ↓0

1
λ

(Wa(x + λx̄,y
∗) −Wa(x,y∗)).

It is easily shown that

F(x̄,x) = lim sup
λ↓0

1
λ

(Wa(x + λx̄ + τx(λ),y
∗) −Wa(x,y∗))

holds independently from the choice of τ(λ). From the definition of Wa(x,y∗) and
from the condition y ∈ a(x,y∗) it follows that
1
λ

(Wa(x + λx̄ + τx(λ),y
∗) −Wa(x,y∗)) ≤

1
λ

(< y + λȳ + τy(λ),y
∗ > − < y,y∗ >) =

< ȳ,y∗ > + <
τy (λ)
λ

,y∗ > .

Then we have

F(x̄,x) = lim sup
λ↓0

1
λ

(Wa(x + λx̄ + τx(λ),y
∗) −Wa(x,y∗))

≤ limsupλ↓0[< ȳ,y∗ > + < λ−1τy (λ),y∗ >] =< ȳ,y∗ > .
It means that

F(x̄,x) ≤ inf
ȳ
{< ȳ,y∗ >: ȳ ∈ az(x̄)}.

In addition, given x̄ �∈ domaz let us put Waz (x̄,y∗) = +∞. Then, by applying
Lemma 1.2 to az, we get

Waz (x̄,y
∗) = sup

x∗
{< x̄,x∗ >: x∗ ∈ a∗z(y∗)}.

But on the other hand, by the definition, a∗(y∗; z) = a∗z(y
∗). Hence

F(x̄,x) ≤ Waz (x̄,y∗) = sup
x∗
{< x̄,x∗ >: x∗ ∈ a∗(y∗; z)},

where Waz (x̄,y
∗) is positive homogenous convex closed function of x̄, i.e. Waz (x̄,y

∗)
is an UCA function of Wa(.,y∗) at the point x. Now to conclude the proof, it remains
only to apply Theorem 3.2.II[1], thus we find

∂Wa(x,y
∗) = ∂h(0,x) = a∗(y∗; z).

Let us investigate the relation between conjugate function and LCM(Locally Con-
jugate Mapping). We need the following two theorems.

Let KM (z) be the cone of tangent directions to a convex set M ⊆ Z = X ×Y at



28 E. N. Mahmudov and G. Çiçek
7, 2(2005)

a point z ∈ M, i.e.

KM (z) = con(M −z) = {z̄ : z̄ = λ(z1 −z),λ > 0,z1 ∈ M}. (11)
Theorem 1.3: Let O+M be the recession cone of a convex closed set M ⊂ Z. Then
we have ⋂

z∈M
KM (z) = O

+M.

Proof. Let us show that

M =
⋂

z∈M
(z + KM (z)). (12)

In fact, let z0 ∈ M be an arbitrary fixed point. It is evident that all vectors as
z̄ = z0 − z (in definition (11) they corresponds to λ = 1) belong to the cone KM (z),
i.e. z0 ∈ z + KM (z),z ∈ M, then z0 ∈

⋂
z∈M

(z + KM (z)). Conversely, if we have the

last inclusion then z0 ∈ z + KM (z) or there are such z1 ∈ M and a number γ > 0,
that z0 − z = γ(z1 − z) ∈ KM (z). Hence z0 = γz1 + (1 − γ)z ∈ M. Formula (12)
follows.

On the other hand, we easily show that

O+[
⋂

z∈M
(z + KM (z))] =

⋂
z∈M

[O+(z + KM (z))].

In fact if z is an arbitrary point of closed convex set M =
⋂

z∈M
(z + KM (z)) then by

the definition of the recession cone, it is evident that, directed ray z + λz̄, ∀λ ≥ 0, is
contained in any cone z + KM (z),z ∈ M. But it means that

z̄ ∈
⋂

z∈M
[O+(z + KM (z))].

Therefore

O+M = O+[
⋂

z∈M
(z + KM (z))] =

⋂
z∈M

[O+(z + KM (z))] =
⋂

z∈M
KM (z).

