Sufficient Fritz John Type Optimality Criteria and Duality for Control Problems ()
1. Introduction
Optimal control models represent a variety of common situations, notably, advertising investment, production and inventory, epidemic, control of a rocket, etc. The optimal planning of a river system which is an invincible resource of nature, where it is needed to make the best use of the water, can also be modelled as an optimal control problem. Optimal control models are also potentially applicable to economic planning and to the world models of the “Limits to Growth” kind in general.
Optimality criteria for any optimization problem are of great significance and lay the foundation of the concept of duality. Fritz John optimality criteria for a control problem were first derived by Berkovitz [1]. Subsequently Mond and Hanson [2], who first investigated duality in optimal control pointed out that from Fritz John optimal criteria, Karush-Kuhn-Tucker optimality criteria can be deduced if normality of the solution of a control problem which replaces a regularity conditions is assumed. Later, treating a nondifferentiable control problem as a nondifferentiable mathematical programming problem in an infinite-dimensional space, Chandra et al. [3], obtained Fritz John as well as Karush-Kuhn-Tucker optimality criteria.
For a nondifferentiable control problem Using KarushKuhn-Tucker optimality criteria, they formulated Wolfe type dual and derived usual duality results under appropriate convexity assumptions.
In this research exposition, sufficient Fritz John criteria are derived for a differentiable control problem in which objective functional is pseudoconvex and constraint functions are quasiconvex or semi-strictly pseudoconvex. A number of duality results are proved for relating the solution of the control problem with that of its proposed dual under suitable generalized convexity requirements. The relationship of our duality results to those of a nonlinear programming problem is indicated.
2. Control Problem and Related Preliminaries
Let
denotes a n-dimensional Euclidean space, 
be a real interval and 
be a continuously differentiable with respect to each of its arguments. For the function 
where 
is differentiable with its derivative 
and 
is the smooth function, denote the partial derivatives of 
by
, 
and
, where
For m-dimensional vector function 
the gredient with respect to 
is
a 
matrix of first order derivatives.
Here 
is the control variable and 
is the state variable, 
is related to 
via the state equation
. Gradients with respect to 
are defined analogously.
A control problem is to transfer the state vector from an initial state 
to a final state 
so as to minimize a functional, subject to constraints on the control and state variables.
A control problem can be stated formally as(CP): 
subject to
(1)
(2)
(3)
1) 
is as before, 
and 
are continuously differentiable functions with respect to each of its arguments.
2) X is the space of continuously differentiable state functions 
such that 
equipped with the norm
, and 
is the space of piecewise continuous control functions 
has the uniform norm 
and 3) The differential Equation (2) for 
with the initial conditions expressed as 
may be written as 
where the map 
being the space of continuous functions from
, defined by 
Following Craven [4], the control problem can be expressed as(ECP): 
subject to
where 
is function from 
into 
given by 
from
, and
; 
is the convex cone of functions in 
whose components are non-negative; thus 
has interior points.
Necessary optimality conditions for existence of extermal solution for a variational problem subject to both equality and inequality constraints were given by valentine [5]. Invoking Valentine’s [5] results, Berkovitz [1] obtained corresponding necessary optimality criteria for the above control problem (CP). Here we state the Fritz John type optimality conditions derived by Chandra et al. [3] in of the following proposition which will be required in the sequel.
Proposition 1
(Necessary Optimality Conditions)
If 
an optimal solution of (CP) and the Frechet derivatives 
is surjective, then there exist Lagrange multipliers
, and piecewise smooth functions 
and 
satisfying, for all
,
(4)
(5)
(6)
(7)
(8)
The above conditions will become Karush-Kuhn-Tucker conditions if
. Therefore, if we assume that the optimal solutions 
is normal, then without any loss of generality, we can set
. Thus from the above we have the Karush-Kuhn-Tucker type optimality conditions
Using these optimality conditions, Mond and Hanson [2] constructed following Wolfe type dual.
(CD): 
subject to
In [6], [CP] and (CD) are shown to from a dual pair if
, 
and 
are all convex in 
and
. Subsequently, Mond and Smart [6] extended this duality under generalized invexity.
As a follows up, Husain et al. [7] formulated the following dual (CD) to the primal problem (CP) in the spirit of Mond and Weir [8].
(CD): Maximize 
subject to
They proved sufficiency of the optimality criteria and duality for the pair of dual problems (CP) and (CD) under pseudoinvexity of 
and quasi-invexity of
.
3. Sufficiency of Fritz Type Optimality Criteria
Before proceeding to the main results of this section, we formulate the following definitions which will be required in the forthcoming analysis:
Definitions: 1) For
the functional 
is said to be strict pseudoconvex, if all 
Equivalently
2) 
the functional 
is semistrictly pseudoconvex if 
is strictly pseudoconvex for all
If 
and 
are independent of t and u then the above definitions reduce to those of [6].
Theorem 1 (Sufficiency): If 
is pseudoconvex,
is semi-strictly pseudoconvex and 
is quasiconvex, and if there exist
and piecewise smooth 
and 
such that from (4)-(8) are satisfied, then 
is an optimal solution of (CP).
Proof: Suppose that 
is not optimal for (CP) i.e. there exist 
such that
This, by pseudoconvexity of 
implies
and
(9)
with strict inequality in (9) if
.
Feasibility of 
for (CP) together with (6) implies,
which by semi-strict pseudoconvexity of 
implies
(10)
with strict inequality in (10) if some
.
