Control and Cybernetics Generalized α-V

Control and Cybernetics Generalized α-V

Control and Cybernetics
vol.
43 (2014) No. 3
Generalized α-V-univex functions for multiobjective
variational control problems∗†
by
1
Sarita Sharma , Anurag Jayswal2 and Sarita Choudhury2
1
Department of Applied Sciences, IIMT Engg. College, Ganga Nagar,
Meerut-250 005, India
[email protected]
2
Department of Applied Mathematics, Indian School of Mines,
Dhanbad-826 004, Jharkhand, India
anurag [email protected], [email protected]
Abstract: The purpose of this paper is to introduce a new class
of α-V-univex/ generalized α-V-univex functions for a class of multiobjective variational control problems. Moreover, sufficient optimality conditions and Mond-Weir type duality results, associated
with the multiobjective variational control problem, are established
under aforesaid assumptions.
Keywords: control problem; α-V-univexity; efficiency; sufficiency; duality
1.
Introduction
Multiobjective optimization deals with solving problems having several conflicting objectives simultaneously, while the control problem consists in transferring
the state variable from an initial state to a final state so as to optimize a given
functional, subject to constraints on the control and state variables.
Thus, multiobjective control problem is a wide field of research having extensive applications in real world situations, ranging from engineering to economics,
and many more. For example, multiobjective control problems are used in flight
control design, in the control of space structures, in industrial process control
and other diverse fields.
Convexity plays a significant role in optimization as it gives global validity
to propositions otherwise only locally true. However, in the real world of mathematical and economic models, convexity appears to be a restrictive condition.
As a result, nonconvex optimization problems have been studied by various authors. Naniewicz and Puchala (2012) studied a nonconvex optimization problem
∗ The research of the second author has been supported by the DST, New Delhi, India
through Grant No.: SR/FTP/MS-007/2011.
† Submitted: April 2014; Accepted: August 2014
404
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
in the case when the functional to be minimized has integrand expressed as a
minimum of two quadratic functions by constructing an appropriate minimizing
sequence. Tabor and Tabor (2012) showed that the Takagi class can serve as an
important source of examples and counterexamples for paraconvex and semiconvex functions. By similar motivation, several researchers have generalized
the concept of convexity.
Firstly, Hanson (1981) introduced the concept of invexity to extend the
validity of the sufficiency of the Kuhn-Tucker conditions. Mond and Smart
(1988) obtained duality results for a control problem using invexity and showed
that for invex functions, the necessary conditions are also sufficient. Bhatia and
Kumar (1995) introduced multiobjective control problems and proved duality
results under generalized ρ-invexity assumptions. Nahak and Nanda (1997,
2007) further extended this concept by proving duality results for multiobjective
variational control problems, under the assumption of (F, ρ)-convexity and Vinvexity, respectively.
Gulati, Husain and Ahmed (2005) derived optimality conditions and duality results for multiobjective control problems involving generalized convexity.
Also, Ahmad and Gulati (2005) proved results for mixed type dual for multiobjective variational problems under generalized (F, ρ)-convexity. Later, Ahmad and Sharma (2010) extended the notion of generalized (F, α, ρ, θ)-V-convex
functions to variational control problems. Recently, Kailey and Gupta (2013)
further extended the concept of generalized (F, α, ρ, d)-convexity and proved duality results for a class of symmetric non-differentiable multiobjective fractional
variational problems.
The concept of univex functions as a generalization of invex functions was introduced by Bector, Suneja and Gupta (1992). Later on, many authors (AranaJiménez, Ruiz-Garzón, Rufián-Lizana and Osuna-Gómez, 2012; Chen, 2002;
de Oliveira, Silva and Rojas-Medar, 2009; Khazafi and Rueda, 2009; Khazafi,
Rueda and Enflo, 2010; Zhian and Qingkai, 2001) extended the concept of generalized convexity. Being inspired by Bector, Suneja and Gupta (1992), Noor
(2004), Nahak and Nanda (1997, 2007), and Preda, Stancu-Minasian, Beldiman
and Stancu (2009), we introduce the concept of α-V-univex functions for a multiobjective variational control problem and obtain sufficient optimality conditions
and duality results.
The rest of the paper is organized as follows: In Section 2, we introduce
the definitions of α-V-univex and generalized α-V-univex functions, and recall
a set of necessary optimality conditions. In Section 3, we prove the sufficient
optimality conditions, and in Section 4, we present the Mond-Weir type multiobjective variational control dual problem and derive weak and strong duality
results. Finally, we conclude our paper in Section 5.
2.
Notations and preliminaries
Let Rn denote the n-dimensional Euclidean space. Let y, z ∈ Rn , we denote:
y ≦ z ⇔ yi ≦ zi , i = 1, 2, . . . , n; y ≤ z ⇔ y ≦ z and y 6= z; y < z ⇔ yi < zi , i =
Generalized α-V-univex functions for multiobjective variational control problems
405
1, 2, . . . , n.
Let I = [a, b] be a real interval. Let fi : I × Rn × Rn × Rm × Rm → R,
i ∈ P = {1, 2, . . . , p}, gj : I × Rn × Rn × Rm × Rm → R, j ∈ M = {1, 2, . . . , m}
and hk : I × Rn × Rn × Rm × Rm → R, k ∈ N = {1, 2, . . . , n} be continuously differentiable functions. Consider the function f (t, x(t), ẋ(t), u(t), u̇(t)),
where t is the independent variable, x : I → Rn is the state variable and
u : I → Rm is the control variable. u(t) is related to x(t) via the state equation
h(t, x(t), ẋ(t), u(t), u̇(t)) = 0, where the dot denotes the derivative with respect
to t. fix , fiẋ , fiu and fiu̇ denote the partial derivatives of fi with respect to
x, ẋ, u and u̇, respectively. For instance,
∂fi
∂fi ∂fi
∂fi
∂fi ∂fi
,
,...,
, fiẋ =
,
,...,
.
fix =
∂x1 ∂x2
∂xn
∂ ẋ1 ∂ ẋ2
∂ ẋn
Similarly, gjx , gj ẋ , gju , gj u̇ and hkx , hkẋ , hku , hku̇ can be defined. For notational convenience, we use x, ẋ, u, u̇ in place of x(t), ẋ(t), u(t), u̇(t), respectively.