Theorem is proved.
Remark 1.3.1: In the statement of the above Theorem, the closedness of M is
essential.
Proof. Actually, let M = {(x,y) : x > 0,y > 0}∪{(0, 0)}⊂ R2. Clearly, O+M = M.
The set M contains points (x0,y0) + λ(0,y0), where x0 > 0, y0 > 0 are fixed. But
(0,y0) �∈ O+M.
Theorem 1.4: Let M be a closed convex set and let K∗M (z) be the conjugate cone
to the cone of tangent directions KM (z),z ∈ M. Then⋃

z∈M
K∗M (z) = (O

+M)∗,



7, 2(2005)
Optimization of differential inclusions ... 29

where the bar denotes closure.
Proof. It is sufficient to show that

⋃
z∈M

K∗M (z) = (
⋂

z∈M
KM (z))

∗. (13)

Get any fixed point z∗0 ∈
⋃

z0∈M
K∗M (z). Then there exists a sequence z

∗
n → z∗0 , z∗n ∈⋃

z∈M
K∗M (z). Let us define sequence {zn} by the relation z∗n ∈ K∗M (zn). Note that

z∗n ∈
⋃

z∈M
K∗M (z) implies the existence of zn ∈ M such that z∗n ∈ K∗M (zn).

On the other hand, since KM (zn) ⊇
⋂

z∈M
KM (z) it is evident that K∗M (zn) ⊆

(
⋂

z∈M
KM (z))∗. So that z∗n ∈ (

⋂
z∈M

KM (z))∗, and therefore z∗0 ∈ (
⋂

z∈M
KM (z))∗.

Let us prove the converse inclusion in (13). Let us z∗1 ∈ (
⋂

z∈M
KM (z))∗ be arbitrary

fixed point and let us assume the contrary i.e. let z∗1 �∈
⋃

z∈M
K∗M (z). Then z

∗
1 �∈ K∗M (z)

for any z ∈ M. In other words, there exists a vector z̄1(z̄1 �= 0) such that

< z∗1, z̄1 >< 0, z̄1 ∈ KM (z), ∀z ∈ M

or

< z∗1, z̄1 >< 0, z̄1 ∈
⋂

z∈M
KM (z), i.e. z

∗
1 �∈ (

⋂
z∈M

KM (z))
∗.

This contradiction shows that

(
⋂

z∈M
KM (z))

∗ ⊆
⋃

z∈M
K∗M (z).

The proof of the theorem is over now.
Theorem 1.5: Let a be a closed convex mapping. Then the conjugate function
a∗(y∗) and the LCM of a implies the following relation

a∗(y∗) =
⋃

z∈gf a
a∗(y∗; z), y ∈ a(x,y∗).

Proof. Setting M = gfa as in the previous theorem, we obtain

a∗(y∗) =
⋃

z∈gf a
a∗(y∗; z).

By the Theorem 2.1.III[1] z = (x,y), y �∈ a(x,y∗), implies a∗(y∗; z) = ∅.



30 E. N. Mahmudov and G. Çiçek
7, 2(2005)

2 Sufficient conditions of the optimization.

According to [1], the LCM (Locally Conjugate Mapping) a∗ of the multi-valued map-
ping a at a point z = (x,y) ∈ gfa(., t), t ∈ [t0, t1], is defined as follows:

a∗(y∗, (x,y), t) = {x∗ : (−x∗,y∗) ∈ K∗a (z,t)},y∗ ∈ R
n
,

where K∗a(z,t) is the conjugate cone to the cone of tangent directions Ka(z,t).
Let us define

Wa(x,y∗, t) =
{

inf{< y,y∗ >: y ∈ a(x,t)}, a(x,t) �= ∅
+∞ a(x,t) = ∅,

a(x,y∗, t) = {y ∈ a(x,t) :< y,y∗ >= Wa(x,y∗, t)} and
WM (x∗) = inf

y∈M
< x∗,y > .

Note that for a convex mapping a, the LCM coincides with the subdifferential[1]
∂xWa(x̃,y∗, t) of the function Wa(.,y∗, t) at the point x̃.It is known that

a∗(y∗, (x̃, ỹ), t) =
{
∂xWa(x̃,y∗, t), ỹ ∈ a(x̃,y∗, t)

∅, ỹ �∈ a(x̃,y∗, t).
Let x̃(t), t ∈ [t0, t1], x̃(t0) = x0, be any admissible solution of the problem (1)-(4).

