Consider the following control problem containing support functions introduced by Husain et al. [1]

subject to
(1)
(2)
(3)
where
1)
is a differentiable state vector function with its derivative
and
is a smooth control vector function.
2)
denotes an n-dimensional Euclidean space and
is a real interval.
3)
,
and
are continuously differentiable.
4)
and
,
are the support function of the compact set K and
respectively.
Denote the partial derivatives of f where by ft, fx and ft,
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where superscript denote the vector components. Similarly we have ht, hx, hu and gt, gx, gu. X is the space of continuously differentiable state functions
. Such that
and
and are equipped with
the norm
and U, the space of piecewise continuous control vector functions ![]()
having the uniform norm
The differential Equation (2) with initial conditions expressed as
may be written as
where
being the space of continuous function from I to Rn defined as
In the derivation of these optimality condition, some constraint qualification to make the equality constraint locally solvable [2] and hence the Fréchét derivative of
(say) with respect to
namely
are required to be surjective. In [1] , Husain et al. derived the following Fritz john type necessary optimality for the existence of optimal solution of (CP).
Proposition 1. (Fritz John Condition): If
is an optimal solution of (CP) and the Fréchét derivative Q' is surjective, then there exist Langrange multipliers
and piecewise smooth
,
,
and
such that for all t,
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As in [3] , Husain et al. [1] pointed out if the optimal solution for (CP) is normal, then the Fritz john type optimal conditions reduce to the following Karush-Kuhn-Tucker optimal conditions.
Proposition 2. If
is an optimal solution and is normal and Q' is surjective, there exist piecewise smooth
with
,
,
and
,
such that
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Using the Karush-Kuhn-Tucker type optimality condition given in Proposition 2, Husain et al. [1] presented the following Wolfe type dual to the control problem (CP) and proved usual duality theorem under the pseudo-
convexity of
for all
, and
,
.
(WCD): Maximize
![]()
subject to
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We review some well known facts about a support function for easy reference. Let
be a compact convex set in
. Then the support function of
denoted by
is defined as
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A support function, being convex and everywhere finite, has a subdifferential in the sense of convex analysis, that is, there exists
such that
for all x. The subdif-
ferential of
is given by
Let
be normal
cone at a point
Then
if and only if
or, equivalently,
is in the subdifferential of s at ![]()
In order to relax the pseudoconvexity in [1] , Mond-Weir type dual to (CP) is constructed and various duality theorems are derived. Particular cases are deduced and it is also indicated that our results can be considered as the dynamic generalization of the duality results for nonlinear programming problem with support functions.
2. Mond-Weir Type Duality
We propose the following Mond-Weir type dual (M-WCD) to the control problem (CP):
Dual (M-WCD): Maximize ![]()
subject to
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Theorem 1. (Weak Duality): Assume that
(A1):
is feasible for (CP),
(A2):
is feasible for the problem (M-WCD),
(A3):
for
is pseudoconvex, and
(A4):
for all
and
are quasiconvex at ![]()
Then
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Proof: Since
we have
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and
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Combining these inequalities with (14) and (15) respectively, we have
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and
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These, because of the hypothesis (A4) yields
(19)
(20)
Combining (19) and (20) and then using (12) and (13), we have
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This, due to the pseudoconvexity of
for
implies
![]()
Since
the above inequality gives
![]()
yielding
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Theorem 2. (Strong Duality): If
is an optimal solution of (CP) and is normal, then there exist piecewise smooth
with
and such that
is feasible for (M-WCD) and the corresponding values of (CP) and (M-WCD) are equal. If also, the hypotheses of Theorem 1 hold, then
is optimal solution of the problem (M-WCD).
Proof: Since
is an optimal solution of (CP) and is normal, it follows by Proposition 2 that there exist piecewise smooth
and
. satisfying for all
the conditions (4)-(10) are satisfied. The conditions (4)-(6) together with (9) and (10) imply that
is feasible for (M-WCD). Using
we obtain,
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The equality of the objective functionals of the problems (CP) and (M-WCD) follows. This along with the hypotheses of Theorem 1, the optimality of
for (M-WCD) follows.