Let the differentiation operator D be given by
Z t
z = Dx ⇔ x(t) = γ +
z(s)ds,
a
where γ is a given boundary value. Therefore, D = d/dt except at discontinuities. Let X denote the space of all piecewise smooth functions x : I 7→ Rn with
norm kxk = kxk∞ + kDxk∞ and Y denote the space of all piecewise smooth
functions u : I 7→ Rm with norm kuk∞ .
In this paper, we consider the following multiobjective variational control problem:
Z
Z
b
(CP) Minimize
Z
subject to
b
f (t, x, ẋ, u, u̇)dt = (
a
b
f2 (t, x, ẋ, u, u̇)dt, . . . ,
a
f1 (t, x, ẋ, u, u̇)dt,
Za
b
fp (t, x, ẋ, u, u̇)dt)
a
x(a) = γ, x(b) = δ,
g(t, x, ẋ, u, u̇) ≦ 0, t ∈ I,
(1)
h(t, x, ẋ, u, u̇) = 0, t ∈ I.
(2)
We denote the set of all feasible solutions to (CP) by X ◦ , i.e.,
X ◦ = {(x, u) ∈ (X, Y ) : x(a) = γ, x(b) = δ, g(t, x, ẋ, u, u̇) ≦ 0, h(t, x, ẋ, u, u̇) = 0}.
Definition 1 A point (x̄, ū) ∈ X ◦ is said to be an efficient solution of (CP),
if there exists no other point (x, u) ∈ X ◦ such that
Z b
Z b
˙ ū, ū)dt.
˙
f (t, x, ẋ, u, u̇)dt ≤
f (t, x̄, x̄,
a
a
406
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
Definition 2 A feasible point (x̄, ū) ∈ X ◦ is said to be a weakly efficient solution of (CP), if there exists no other point (x, u) ∈ X ◦ such that
Z b
Z b
˙ ū, ū)dt.
˙
f (t, x, ẋ, u, u̇)dt <
f (t, x̄, x̄,
a
a
p
˙ ū, ū)
˙ ∈ R+ , α(t, x, ẋ, u, u̇, x̄, x̄,
˙ ū, ū)
˙ ∈ R+
Let b◦ (t, x, ẋ, u, u̇, x̄, x̄,
\{0}, η(t, x, ẋ, u,
n
m
˙ ū, ū)
˙ ∈ R , ξ(t, x, ẋ, u, u̇, x̄, x̄,
˙ ū, ū)
˙ ∈ R and φ◦ : R → R. For notau̇, x̄, x̄,
˙ ū, ū),
˙
tional convenience, we use b◦ for b◦ (t, x, ẋ, u, u̇, x̄, x̄,
˙ ū, ū),
˙ η for η(t, x, ẋ, u, u̇, x̄, x̄,
˙ ū, ū),
˙ and ξ for
αi for αi (t, x, ẋ, u, u̇, x̄, x̄,
˙ ū, ū).
˙ Let ψ : I × X × X × Y × Y 7→ Rp be a vector functional.
ξ(t, x, ẋ, u, u̇, x̄, x̄,
Now, we introduce the concept of α-V-univexity as follows.
Rb
Definition 3 A vector functional a ψ(t, x, ẋ, u, u̇)dt is said to be α-V-univex
at (x̄, ū) with respect to the functions b◦ , φ◦ , α, η and ξ, if for all (x, u) ∈ (X, Y )
and i ∈ P ,
Z b
˙ ū, ū)]dt
˙
b◦
φ◦ [ψi (t, x, ẋ, u, u̇)−ψi (t, x̄, x̄,
a
≧
Z
b
˙ ū, ū)
˙ − Dψiẋ (t, x̄, x̄,
˙ ū, ū))η
˙
αi [(ψix (t, x̄, x̄,
a
˙ ū, ū)
˙ − Dψiu̇ (t, x̄, x̄,
˙ ū, ū))ξ]dt.
˙
+ (ψiu (t, x̄, x̄,
Remark 1 If we take b◦ = 1, φ◦ (a) = a and Dψiu̇ = 0, then α-V-univex
function reduces to V -invex function given by Nahak and Nanda (2007).
Now, we present the following example which is α-V-univex but not V -invex.
Example 1 Let I = [0, 1] and X = Y = C([0, 1], R+ ). We define the function
ψ : I × X × X × Y × Y 7→ R2 as
ψ(t, x, ẋ, u, u̇) = (−4(x2 (t) + x(t) + u(t)), −x2 (t) − 2x(t) − 3u(t)).
Further, let φ◦ : R 7→ R be given as φ◦ (a) = −2a. Define
x̄(t)ū(t) + 1
x̄(t)ū(t) + 3
, α2 =
,
2
2
2
2
2x (t) + 3x(t) + x̄(t)ū(t)
x (t) + 3u(t)
η=
, ξ=
,
4
4
R1
and take b◦ = 2, x̄(t) = t and ū(t) = t2 . Then, 0 ψ(t, x, ẋ, u, u̇)dt is α-Vunivex at (x̄, ū) = (0, 0) but not V -invex for the same functions α, η and ξ as
can be seen below.