Let us construct the conjugate differential inclusion of the conjugate variable x∗(t) by

a) −ẋ∗(t) ∈ a∗(x∗(t); (x̃(t), ˙̃x(t)), t) + ∂g(x̃, t), t ∈ [t0, t1], a.e;
˙̃x(t) ∈ a(x̃(t),x∗(t), t), t ∈ [t0, t1] a.e;

which should be fulfilled for all x ∈ F(t). The solution x∗(t), t ∈ [t0, t1], satisfies the
conjugate differential inclusion a) almost everywhere and is in the form of the sum of
absolutely continuous functions and jump functions. Let us denote points of jumps
and values of jumps x∗(t) by

τi(i = 1, 2, . . .), t0 < τi < t1,
x∗i = x

∗(τi + 0) −x∗(τi − 0) (i = 1, 2, . . .),
respectively.

If the following condition

− < x∗(t), x̃(t) >< WM∩F (t)(−x∗(t)), t0 ≤ t < t1
holds, the admissible trajectory x̃(t) would be called strictly transversal on the set
M. Note that this definition guarantees that point x̃(t) /∈ M for every t ∈ [t0, t1).

If the inequality

I(x(.),θ) < I(x(.),θ
′
)

holds for any θ,θ
′ ∈ [t0, t1] with θ < θ

′
and for any admissible trajectory of the dif-

ferential inclusion (2) with initial condition x(t0) = x0, then the function I(x(.), t) is
called monotone increasing with respect to argument t.
Theorem 2.1: Let x̃(t), t ∈ [t0, t1], be any admissible trajectory of the problem



7, 2(2005)
Optimization of differential inclusions ... 31

(1)-(4) and let there exists absolutely continuous function x∗(t) which satisfies the in-
clusion a). Furthermore assume that I(x(.), t) is monotone increasing with respect to
argument t for any admissible trajectory x(t), t ∈ [t0, t1], of the differential inclusion
(2) and the following conditions are satisfied:

1) x∗(t1) ∈ ∂ϕ(x̃(t1), t1),x∗(t1) ∈ K∗M (x̃(t1));
2) the jumps x∗i satisfy < x̃(τi),x

∗
i >= WF (τi)(x

∗
i );

3) x̃(t) is strictly transversal on M.
Then trajectory x̃(t) is optimal.

Proof. Let x(t) ∈ F(t) be an arbitrary admissible trajectory, realising the transition
from the interval [t0,θ] to the set M. Let us show that

I(x(.),θ) ≥ I(x̃(.), t1).
Using ∂xWa(.,x∗(t), t) as the representation of LCM and by the Moreau-Rockafellar

Theorem[4] we can rewrite the inclusion a) as follows:

−ẋ∗(t) ∈ ∂x[Wa(x̃(t),x∗(t), t) + g(x̃(t), t)],
i.e.

Wa(x(t),x
∗(t), t) −Wa(x̃(t),x∗(t), t) + g(x(t), t) −g(x̃(t), t) ≥

< −ẋ∗(t),x(t) − x̃(t) >, t ∈ [t0, t1], (14)
Wa(x̃(t),x∗(t), t) =< ˙̃x(t),x∗(t) > .

Since Wa(x(t),x∗(t), t) ≤< ẋ(t),x∗(t) >, from (14) we have

dψ(t)/dt ≥ g(x̃(t), t) −g(x(t), t) (15)
for almost every t ∈ [t0, t1], where ψ(t) =< x(t) − x̃(t),x∗(t) > .
Then integrating (15) we find

∫ t1
t0

ψ̇(t)dt =< x(t1) − x̃(t1),x∗(t1) >≥
∫ t1

t0

[g(x̃(t), t) −g(x(t), t)]dt. (16)

x(t), x̃(t) are absolutely continuous, therefore ψ(t) can be represented by the sum of
absolutely continuous functions and jump functions (see [9]).

ψ(θ) = ψ(t0) +
∫ θ

t0

ψ̇(t)dt +
∑

i∈J(θ)
[ψ(τi + 0) −ψ(τi − 0)],

J(t) = {i : τi ∈ [t0, t]}. (17)
Let us compute the values of the jumps of the function ψ(t) at points τi(i = 1, 2, . . .).
Using the condition 2) of the theorem, we find

ψ(τi + 0) −ψ(τi − 0) =< x(τi) − x̃(τi),x∗i >=< x(τi),x∗i > −WF (τi)(x∗i ).