The following gives the Mangasarian type strict converse duality theorem:
Theorem 3. (Strict Converse Duality): Assume that
(A1):
is an optimality solution of (CP) and is normal;
(A2):
is an optimal solution of (M-WCD),
(A3):
in strictly is pseudoconvex for all
and
(A4):
for all
and
are quasi convex.
Then
i.e.
is an optimal solution of (CP).
Proof: Assume that
and exhibit a contradiction. Since
is an optimality solution of
(CP). By Theorem 2 there exist
with
such that
is an optimal solution of (M-WCD).
Thus
(21)
Since
is feasible for (CP) and
for (M-WCD), we have
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and
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These, because of the hypothesis (A4) imply the merged inequality
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This, by using the equality constraints (12) and (13) of (M-WCD) gives
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By the hypothesis (A2), this implies
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(using (21)). Consequently, we have
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Since
for
and
for
this yields,
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This cannot happen. Hence ![]()
3. Converse Duality
The problem (M-WCD) can be written as the follows:
Maximize: ![]()
subject to
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where
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Consider
and
as defining a map- pings
and
respectively where
is the space of piecewise smooth
, V is space of piececewise smooth
, Wj is the space of piecewise of smooth Wj,
B1 and B2 are Banach spaces.
and
with
Here some restrictions are required on the equality constraints. For this, it suffices that if the
derivatives
and
have weak * closed range.
Theorem 4. (Converse Duality): Assume that
(A1):
and h are twice continuously differentiable.
(A2):
is an optimal solution of (CP).
(A3):
and
have weak * closed ranges.
(A4): ![]()
for some
, and
![]()
(A5): 1) The gradient vectors
and
are linearly independent, or
2) The gradient vectors
and
are linearly independent.
(A6): ![]()
Proof: Since
is an optimal solution of (M-WCD), by Proposition 1 there exists
and
and piecewise smooth functions
,
, such that
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
Multiplying (24) by
and summing over i and then integrating using (28), we have
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which can be written as,
(33)
Multiplying (25) by
and then integrating and using (29), we have
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This implies
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or
(34)
Using the equality constraints (12) and (13) of the problem (M-WCD) in (22) and (23), we have
(35)
(36)
Combining (35) and (36), we have
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This by premultiplying by
and then integrating, we have
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Using (33) and (34), we have
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This because of hypothesis (A4) implies
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Using
gives
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This, because of hypothesis (A5) implies
(37)
Assume
(37) gives
from (24) it follows
Consequently we have
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contradicting (32). Hence
and
The relations (26) and (27) gives
and ![]()
yielding
and
.
From (24), we have
(38)
and
(39)
From (25), we have
(40)
and
(41)
The feasibility of
for (CP) follows from (38) and (40).
Consider
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(by using
along (39) and (41)).
This along with the generalized convexity hypotheses implies that
is an optimal solution of (M-WCD).
4. Special Cases
Let for
and
be positive semidefinite matrices and continuous on I. Then
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where
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and
![]()
where
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Replacing the support function by their corresponding square root of a quadratic form, we have:
Primal (CP0): Minimize ![]()
subject to
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(M-WCD0): Maximize ![]()
subject to
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The above pair of nondifferentiable dual control problem has not been explicitly reported in the literature but the duality amongst (CP0) and (M-WCD0) readily follows on the lines of the analysis of the preceding section.
5. Related Nonlinear Programming Problems
If the time dependency of the problem (CP) and (M-WCD) is removed, then these problems reduce to the following problem (NP), its Mond-Weir dual (M-WND):
Primal (NP0): Minimize ![]()
subject to
![]()
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Dual (M-WND0): Maximize ![]()
subject to
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The above nonlinear programming problems with support functions do not appear in the literature. However, if
and
are replaced by
and
respectively in (NP0), then problems reduced to following studied by Hussain et al. [4] .
(P1): Minimize ![]()
subject to
![]()
(M-WCD): Maximize ![]()
subject to
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6. Conclusion
Mond-Weir type duality for a control problem having support functions is studied under generalized convexity assumptions. Special cases are deduced. The linkage between the results of this research and those of nonlinear programming problem with support functions is indicated. The problem of this research can be revisited in multiobjective setting.