α1 =
Generalized α-V-univex functions for multiobjective variational control problems
407
Explanation: Firstly, we have to show that
Z 1
˙ ū, ū)]dt
˙
b◦
φ◦ [ψ1 (t, x, ẋ, u, u̇) − ψ1 (t, x̄, x̄,
0
≧
Z
1
˙ ū, ū)
˙ − Dψ1ẋ (t, x̄, x̄,
˙ ū, ū))η
˙
α1 [(ψ1x (t, x̄, x̄,
0
˙ ū, ū)
˙ − Dψ1u̇ (t, x̄, x̄,
˙ ū, ū))ξ]dt,
˙
+ (ψ1u (t, x̄, x̄,
at (x̄, ū) = (0, 0).
L.H.S.:
1
Z
b◦
˙ ū, ū)]dt
˙
φ◦ [ψ1 (t, x, ẋ, u, u̇) − ψ1 (t, x̄, x̄,
0
=2
Z
1
Z
1
φ◦ [−4(x2 (t) + x(t) + u(t)) + 4(x̄2 (t) + x̄(t) + ū(t))]dt
0
=2
(−2)[−4(x2 (t) + x(t) + u(t)) + 4(x̄2 (t) + x̄(t) + ū(t))]dt
0
= 16
Z
1
(x2 (t) + x(t) + u(t))dt.
0
R.H.S.:
Z
1
˙ ū, ū)
˙ − Dψ1ẋ (t, x̄, x̄,
˙ ū, ū))η
˙
α1 [(ψ1x (t, x̄, x̄,
0
˙ ū, ū)
˙ − Dψ1u̇ (t, x̄, x̄,
˙ ū, ū))ξ]dt
˙
+(ψ1u (t, x̄, x̄,
Z 1
x2 (t) + 3u(t)
2x2 (t) + 3x(t) + x̄(t)ū(t)
x̄(t)ū(t) + 1
[−4(2x̄(t) + 1)
−4
]dt
=
2
4
4
0
Z 1
x̄(t)ū(t) + 1
=−
[(2x̄(t) + 1)(x2 (t) + 3u(t)) + 2x2 (t) + 3x(t) + x̄(t)ū(t)]dt
2
0
Z
3 1 2
=−
(x (t) + x(t) + u(t))dt.
2 0
Therefore, it follows that
Z 1
˙ ū, ū)]dt
˙
b◦
φ◦ [ψ1 (t, x, ẋ, u, u̇)−ψ1 (t, x̄, x̄,
0
≧
Z
1
˙ ū, ū)
˙ − Dψ1ẋ(t, x̄, x̄,
˙ ū, ū))η
˙
α1 [(ψ1x (t, x̄, x̄,
0
˙ ū, ū)
˙ − Dψ1u̇ (t, x̄, x̄,
˙ ū, ū))ξ]dt,
˙
+ (ψ1u (t, x̄, x̄,
Similarly, it can be shown that
Z 1
˙ ū, ū)]dt
˙
b◦
φ◦ [ψ2 (t, x, ẋ, u, u̇) − ψ2 (t, x̄, x̄,
0
at (x̄, ū) = (0, 0).
408
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
Z
≧
1
˙ ū, ū)
˙ − Dψ2ẋ (t, x̄, x̄,
˙ ū, ū))η
˙
α2 [(ψ2x (t, x̄, x̄,
0
˙ ū, ū)
˙ − Dψ2u̇ (t, x̄, x̄,
˙ ū, ū))ξ]dt,
˙
+ (ψ2u (t, x̄, x̄,
at (x̄, ū) = (0, 0).
R1
Therefore, 0 ψ(t, x, ẋ, u, u̇)dt is α-V-univex at (x̄, ū) = (0, 0). Again,
Z 1
˙ ū, ū)]dt
˙
[ψ1 (t, x, ẋ, u, u̇) − ψ1 (t, x̄, x̄,
0
=
1
Z
[−4(x2 (t) + x(t) + u(t)) + 4(x̄2 (t) + x̄(t) + ū(t))]dt
0
= −4
Z
1
(x2 (t) + x(t) + u(t))dt.
0
Also,
1
Z
=
˙ ū, ū)
˙ − Dψ1ẋ (t, x̄, x̄,
˙ ū, ū))η
˙
˙ ū, ū)ξ]dt
˙
α1 [(ψ1x (t, x̄, x̄,
+ ψ1u (t, x̄, x̄,
0
1
Z
x̄(t)ū(t) + 1
x2 (t) + 3u(t)
2x2 (t) + 3x(t) + x̄(t)ū(t)
[−4(2x̄(t) + 1)
−4
]dt
2
4
4
0
Z
1
3
=−
2
Z
=−
0
x̄(t)ū(t) + 1
[(2x̄(t) + 1)(x2 (t) + 3u(t)) + 2x2 (t) + 3x(t) + x̄(t)ū(t)]dt
2
1
(x2 (t) + x(t) + u(t))dt.
0
Hence, it follows that
Z 1
˙ ū, ū)]dt
˙
[ψ1 (t, x, ẋ, u, u̇)−ψ1 (t, x̄, x̄,
0
6≧
Z
1
˙ ū, ū)
˙ − Dψ1ẋ(t, x̄, x̄,
˙ ū, ū))η
˙
˙ ū, ū)ξ]dt,
˙
α1 [(ψ1x (t, x̄, x̄,
+ ψ1u (t, x̄, x̄,
0
R1
which shows that
ψ(t, x, ẋ, u, u̇)dt is not V -invex at (x̄, ū) = (0, 0).
Rb
Definition 4 A vector functional a ψ(t, x, ẋ, u, u̇)dt is said to be (strictly) αV-pseudounivex at (x̄, ū) with respect to the functions b◦ , φ◦ , α, η and ξ, if for
all (x, u) ∈ (X, Y ) and i ∈ P ,
Z b X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
ψix (t, x̄, x̄,
ψiẋ (t, x̄, x̄,
a
0
i∈P
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))ξ]dt
˙
+(
ψiu (t, x̄, x̄,
ψiu̇ (t, x̄, x̄,
≧0
i∈P
⇒ b◦
Z
a
b
φ◦ [
X
i∈P
i∈P
αi ψi (t, x, ẋ, u, u̇) −
X
i∈P
˙ ū, ū)]dt(>)
˙
αi ψi (t, x̄, x̄,
≧ 0.