32 E. N. Mahmudov and G. Çiçek
7, 2(2005)

Then by the relation x(τi) ∈ F(τi), it is evident that

ψ(τi + 0) −ψ(τi − 0) ≥ 0 ∀τi ∈ [t0,θ],
i.e.

∑
i∈J(θ)

[ψ(τi + 0) −ψ(τi − 0)] ≥ 0.

By the condition 1) of the theorem and definition of dual cone the inequality
< x(t1) − x̃(t1),x∗(t1) >≥ 0 holds. Since t1 is free the last inequality is correct
for any t1 = θ.

Obviously, the inequality (16) is correct for any t1 = θ. Therefore from (17), it is
evident that ψ(θ) ≥ ψ(t0) i.e.

< x(θ) − x̃(θ),x∗(θ) >≥< x(t0) − x̃(t0),x∗(t0) >= 0.
From the last inequality and condition 3) of the Theorem

− < x(θ),x∗(θ) > ≤− < x̃(θ),x∗(θ) > < WM∩F (θ)(−x∗(θ)). (18)
Let �I = I(x(.),θ) − I(x̃(.), t1) be the increment of the target functional I, ob-

tained by the transition from the trajectory x̃(t) to the trajectory x(t). Then

�I = ϕ(x(θ),θ) +
∫ θ

t0

g(x(t), t)dt−ϕ(x̃(t1), t1) −
∫ t1

t0

g(x̃(t), t)dt

= ϕ(x(θ),θ) +
∫ θ

t0

g(x(t), t)dt−ϕ(x(t1), t1) −
∫ t1

t0

g(x(t), t)dt + ϕ(x(t1), t1)+

+
∫ t1

t0

g(x(t), t)dt−ϕ(x̃(t1), t1) −
∫ t1

t0

g(x̃(t), t)dt.

On the other hand from the inequality (16) and by condition 1) of the Theorem
we obtain:

∫ t1
t0

[g(x(t), t) −g(x̃(t), t)]dt + ϕ(x(t1), t1) −ϕ(x̃(t1), t1) ≥ 0.

Since (16) is correct for any t ∈ [t0, t1], the last relation implies

�I ≥ ϕ(x(θ),θ) +
∫ θ

t0

g(x(t), t)dt−ϕ(x(t1), t1) −
∫ t1

t0

g(x(t), t)dt. (19)

To prove the optimality of x̃(t) let us assume the contrary, i.e. let for any ad-
missible trajectory x(t), t ∈ [t0,θ],x(t0) = x0,x(θ) ∈ M, �I < 0, i.e. I(x(.),θ) <
I(x̃(.), t1). Then by the inequality (19), we have I(x(.),θ) < I(x(.), t1). Since I(x(.), t)
is monotone we conclude that

θ < t1. (20)



7, 2(2005)
Optimization of differential inclusions ... 33

Thus by the inequalities (18) and (20) we have x(θ) /∈ M∩F(θ). Hence x(θ) /∈ M,
i.e. the trajectory x(t) cannot realize the transition from the interval [t0,θ] to the set
M. It means that, x̃(t) is the optimal trajectory.
Remark 2.1.1: If t1 is fixed then θ = t1, and then �I ≥ 0 (see (19)), i.e. x̃(t) is
optimal. Moreover, in that case the condition of monotone increasingness of I(x(.), t)
on t is superfluous .
Remark 2.1.2: Condition of monotonicity of I(x(.), t) on t for any admissible tra-
jectory x(t) is not very restrictive and we can verify it. For example it is fulfilled for
high speed problems and for problems with quadratic criteria of quality and in case
when ϕ(x,t) ≡ 0,g(x,t) ≥ 0.
Remark 2.1.3: Suppose a is a convex-valued closed bounded continuous mapping
and Wa(x,y∗) is continuous differentiable on x. Theorem 1.1 and the condition a) of
Theorem 2.1 imply

−ẋ∗(t) ∈ ∂Wa(x̃(t),x
∗(t), t)

∂x
+ ∂g(x̃(t), t).

