ABSTRACT

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.

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

Letdenotes 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 (F_rCD).

(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 (F_rCD), 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): Ifis an optimal solution of (CP), then there exist and piecewise smooth and such that is feasible for (F_rCD) and objective values are equal. If hypotheses of Theorem 2 hold, then is an optimal solution of (F_rCD).

Proof: Sinceis 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 (F_rCD). Equality of objective functionals of (CP) and (F_rCD) is obvious from their formulations.

Consequently the optimality for (F_rCD) follows, given the pseudo-convexity of thesemi-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 (F_rCD).

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 ofand 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 (F_rCD), hence is an optimal solution of (CP) and.

Theorem 5 (Converse duality): Let be an optimal solution of (F_rCD), 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) byand 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 (F_rCD) are independent of t and x, these problems reduce to essentially to the static cases of nonlinear programming problem. Letting, the problems (CP) and (F_rCD) become the pair of dual nonlinear programming problems formulated by Husain and Srivastav [9].

(CD₀): Minimize

subject to

(FrCD): Maximize

subject to

whereis pseudoconvex, is semi-strictly pseudoconvex and is quasi-convex. If only inequality constraint in (CD₀) is given, then (CP₀) and (F_rCD₀) 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.

REFERENCES