Generalized α-V-univex functions for multiobjective variational control problems
409
Rb
Definition 5 A vector functional a ψ(t, x, ẋ, u, u̇)dt is said to be (strictly) αV-quasiunivex at (x̄, ū) with respect to the functions b◦ , φ◦ , α, η and ξ, if for all
(x, u) ∈ (X, Y ) and i ∈ P ,
b◦
⇒
Z
b
φ◦ [
a
αi ψi (t, x, ẋ, u, u̇)−
i∈P
b
Z
X
a
X
˙ ū, ū)]dt
˙
αi ψi (t, x̄, x̄,
≦0
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
ψix (t, x̄, x̄,
ψiẋ (t, x̄, x̄,
i∈P
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))ξ]dt(<)
˙
+(
ψiu (t, x̄, x̄,
ψiu̇ (t, x̄, x̄,
≦ 0.
i∈P
i∈P
Rb
Definition 6 A vector functional a ψ(t, x, ẋ, u, u̇)dt is said to be prestrictly
α-V-quasiunivex at (x̄, ū) with respect to the functions b◦ , φ◦ , α, η and ξ, if for
all (x, u) ∈ (X, Y ) and i ∈ P ,
b◦
⇒
Z
b
φ◦ [
a
αi ψi (t, x, ẋ, u, u̇)−
i∈P
b
Z
X
a
X
˙ ū, ū)]dt
˙
αi ψi (t, x̄, x̄,
<0
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
ψix (t, x̄, x̄,
ψiẋ (t, x̄, x̄,
i∈P
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))ξ]dt
˙
+(
ψiu (t, x̄, x̄,
ψiu̇ (t, x̄, x̄,
≦ 0.
i∈P
i∈P
Remark 2 In the proofs of theorems, sometimes it may be more convenient
to use certain alternative, but equivalent, forms of the above definitions. For
Rb
example: a vector functional a ψ(t, x, ẋ, u, u̇)dt is said to be α-V-pseudounivex
with respect to the functions b◦ , φ◦ , α, η and ξ, if for all (x, u) ∈ (X, Y ) and
i ∈ P,
b◦
⇒
Z
Z
b
φ◦ [
a
a
b
X
αi ψi (t, x, ẋ, u, u̇)−
i∈P
X
˙ ū, ū)]dt
˙
αi ψi (t, x̄, x̄,
<0
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
ψix (t, x̄, x̄,
ψiẋ (t, x̄, x̄,
i∈P
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))ξ]dt
˙
+(
ψiu (t, x̄, x̄,
ψiu̇ (t, x̄, x̄,
< 0.
i∈P
i∈P
Lemma 1 (Kuhn-Tucker type necessary conditions) (Ahmad and Sharma, 2010)
Let (x̄, ū) solve the following single objective problem:
410
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
Minimize
subject to
Z
b
φ(t, x, ẋ, u, u̇)dt
a
x(a) = γ, x(b) = δ,
g(t, x, ẋ, u, u̇) ≦ 0, t ∈ I,
h(t, x, ẋ, u, u̇) = 0, t ∈ I.
If the Frëchet derivative [D − Hx (x̄, ū)] is surjective and the optimal solution
(x̄, ū) is normal, then there exist piecewise smooth functions µ̄ : I 7→ Rm and
ν̄ : I 7→ Rn satisfying the following conditions:
X
X
˙ ū, ū)
˙ +
˙ ū, ū)
˙ +
˙ ū, ū)
˙
φx (t, x̄, x̄,
µ̄j (t)gjx (t, x̄, x̄,
ν̄k (t)hkx (t, x̄, x̄,
j∈M
k∈N
˙ ū, ū)
˙ +
= D[φẋ (t, x̄, x̄,
X
˙ ū, ū)
˙ +
µ̄j (t)gj ẋ (t, x̄, x̄,
j∈M
X
˙ ū, ū)
˙ +
φu (t, x̄, x̄,
j∈M
X
˙ ū, ū)
˙
ν̄k (t)hku (t, x̄, x̄,
k∈N
X
˙ ū, ū)
˙ +
µ̄j (t)gj u̇ (t, x̄, x̄,
j∈M
X
˙ ū, ū)],
˙
ν̄k (t)hkẋ (t, x̄, x̄,
k∈N
˙ ū, ū)
˙ +
µ̄j (t)gju (t, x̄, x̄,
˙ ū, ū)
˙ +
= D[φu̇ (t, x̄, x̄,
X
X
˙ ū, ū)],
˙
ν̄k (t)hku̇ (t, x̄, x̄,
k∈N
˙ ū, ū)
˙ = 0,
µ̄j (t)gj (t, x̄, x̄,
j∈M
µ̄(t) ≧ 0, ∀t ∈ I.
Lemma 2 (Chankong and Haimes, 1983) (x̄, ū) is an efficient solution for (CP)
if and only if (x̄, ū) solves
(CP)s Minimize
subject to
Z
b
fk (t, x, ẋ, u, u̇)dt
a
x(a) = γ, x(b) = δ,
Z
Z b
fi (t, x, ẋ, u, u̇)dt ≤
a
b
˙ ū, ū)dt,
˙
fi (t, x̄, x̄,
∀i ∈ P, i 6= k,
a
g(t, x, ẋ, u, u̇) ≤ 0, t ∈ I,
h(t, x, ẋ, u, u̇) = 0, t ∈ I.
3.