3 Duality.

Let us reconsider the problem (1)-(4), given in the Introduction. This problem is
called a convex problem if the functions, multivalued mapping and the set are convex
and the function F is convex-valued. Now consider (1)-(4) as a convex problem. Let
us denote by ϕ∗(., t1) and g∗(., t) conjugate functions [1,10] to functions ϕ(., t1) and
g(., t), respectively. Let us recall the equality

Ωa(x
∗,y∗, t) = inf{− < x,x∗ > + < y,y∗ >: (x,y) ∈ gfa}.

Evidently Ωa(x∗,y∗, t) = inf
x
{− < x,x∗ > +Wa(x,y∗, t)}. The following problem is

called the dual problem to (1)-(4):

sup
x∗(t),ξ∗(t),u∗(t),v∗(t1)

{−ϕ∗(v∗(t1) − ξ∗(t1), t1) −
∫ t1

t0

g∗(u∗(t), t)dt

+ < x(t0),x
∗(t0) > +

∫ t1
t0

Ωa(−ξ∗(t) − ẋ∗(t) −u∗(t),x∗(t), t)dt

−
∫ t1

t0

WF (t)(ξ
∗(t))dt + WM (v

∗(t1) −x∗(t1) − ξ∗(t1))}. (21)

Here x∗(t),ξ∗(t),u∗(t) and v∗(t1), are absolutely continuous functions. Let us denote
the expression in curly brackets by I∗(x∗(t),ξ∗(t),u∗(t),v∗(t1), t1).
Theorem 3.1. For any admissible solutions x(t) and {x∗(t),ξ∗(t),u∗(t),v∗(t1)} of
the direct problem (1)-(4) and the dual problem (21), respectively, the relation

I(x(t), t1) ≥ I∗(x∗(t),ξ∗(t),u∗(t),v∗(t1), t1)



34 E. N. Mahmudov and G. Çiçek
7, 2(2005)

holds.
Proof. By the definitions of the conjugate function, Ωa,WM and WF (t), we have

I∗(x
∗(t),ξ∗(t),u∗(t),v∗(t1), t1) ≤− < x(t1),v∗(t1) − ξ∗(t1) > +ϕ(x(t), t) −∫ t1

t0

[< x(t),u∗(t) > −g(x(t), t)]dt+ < x(t0),x∗(t0) > +

+
∫ t1

t0

[− < x(t),−ξ∗(t) − ẋ∗(t) −u∗(t) > + < ẋ(t),x∗(t) >]dt−
∫ t1

t0

< x(t),ξ∗(t) > dt+ < x(t1),v
∗(t1) −x∗(t1) − ξ∗(t1) >=

I(x(t), t1)+ < x(t0),x
∗(t0) > +

∫ t1
t0

d < x(t),x∗(t) > − < x(t1),x∗(t1) >=
I(x(t), t1). (22)

Theorem is proved.
Theorem 3.2: Let the trajectory x(t), t ∈ [t0, t1], be a solution of the direct convex
problem (1)-(4). Further, let x∗(t), ξ∗(t),u∗(t) and v∗(t1) be functions such that x∗(t)
satisfies the dual differential inclusion a), u∗(t) ∈ ∂g(x̃(t), t),
v∗(t1) − ξ∗(t1) ∈ ∂ϕ(x̃(t1), t1), ξ∗(t) ∈ K∗F (t)(x̃(t)) and v∗(t1) − x∗(t1) − ξ∗(t1) ∈
K∗M (x̃(t1)).

Then {x∗(t),ξ∗(t),u∗(t),v∗(t1)} is a solution of the dual problem and in this case,
the values of the two problems coincide.
Proof. By the definitions of locally conjugate mapping and conjugate cone we have

< ξ∗(t) + ẋ∗(t) + u∗(t),x− x̃(t) > + < x∗(t),y − ˙̃x(t) >≥ 0
at almost every t ∈ [t0, t1] and all x ∈ F(t), (x,y) ∈ gfa(., t). It means that
(−ξ∗(t) − ẋ∗(t) −u∗(t),x∗(t)) ∈ domΩa, t ∈ [t0, t1].

If we consider ∂xg(x,t) ⊂ domg∗(., t) and ∂xϕ(x,t1) ⊂ domϕ∗(., t1) then we may
conclude that {x∗(t),ξ∗(t),u∗(t),v∗(t1)} is an admissible solution of the dual problem.

