Sufficient Fritz John Type Optimality Criteria and Duality for Control Problems

Sufficient Fritz John optimality conditions are obtained for a control problem in which objective functional is pseudoconvex and constraint functions are quasiconvex or semi-strictly quasiconvex. A dual to the control problem is formulated using Fritz John type optimality criteria instead of Karush-Kuhn-Tucker optimality criteria and hence does not require a regularity condition. Various duality results amongst the control problem and its proposed dual are validated under suitable generalized convexity requirements. The relationship of our duality results to those of a nonlinear programming problem is also briefly outlined.


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 Karush-Kuhn-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.

Control Problem and Related Preliminaries
Let denotes a n-dimensional Euclidean space, n R   , I a b  be a real interval and :  be a continuously differentiable with respect to each of its arguments.For the function   , , f t x u where : n x I  R is differentiable with its derivative x and is the smooth function, denote the partial derivatives of : .
1) f 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 : equipped with the norm x x Dx   , and is the space of piecewise continuous control functions has the uniform norm 3) The differential Equation (2) for

 and
x with the initial conditions expressed as where the map , defined by

  
, , H x u h t x t  u t Following Craven [4], the control problem can be expressed as, (ECP): , and ; is the convex cone of func- whose components are non-negative; thus has interior points.S 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)
an optimal solution of (CP) and the Frechet derivatives x rf t x u y t g t x u The above conditions will become Karush-Kuhn-Tucker conditions if .Therefore, if we assume that the optimal solutions 0 r    , x u is normal, then without any loss of generality, we can set .Thus from the above we have the Karush-Kuhn-Tucker type optimality conditions In [6], [CP] and (CD) are shown to from a dual pair if f , g and are all convex in h x and .Subsequently, Mond and Smart [6] extended this duality under generalized invexity.u 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]. .

u y t g t x u z t h t x u z t t I f t x u y t g t x u z t h t x u t I y t g t x u z t h t x u x t y t t I
They proved sufficiency of the optimality criteria and duality for the pair of dual problems (CP) and (CD) under pseudoinvexity of

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 : the functional For : , , , g t x u are independent of t and u then the above definitions reduce to those of [6].
Theorem 1 (Sufficiency): is quasiconvex, and if there exist r R  and piecewise smooth : such that from ( 4)-( 8) are satisfied, then   , x u is an optimal solution of (CP).
Proof: Suppose that   , x u is not optimal for (CP) i.e.

x rf t x u y t g t x u z t h t x u z t u u rf t x u y t g t x u z t h t x u z t h t x u t
This contradicts (4) and (5).Hence   , x u is an op- timal solution of (CP).

Fritz Type Duality
The following is the Fritz john type dual to the problem (CP): This, because of pseudo-convexity of

 
, , d , , , , , , d 0 with strict inequality in the above with .From the constraints of (CP) and (F r CD), we have with strict inequality with which as earlier becomes From ( 13) and ( 14), we get The relation ( 22) and ( 23) are in contradiction, thus

 
, , , , x u r y z is an optimal solution of (F r CD).
Proof: Since   , x u is an optimal solution of (CP) by Proposition 1, there exist r R  , piecewise smooth and such that : T T 0, The relation (26) implies
Consequently the optimality for (F r CD) follows, given the pseudo-convexity of the d , , x u is an optimal solution of (CP) and (A 3 ):   , , , , x u r y z is an optimal solution of (F r CD).
Then   , x u is an optimal solution of (CP) with and exhibit a contradiction.Since   0 0 , x u is an optimal solution of (CP) by theorem (Strong Duality) that there exist ,t I  r y t z t  where and piecewise smooth and piecewise smooth and such that r R  : , , , , x u r y z is also an optimal solution for (FrCD), it follows that By strict pseudo-convexity of d with strict inequality if From the constraints of (CP) and (FrCD), we have By semi-strict pseudoconvexity of and from (34), we have with strict inequality in the above if, This contradicts the feasibility of   , , , , x u r y z for (F r CD), hence   , x u is an optimal solution of (CP) and By quasi-convexity of and from (35), we get x r y z be an optimal solution of (F r CD), Assume  d y t g z t h z t  or combining (33), (36), and (37), we have   , x u is optimal for (CP).Then Multiplying (41) by   y t and integrating, and then using ( 43) and (46), we have which can be written as Multiplying (42) by  , z t and then integrating we get This can be written as Using ( 13) in ( 38) and ( 14) in (39), we have These can be combined as and then integrating we have Using (49) and (50) , this implies In view of (A 3 ), the equality constraint implies .Consequently, we have Using (52), along with , we have This, in view of (A 3 ), These relations yield the feasibility of  ,  x u for (CP) and objective functionals of (CP) and (FrCD) are equal there.Hence under the stated convexity hypotheses, by Theorem 2,   , x u is an optimal solution of (CP).

Mathematical Programming Problems
If the problems (CP) and (F r CD) are independent of t and x, these problems reduce to essentially to the static cases of nonlinear programming problem.Letting where f is pseudoconvex,

 
T y g  is semi-strictly pseudoconvex and is quasi-convex.If only inequality constraint in (CD 0 ) is given, then (CP 0 ) and (F r CD 0 ) become a pair of dual the nonlinear programming problems considered by Weir and Mond [10].
T z h

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 Karush-Kuhn-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.
y z is feasible for (F r CD) and objective values are equal.If hypotheses of Theorem 2 hold, then problems (CP) and (F r CD) become the pair of dual nonlinear programming problems formulated by Husain and Srivastav[9].(CD0 ): Minimize   of first order derivatives. . (CD):