Sufficient optimality conditions
Rb
Rb
˙ ū, ū)dt
˙
˙
denotes the vector ( a λ̄1 f1 (t, x̄, x̄,
In the sequel of the paper, a λ̄f (t, x̄, x̄,
Rb
Rb
Rb
˙ ū, ū)dt,
˙
˙ ū, ū)dt).
˙
˙
Similarly, a µ̄(t)g(t,
ū, ū)dt,
λ̄ f (t, x̄, x̄,
. . . , a λ̄p fp (t, x̄, x̄,
a 2 2 R
b
˙
˙
˙
˙
x̄, x̄, ū, ū)dt and a ν̄(t)h(t, x̄, x̄, ū, ū)dt can be defined.
Generalized α-V-univex functions for multiobjective variational control problems
411
Theorem 1 (Sufficiency) Let
P (x̄, ū) be a feasible solution to (CP). Suppose that
there exist scalars λ̄i ≧ 0,
λ̄i = 1, µ̄j (t) ≧ 0, j ∈ M , such that for all t ∈ I,
i∈P
X
˙ ū, ū)
˙ +
λ̄i fix (t, x̄, x̄,
i∈P
X
˙ ū, ū)
˙ +
µ̄j (t)gjx (t, x̄, x̄,
j∈M
= D[
X
˙ ū, ū)
˙
ν̄k (t)hkx (t, x̄, x̄,
k∈N
X
˙ ū, ū)+
˙
λ̄i fiẋ (t, x̄, x̄,
X
˙ ū, ū)
˙ +
µ̄j (t)gj ẋ (t, x̄, x̄,
i∈P
j∈M
X
X
˙ ū, ū)],
˙
ν̄k (t)hkẋ (t, x̄, x̄,
˙ ū, ū)
˙ +
λ̄i fiu (t, x̄, x̄,
i∈P
X
˙ ū, ū)
˙ +
µ̄j (t)gju (t, x̄, x̄,
j∈M
= D[
X
˙ ū, ū)
˙ +
λ̄i fiu̇ (t, x̄, x̄,
i∈P
+
(3)
k∈N
X
X
˙ ū, ū)
˙
ν̄k (t)hku (t, x̄, x̄,
k∈N
X
˙ ū, ū)
˙
µ̄j (t)gj u̇ (t, x̄, x̄,
j∈M
˙ ū, ū)],
˙
ν̄k (t)hku̇ (t, x̄, x̄,
(4)
k∈N
Z
b
Z
b
X
˙ ū, ū)dt
˙
µ̄j gj (t, x̄, x̄,
= 0,
(5)
X
˙ ū, ū)dt
˙
ν̄k hk (t, x̄, x̄,
= 0.
(6)
a j∈M
a k∈N
Further, assume that
Rb
(i) a λ̄f (t, ·, ·, ·, ·)dt is α̂-V-pseudounivex at (x̄, ū) with respect to b◦ , φ◦ , α̂,
η and ξ;
Rb
(ii) a µ̄(t)g(t, ·, ·, ·, ·)dt is α̃-V-quasiunivex at (x̄, ū) with respect to b1 , φ1 , α̃,
η and ξ;
Rb
(iii) a ν̄(t)h(t, ·, ·, ·, ·)dt is α∗ -V-quasiunivex at (x̄, ū) with respect to b2 , φ2 , α∗ ,
η and ξ;
(iv) φ2 (0) = 0 and for any scalar function p(t),
Rb
Rb
p(t)dt < 0 ⇒ a φ◦ (p(t))dt < 0,
a
Rb
Rb
φ (p(t))dt > 0 ⇒ a p(t)dt > 0;
a 1
(v) b◦ > 0, b1 > 0.
Then (x̄, ū) is a weakly efficient solution of (CP).
Proof. Suppose, contrary to the result, that (x̄, ū) ∈ X ◦ is not a weakly efficient
solution to (CP). Then there exists (x, u) ∈ X ◦ such that
Z
a
b
f (t, x, ẋ, u, u̇)dt <
Z
a
b
˙ ū, ū)dt,
˙
f (t, x̄, x̄,
412
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
which by λ̄i ≧ 0,
P
λ̄i = 1, α̂i > 0, i ∈ P, gives
i∈P
b
Z
X
Z
α̂i λ̄i fi (t, x, ẋ, u, u̇)dt <
a i∈P
b
X
˙ ū, ū)dt.
˙
α̂i λ̄i fi (t, x̄, x̄,
(7)
a i∈P
From the assumptions (iv), (v) and inequality (7), it follows that
Z b
X
X
˙ ū, ū))dt
˙
b◦
φ◦ (
α̂i λ̄i fi (t, x, ẋ, u, u̇) −
α̂i λ̄i fi (t, x̄, x̄,
< 0.
a
i∈P
i∈P
Therefore, by hypothesis (i) and the above inequality, we get
Z b X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
λ̄i fix (t, x̄, x̄,
λ̄i fiẋ (t, x̄, x̄,
a
i∈P
i∈P
X
X
˙ ū, ū)
˙ −D
˙ ū, ū))ξ]dt
˙
λ̄i fiu (t, x̄, x̄,
λ̄i fiu̇ (t, x̄, x̄,
< 0.
+(
i∈P
i∈P
(8)
Now, from the feasibility of (x, u) to (CP), hypothesis (6), and using ν̄ 6= 0, we
have
Z bX
Z bX
˙ ū, ū)dt.