Further, by Lemma 1.3 and from the conjugate differential inclusion a) it is clear
that

Ωa(ξ
∗(t) − ẋ∗(t) −u∗(t),x∗(t), t) =

− < x̃(t),−ξ∗(t) − ẋ∗(t) −u∗(t) > +Wa(x̃(t),x∗(t), t), t ∈ [t0, t1]. (23)
From conditions of the theorem and from the fact that ˙̃x(t) ∈ a(x̃(t),x∗(t), t),
t ∈ [t0, t1], it follows that

g∗(u∗(t), t) =< x̃(t),u∗(t) > −g(x̃(t), t),
ϕ∗(v∗(t1) − ξ∗(t1), t1) =< x̃(t1),v∗(t1) − ξ∗(t1) > −ϕ(x̃(t1), t1),

WF (t)(ξ
∗(t)) =< ξ∗(t), x̃(t) >, t ∈ [t0, t1], (24)

WM (v
∗(t1) −x∗(t1) − ξ∗(t1)) =< v∗(t1) −x∗(t1) − ξ∗(t1), x̃(t1) >,

Wa(x̃(t),x
∗(t), t) =< ˙̃x(t),x∗(t) >, t ∈ [t0, t1].



7, 2(2005)
Optimization of differential inclusions ... 35

From relations (23), (24) and the proof of Theorem 3.1(see (22)), we get the required
result.

4 Examples about the construction of the dual
problem.

Let consider the following problem

I(x(.), t1) = ϕ(x(t1), t1) → inf
ẋ(t) = f(x(t),u(t)),u(t) ∈ U ⊂ Rn, t ∈ [t0, t1], (25)

x(t0) = x0, x(t1) ∈ M = {x1},
where f(x,u) is differentiable on x and a(x) = f(x,U) is convex. Let us replace the
problem (25) with the following:

I(x(.), t1) → inf
ẋ(t) ∈ a(x(t)) (26)

x(t0) = x0, x(t1) = x1.

It is obvious that

Wa(x,y
∗) = inf

u∈U
< y∗,f(x,u) > . (27)

Then,if ũ is a solution of the problem (27), and x̃ is a solution of the problem which
is formulated in Lemma 1.3, then the following relation is valid

x∗ = f
′∗
x (x̃, ũ)y

∗, (28)

where f
′∗
x is matrix conjugate to the matrix f

′
x. When F(t) ≡ R

n
and M = R

n
,

WF (t) and WM in (24) show that

ξ∗(t) = 0,v∗(t1) = x
∗(t1). (29)

Since g(x,t) ≡ 0 in the problem (25), then

g∗(u∗, t) =
{

0 ,u∗ = 0
∞ ,u∗ �= 0. (30)

Considering Ωa in various intervals (see(21)) and using (29) and (30), we obtain

sup
x∗(t)
{−ϕ∗(x∗(t1), t1) +

∫ t1
t0

Ωa(−ẋ∗(t),x∗(t), t)dt}. (31)

From relations (28)-(30) we have

−ẋ∗(t) = f ′∗x (x̃(t), ũ(t))x∗(t), t ∈ [t0, t1], (32)



36 E. N. Mahmudov and G. Çiçek
7, 2(2005)

Wa(x̃(t),x
∗(t)) =< x∗(t),f(x̃(t), ũ(t)) > .

Thus the dual problem is defined by the formulas (31) and (32).
Let us consider the problem with polyhedral mapping[1]

a(x) = {y : Ax−By ≤ d},
where A,B are (m×n) matrices and d is an m-dimensional column-vector. We com-
pute easily that the LCM is given by

a∗(y∗; (x̃, ỹ)) = {A∗λ : y∗ = B∗λ,λ ≥ 0,< Ax̃−Bỹ −d,λ >= 0}.
Using the last formula it is easy to show that the dual problem consists of the following:

I∗(x
∗(.), t1) → sup,

−ẋ∗(t) = A∗λ(t), t ∈ [t0, t1],
x∗(t) = B∗λ(t), t ∈ [t0, t1],

< Ax̃(t) −B ˙̃x(t) −d,λ(t) >= 0,
λ(t) ≥ 0, t ∈ [t0, t1].

Received: April 2003. Revised: December 2003.

References

[1] B. N. PSHENICHNYI, Convex Analysis and Extremal Problems, M., Nauka,
(1980).

[2] R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, New
Jersey, (Princeton 1972).