Also
This, in view of quasiconvexity of 
yields
(By integrating by parts)
(11)
(Using (1))
Combining (9)-(11), we have
This contradicts (4) and (5). Hence 
is an optimal solution of (CP).
4. Fritz Type Duality
The following is the Fritz john type dual to the problem (CP):
Maximize 
subject to
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Theorem 2 (Weak Duality): Assume that
(A1) satisfies 
is feasible for (CP) and 
is feasible for (FrCD).
(A2): 
is pseudo-convex,
is semi-strictly pseudo-convex and
is quasi-convex.
Then
Proof: Suppose
This, because of pseudo-convexity of 
yields 
and
(19)
with strict inequality in the above with
. From the constraints of (CP) and (FrCD), we have
which by semi-strictly pseudo-convexity of
implying
(20)
with strict inequality with
, 
Also, we have
Using quasi-convexity of 
in the above, we have
(21)
which as earlier becomes
Combining (19)-(21), we have
(22)
From (13) and (14), we get
i.e.
(23)
The relation (22) and (23) are in contradiction, thus
Implying
Theorem 3 (Strong Duality): If
is an optimal solution of (CP), then there exist 
and piecewise smooth 
and 
such that 
is feasible for (FrCD) and objective values are equal. If hypotheses of Theorem 2 hold, then 
is an optimal solution of (FrCD).
Proof: Since
is an optimal solution of (CP) by Proposition 1, there exist
, piecewise smooth 
and 
such that
(24)
(25)
(26)
(27)
(28)
(29)
(30)
The relation (26) implies
(31)
and the relation (28) along 
gives
(32)
The relation (24), (25), (29)-(32), yields the feasibility of 
for (FrCD). Equality of objective functionals of (CP) and (FrCD) is obvious from their formulations.
Consequently the optimality for (FrCD) follows, given the pseudo-convexity of the
semi-strict pseudoconvexity of 
and quasi-convexity of 
by Theorem 2.
Theorem 4 (Strict-Converse duality): Assume that
(A1): 
is strictly pseudo-convex, 
is semi-strictly pseudo-convex and 
is quasi-convex.
(A2): 
is an optimal solution of (CP) and
(A3): 
is an optimal solution of (FrCD).
Then 
is an optimal solution of (CP) with
.
Proof: we suppose 
and exhibit a contradiction. Since 
is an optimal solution of (CP) by theorem (Strong Duality) that there exist 
where 
and piecewise smooth 
and piecewise smooth 
and 
such that 
is also an optimal solution for (FrCD), it follows that
By strict pseudo-convexity of
gives, this implies
and multiplying the above by 
(33)
with strict inequality if 
From the constraints of (CP) and (FrCD), we have
(34)
Also
(35)
By semi-strict pseudoconvexity of
and from (34), we have
(36)
with strict inequality in the above if,
By quasi-convexity of 
and from (35), we get
(37)
As earlier, this reduces to
combining (33), (36), and (37), we have
This contradicts the feasibility of 
for (FrCD), hence 
is an optimal solution of (CP) and
.
Theorem 5 (Converse duality): Let 
be an optimal solution of (FrCD), Assume
(A1) 
is pseudo-convex, 
is semi-strictly pseudo-convex and 
is quasiconvex.
(A2) The set 
or
is linearly independent.
(A3) 
for some column vector 
and where
and (A4) 
Then 
is optimal for (CP).
Proof: By Proposition 1, there exist 
piecewise smooth 
and 
such that
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
Multiplying (41) by 
and integrating, and then using (43) and (46), we have
which can be written as
(49)
Multiplying (42) by
and then integrating we get
using (44), this yields
(by integrating by parts)
which in view of (A4), implies
This can be written as
(50)
Using (13) in (38) and (14) in (39), we have
These can be combined as
Pre-multiplying this by 
and then integrating we have
Using (49) and (50) , this implies
which in view of (A2) implies
implies
(51)
In view of (A3), the equality constraint implies 
Consequently, we have
(52)
Using (52), along with
, we have
This, in view of (A3),
(53)
If 
(53), implies
. Thus
contradiction.
Hence 
and consequently 
and
.
From (41) and (42), we have
These relations yield the feasibility of 
for (CP) and objective functionals of (CP) and (FrCD) are equal there. Hence under the stated convexity hypotheses, by Theorem 2, 
is an optimal solution of (CP).
5. Mathematical Programming Problems
If the problems (CP) and (FrCD) are independent of t and x, these problems reduce to essentially to the static cases of nonlinear programming problem. Letting
, the problems (CP) and (FrCD) become the pair of dual nonlinear programming problems formulated by Husain and Srivastav [9].
(CD0): Minimize 
subject to
(FrCD): Maximize 
subject to
where
is pseudoconvex, 
is semi-strictly pseudoconvex and 
is quasi-convex. If only inequality constraint in (CD0) is given, then (CP0) and (FrCD0) become a pair of dual the nonlinear programming problems considered by Weir and Mond [10].
6. Conclusion
In this paper, sufficient optimality conditions are derived for a control problem which appears in various real life situations under generalized convexity assumptions. In order to formulate the dual to this control problem, Fritz John optimality conditions are used instead of KarushKuhn-Tucker optimality condition and hence the requirement of regularity condition is eliminated. Various duality results are obtained and the linkage of our duality results to those of a nonlinear programming problem is indicated. Our results can be seen in the setting of multiobjective control problems.
7. Acknowledgements
The authors acknowledge anonymous referees for their valuable comments which have improved the presentation of this research paper.