˙
ν̄k (t)hk (t, x, ẋ, u, u̇)dt =
ν̄k (t)hk (t, x̄, x̄,
(9)
a k∈N
a k∈N
Again, by hypothesis (iii) and (9), we have
Z b X
X
˙ ū, ū)
˙ −D
˙ ū, ū))η
˙
[(
ν̄k (t)hkx (t, x̄, x̄,
ν̄k (t)hkẋ (t, x̄, x̄,
a
k∈N
+(
X
k∈N
˙ ū, ū)
˙ −D
ν̄k (t)hku (t, x̄, x̄,
k∈N
X
˙ ū, ū))ξ]dt
˙
ν̄k (t)hku̇ (t, x̄, x̄,
≦ 0. (10)
k∈N
On adding inequalities (8) and (10), we obtain
Z b X
X
X
˙ ū, ū)+
˙
˙ ū, ū)−D(
˙
˙ ū, ū)
˙
[(
λ̄i fix (t, x̄, x̄,
ν̄k (t)hkx (t, x̄, x̄,
λ̄i fiẋ (t, x̄, x̄,
a
+
i∈P
X
k∈N
i∈P
k∈N
X
X
˙ ū, ū)))η+(
˙
˙ ū, ū)+
˙
˙ ū, ū)
˙
ν̄k (t)hkẋ (t, x̄, x̄,
λ̄i fiu (t, x̄, x̄,
ν̄k (t)hku (t, x̄, x̄,
i∈P
k∈N
X
X
˙ ū, ū)
˙ +
˙ ū, ū)))ξ]dt
˙
−D(
λ̄i fiu̇ (t, x̄, x̄,
ν̄k (t)hku̇ (t, x̄, x̄,
< 0.
i∈P
k∈N
The above inequality together with relations (3) and (4), yields
Z b X
X
˙ ū, ū)−D
˙
˙ ū, ū))η
˙
[(
µ̄j (t)gjx (t, x̄, x̄,
µ̄j (t)gj ẋ (t, x̄, x̄,
a
j∈M
j∈M
Generalized α-V-univex functions for multiobjective variational control problems
+(
X
˙ ū, ū)
˙ −D
µ̄j (t)gju (t, x̄, x̄,
j∈M
X
413
˙ ū, ū))ξ]dt
˙
µ̄j (t)gj u̇ (t, x̄, x̄,
> 0,
j∈M
which along with the hypothesis (ii), gives
b1
Z
b
φ1 (
a
X
α˜j µ̄j (t)gj (t, x, ẋ, u, u̇) −
j∈M
X
˙ ū, ū))dt
˙
α˜j µ̄j (t)gj (t, x̄, x̄,
> 0. (11)
j∈M
Hence, it follows from inequality (11), assumptions (iv), (v) and α˜j > 0, j ∈ M ,
that
Z b X
X
˙ ū, ū))dt
˙
(
µ̄j (t)gj (t, x, ẋ, u, u̇) −
µ̄j (t)gj (t, x̄, x̄,
> 0.
(12)
a
j∈M
j∈M
On the other hand, from the feasibility of (x, u) to (CP) and (5), we have
Z
b
X
µ̄j (t)gj (t, x, ẋ, u, u̇)dt ≦
a j∈M
Z
b
X
˙ ū, ū)dt,
˙
µ̄j (t)gj (t, x̄, x̄,
a j∈M
which contradicts (12). Hence (x̄, ū) is a weakly efficient solution to (CP). This
completes the proof.
The proofs of the following two theorems are similar to that of Theorem 1 and
hence are being omitted.
Theorem 2 Let P
(x̄, ū) be a feasible solution of (CP). Suppose that there exist
scalars λ̄i ≧ 0,
λ̄i = 1, µ̄j (t) ≧ 0, j ∈ M satisfying the conditions (3) to
i∈P
(6).
Further, assume that
Rb
(i) a λ̄f (t, ·, ·, ·, ·)dt is prestrictly α̂-V-quasiunivex at (x̄, ū) with respect to b◦ ,
φ◦ , α̂, η and ξ;
Rb
(ii) a µ̄(t)g(t, ·, ·, ·, ·)dt is α̃-V-quasiunivex at (x̄, ū) with respect to b1 , φ1 , α̃,
η and ξ;
Rb
(iii) a ν̄(t)h(t, ·, ·, ·, ·)dt is strictly α∗ -V-quasiunivex at (x̄, ū) with respect to b2 ,
φ2 , α∗ , η and ξ;
(iv) φ2 (0) = 0 and for any scalar function p(t),
Rb
Rb
p(t)dt < 0 ⇒ a φ◦ (p(t))dt < 0,
a
Rb
Rb
a φ1 (p(t))dt > 0 ⇒ a p(t)dt > 0;
(v) b◦ > 0, b1 > 0.
Then (x̄, ū) is a weakly efficient solution of (CP).
414
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
Theorem 3 Let
P(x̄, ū) be a feasible solution of (CP). Suppose that there exist
scalars λ̄i > 0,
λ̄i = 1, µ̄j (t) ≧ 0, j ∈ M satisfying the conditions (3) to (6).
i∈P
Further, assume that
Rb
(i) a λ̄f (t, ·, ·, ·, ·)dt is α̂-V-pseudounivex at (x̄, ū) with respect to b◦ , φ◦ , α̂,
η and ξ;
Rb
(ii) a µ̄(t)g(t, ·, ·, ·, ·)dt is α̃-V-quasiunivex at (x̄, ū) with respect to b1 , φ1 , α̃,
η and ξ;
Rb
(iii) a ν̄(t)h(t, ·, ·, ·, ·)dt is α∗ -V-quasiunivex at (x̄, ū) with respect to b2 , φ2 , α∗ ,
η and ξ;
(iv) φ2 (0) = 0 and for any scalar function p(t),
Rb
Rb
p(t)dt < 0 ⇒ a φ◦ (p(t))dt < 0,
Rab
Rb
φ (p(t))dt > 0 ⇒ a p(t)dt > 0;
a 1
(v) b◦ > 0, b1 > 0.
Then (x̄, ū) is an efficient solution of (CP).
4.
Duality
In this section, we present the following Mond-Weir type dual program (Ahmad
and Sharma, 2010) for (CP) and prove some duality results.