[3] L. S. PONTRYAGIN, V. G. BOLTJANSKI, R. V. GAMKRELIDZE , E. F.
MISHENKO, Mathematical Theory of Optimal Processes, Nauka, (1969).

[4] V. M. TIHOMIROV, Some Problems in Theory of Approximation, Izd-vo
MGU, (1976).

[5] A. M. RUBINOV, Sublinear operatuses and their applications, UMN, (1977),
32, 4, p. 113-174.

[6] E. N. MAHMUDOV, Duality in problems of optimal control, describing
convex discrete and differential inclusions, Avtomatica and Telemehanica,
(1987), no. 2, 13-25.



7, 2(2005)
Optimization of differential inclusions ... 37

[7] E. N. MAHMUDOV, Necessary and sufficient conditions of extremum for
discrete and differential inclusions with distributed parameters, Sib. Mat.
Jurn. , 30 (1989), no. 2, 122-137.

[8] E. N. MAHMUDOV, Duality in problems of theory of convex difference in-
clusions with persistence, Differenc. Uravnenia, 23(1987), no. 8, 1315-1324.

[9] A. N. KOLMOGOROV, S. V. FOMIN, Elements of Theory of Functions
and Functional Analysis, Nauka, (1968).

[10] A. D. IOFFE, V. M. TIHOMIROV, Theory of Extremal Problems, M. :
Nauka, (1974).

[11] R. VINTER, Optimal Control Systems & Control Foundations & Applica-
tion, Birkhäuser Boston Inc. , MA. , (Boston 2000).

[12] B. S. MORDUKHOVICH, Optimal control of difference, differential-
difference inclusions, J. Math. Sci. , (New York)100(2000), no. 6, 2613-2632.

[13] F. H. CLARKE, YU. S. LEDYAEV, M. L. RADULESEV, Approximate
invariance and differential inclusions in Hilbert spaces, J. Dynam. Control
Systems, 3(1997), no. 4, 493-518.

[14] E. N. MAHMUDOV, Optimization of discrete inclusions with distributed
parameters, Optimization, 21(1990), no. 2, 197-207, Berlin.

[15] YU. G. BORISOVICH ET. AL. , Multivalued mappings, Itogi Nauki i
Tekniki: Math. Anal. , vol. 19, VINITI Moscow, (1982), 127-230, English
transl. In J. Soviet Mat. , 26(1984), no. 4

[16] J. V. OUTRATA, Minimization of nonsmooth nonregular functions. Appli-
cation to discrete-time optimal control problems, Problems of Control and
Information Theory, 13(1984), no. 6, 413-424.

[17] R. VINTER, H. H. ZHENG, Necessary conditions for optimal control prob-
lems with state constraints, Transactions of the American Mathematical So-
ciety, 350(3)(1998), 1181-1204.

[18] R. VINTER, H. H. ZHENG, Necessary conditions for free end-time mea-
surably time dependent optimal control problems with state constraints, Set-
Valued Anal. 8(2000), no. 1-2, 11-29.

[19] B. S. MORDUKHOVICH, Optimal control of nonconvex discrete and differ-
ential inclusions, Sociedad Matematica Mexicana, Mexico, (1998), vi+324.

[20] F. H. CLARCE, Optimization and nonsmooth analysis, (1983), John Wiley
& Sons Inc.

[21] P. D. LOEWEN, R. T. ROCKAFELLAR, Bolza problems with general time
constraints, SIAM J. Control Optim. , 35(6)(1997), 2050-2069.



38 E. N. Mahmudov and G. Çiçek
7, 2(2005)

[22] P. D. LOEWEN, R. T. ROCKAFELLAR, New necessary conditions for the
generalized problem of Bolza, SIAM J. Control Optim. , 34(5)(1996), 1496-
1511.

[23] P. D. LOEWEN, R. T. ROCKAFELLAR, Optimal-Control of unbounded
differential inclusions, SIAM J. Control Optim. , 32(2)(1994), 442-470.

[24] A. V. KRYAZHIMSKII, Convex optimization via feedbacks, SIAM J. Control
Optim. , 37(1)(1998), 37(1), 278-302.

[25] E. N. MAHMUDOV, Duality in the problems of Optimal control for systems
described by convex differential inclusions with delay, Problems of Control
and information theory, 16(6), (1987), 411-422, Budapest.