Z b
(MD)
Maximize
f (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt
a
subject to
x◦ (a) = γ, x◦ (b) = δ,
X
λi fix◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) +
i∈P
+
X
µj (t)gjx◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
j∈M
X
◦
◦
◦
νk (t)hkx◦ (t, x , ẋ , u , u̇◦ ) = D[
◦
◦
◦
+
λi fiẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
i∈P
k∈N
X
X
◦
µj (t)gj ẋ◦ (t, x , ẋ , u , u̇ ) +
j∈M
X
νk (t)hkẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )], t ∈ I,
k∈N
(13)
X
λi fiu◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) +
i∈P
+
X
X
µj (t)gju◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
j∈M
◦
◦
◦
νk (t)hku◦ (t, x , ẋ , u , u̇◦ ) = D[
◦
◦
◦
+
j∈M
λi fiu̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
i∈P
k∈N
X
X
◦
µj (t)gj u̇◦ (t, x , ẋ , u , u̇ ) +
X
νk (t)hku̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )], t ∈ I,
k∈N
(14)
Generalized α-V-univex functions for multiobjective variational control problems
Z
Z
b
415
X
µj (t)gj (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt ≧ 0,
(15)
X
νk (t)hk (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt = 0,
(16)
a j∈M
b
a k∈N
λi ≧ 0,
X
λi = 1, µj (t) ≧ 0, j ∈ M, t ∈ I.
i∈P
Theorem 4 (Weak duality) Let (x, u) and (x◦ , u◦ , λ, µ(t), ν(t)) be the feasible
solutions to (CP) and (MD), respectively. If
Rb
(i) a λf (t, ·, ·, ·, ·)dt is α̂-V-pseudounivex at (x◦ , u◦ ) with respect to b◦ , φ◦ , α̂,
η and ξ;
Rb
(ii) a µ(t)g(t, ·, ·, ·, ·)dt is α̃-V-quasiunivex at (x◦ , u◦ ) with respect to b1 , φ1 , α̃,
η and ξ;
Rb
(iii) a ν(t)h(t, ·, ·, ·, ·)dt is α∗ -V-quasiunivex at (x◦ , u◦ ) with respect to b2 , φ2 , α∗ ,
η and ξ;
(iv) φ2 (0) = 0 and for any scalar function p(t),
Rb
Rb
φ◦ (p(t))dt = a p(t)dt,
a
Rb
Rb
a φ1 (p(t))dt = a p(t)dt;
(v) b◦ > 0, b1 > 0.
Then
Z
Z
b
b
f (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt.
f (t, x, ẋ, u, u̇)dt 6<
a
a
Proof. We proceed by contradiction. Suppose that
Z b
Z b
f (t, x, ẋ, u, u̇)dt <
f (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt,
a
which, by λi ≧ 0,
a
P
λi = 1, α̂i > 0, i ∈ P , implies
i∈P
Z
b
X
α̂i λi fi (t, x, ẋ, u, u̇)dt <
a i∈P
Z
b
X
α̂i λi fi (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt.
a i∈P
From the assumptions (iv), (v) and the above inequality, it follows that
Z b
X
X
b◦
φ◦ (
α̂i λi fi (t, x, ẋ, u, u̇) −
α̂i λi fi (t, x◦ , ẋ◦ , u◦ , u̇◦ ))dt < 0.
a
i∈P
i∈P
Therefore, by hypothesis (i) and inequality (17), we get
Z b X
X
[(
λi fix◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )−D
λi fiẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))η
a
i∈P
i∈P
(17)
416
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
X
X
+(
λi fiu◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) − D
λi fiu̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))ξ]dt < 0.
i∈P
(18)
i∈P
Now, from the feasibility of (x, u) and (x◦ , u◦ , λ, µ(t), ν(t)) to (CP) and (MD),
respectively, and ν 6= 0, we have
Z bX
Z bX
νk (t)hk (t, x, ẋ, u, u̇)dt =
νk (t)hk (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt.
(19)
a k∈N
a k∈N
Again, by hypothesis (iii), (iv) and (19), we have
Z b X
X
[(
νk (t)hkx◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )−D
νk (t)hkẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))η
a
+(
k∈N
X
k∈N
X
νk (t)hku◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) − D
k∈N
νk (t)hku̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))ξ]dt ≦ 0.
k∈N
(20)
On adding inequalities (18) and (20), we obtain
Z b X
X
[(
λi fix◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )+
νk (t)hkx◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
a
i∈P
k∈N
X
X
−D(
λi fiẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )+
νk (t)hkẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )))η
i∈P
k∈N
X
X
+(
λi fiu◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )+
νk (t)hku◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )
i∈P
k∈N
X
X
−D(
λi fiu̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) +
νk (t)hku̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ )))ξ]dt < 0.
i∈P
k∈N
The above inequality, together with relations (13) and (14), yields
Z b X
X
[(
µj (t)gjx◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) − D
µj (t)gj ẋ◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))η
a
j∈M
j∈M
+(
X
X
µj (t)gju◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ) − D
j∈M
µj (t)gj u̇◦ (t, x◦ , ẋ◦ , u◦ , u̇◦ ))ξ]dt > 0,
j∈M
which, along with the hypothesis (ii), gives
Z b
X
X
b1
φ1 (
α˜j µj (t)gj (t, x, ẋ, u, u̇) −
α˜j µj (t)gj (t, x◦ , ẋ◦ , u◦ , u̇◦ ))dt > 0.
a
j∈M
j∈M
(21)
Hence, it follows from inequality (21), assumptions (iv), (v) and α˜j > 0, j ∈ M ,
that
Z b X
X
(
µj (t)gj (t, x, ẋ, u, u̇) −
µj (t)gj (t, x◦ , ẋ◦ , u◦ , u̇◦ ))dt > 0.
(22)
a
j∈M
j∈M
Generalized α-V-univex functions for multiobjective variational control problems
417
On the other hand, from the feasibility of (x, u) and (x◦ , u◦ , λ, µ(t), ν(t)) to
(CP) and (MD), respectively, we have
Z bX
Z bX
µj (t)gj (t, x, ẋ, u, u̇)dt ≦
µj (t)gj (t, x◦ , ẋ◦ , u◦ , u̇◦ )dt,
a j∈M
a j∈M
which contradicts (22). This completes the proof.
◦
◦
Theorem 5 (Strong duality) Let (x , u ) be an efficient solution to (CP) at
which a constraint qualification is satisfied. Then there exist piecewise smooth
λ ∈ Rp , µ : I 7→ Rm and ν : I 7→ Rn such that (x◦ , u◦ , λ, µ(t), ν(t)) is feasible
for (MD). Furthermore, if weak duality (Theorem 4) holds between (CP) and
(MD), then (x◦ , u◦ , λ, µ(t), ν(t)) is an efficient solution of the problem (MD).
Proof. Since (x◦ , u◦ ) is an efficient solution for (CP), then from Lemma 2,
(x◦ , u◦ ) solves (CP)s . As (x◦ , u◦ ) satisfies the constraint qualification for (CP)s ,
it follows from Lemma 1 that there exist piecewise smooth λ̄ ∈ Rp−1 , µ̄ : I 7→ Rm
and ν̄ : I 7→ Rn such that for all t ∈ I,
fkx (t, x , x˙◦ , u◦ , u˙◦ ) +
◦
p
X
λ̄i fix (t, x◦ , x˙◦ , u◦ , u˙◦ )
i=1
i6=k
+
X
µ̄j gjx (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
j∈M
X
ν̄k hkx (t, x◦ , x˙◦ , u◦ , u˙◦ )
k∈N
h
= D fkẋ (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
p
X
λ̄i fiẋ (t, x◦ , x˙◦ , u◦ , u˙◦ )
i=1
i6=k
+
X
µ̄j gj ẋ (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
j∈M
X
i
ν̄k hkẋ (t, x◦ , x˙◦ , u◦ , u˙◦ ) ,
X
ν̄k hku (t, x◦ , x˙◦ , u◦ , u˙◦ )
k∈N
p
X
fku (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
λ̄i fiu (t, x◦ , x˙◦ , u◦ , u˙◦ )
i=1
i6=k
+
X
µ̄j gju (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
j∈M
h
k∈N
= D fku̇ (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
p
X
λ̄i fiu̇ (t, x◦ , x˙◦ , u◦ , u˙◦ )
i=1
i6=k
+
X
µ̄j gj u̇ (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
j∈M
Z
b
X
a j∈M
µ̄ ≥ 0.
µ̄j gj (t, x◦ , x˙◦ , u◦ , u˙◦ )dt = 0,
X
k∈N
i
ν̄k hku̇ (t, x◦ , x˙◦ , u◦ , u˙◦ ) ,
418
S. SHARMA, A. JAYSWAL, S. CHOUDHURY
1
α
Let
X
=1+
p
P
λ̄i . Then, we get
i=1
i6=k
λi fix (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
i∈P
+
X
µj gjx (t, x◦ , x˙◦ , u◦ , u˙◦ )
j∈M
X
νk hkx (t, x , x˙◦ , u , u˙◦ ) = D
◦
◦
k∈N
+
X
µj gj ẋ (t, x◦ , x˙◦ , u◦ , u˙◦ ) +
X
µj gju (t, x◦ , x˙◦ , u◦ , u˙◦ )
j∈M
X
νk hku (t, x , x˙◦ , u , u˙◦ ) = D
X
µj gj u̇ (t, x , x˙◦ , u , u˙◦ ) +
◦
◦
k∈N
+
b
hX
λi fiu̇ (t, x◦ , x˙◦ , u◦ , u˙◦ )
i∈P
◦
j∈M
Z
i
νk hkẋ (t, x◦ , x˙◦ , u◦ , u˙◦ ) ,
X
k∈N
λi fiu (t, x , x˙◦ , u◦ , u˙◦ ) +
◦
i∈P
+
λi fiẋ (t, x◦ , x˙◦ , u◦ , u˙◦ )
i∈P
j∈M
X
hX
X
◦
X
k∈N
i
νk hku̇ (t, x◦ , x˙◦ , u◦ , u˙◦ ) ,
µj gj (t, x◦ , x˙◦ , u◦ , u˙◦ )dt = 0,
a j∈M
µ ≥ 0,
P
P
P
P
P
where λk = α > 0, λi = α
λ̄i , i 6= k,
µj = α
µ̄j ,
νk = α
ν̄k .
i∈P
j∈M
j∈M
k∈N
k∈N
Rb P
Also we have a
νk hk (t, x◦ , x˙◦ , u◦ , u˙◦ )dt = 0. Therefore (x◦ , u◦ , λ, µ(t), ν(t))
k∈N
is a feasible solution for (MD).
Moreover, if we assume that (x◦ , u◦ , λ, µ(t), ν(t)) is not an efficient solution
to (MD), then there exists a feasible solution (x, u, λ◦ , µ◦ , ν ◦ ) to (MD) such that
Z
b
f (t, x, ẋ, u, u̇)dt ≥
a
Z
b
f (t, x◦ , x˙◦ , u◦ , u˙◦ )dt,
a
which contradicts the weak duality Theorem 4. Hence, (x◦ , u◦ , λ, µ(t), ν(t)) is
an efficient solution to (MD).
5.
Conclusion
In this paper, we have considered a multiobjective variational control problem
and its Mond-Weir type dual problem. Using the concept of efficiency, weak and
strong duality theorems have been proved under the assumptions of generalized
α-V-univexity. There is a rich scope to extend these notions to the class of nondifferentiable multiobjective variational problems. This will orient the future
research of the authors.
Generalized α-V-univex functions for multiobjective variational control problems
419
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