Some Aspects Of Duality In Variational Problems And Optimal Control

This thesis is divided into six chapters. In the Ist chapter we present a brief survey of related work done in the area of multiobjective mathematical programming, optimal control and game theory. Chapter Two: In this chapter sufficient optimality criteria are derived for a control problem under generalized invexity. A Mond-Weir type dual to the control problem is proposed and various duality theorems are validated under generalized invexity assumptions on functionals appearing in the problems. It is pointed out that these results can be applied to the control problem with free boundary conditions and have linkage with results for nonlinear programming problems in the presence of inequality and equality constraints already established in the literature. Chapter Three: In this chapter a mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results. Chapter Four: In this chapter two types of duals are considered for a class of variational problems involving higher order derivative. The duality results are derived without any use of optimality conditions. One set of results is based on Mond-Weir type dual that has the same objective functional as the primal problem but different constraints. The second set of results is based on a dual of an auxiliary primal with single objective function. Under various convexity and generalized convexity assumptions, duality relationships between primal and its various duals are established. Problems with natural boundary values are considered and the analogues of our results in nonlinear programming are also indicated. Chapter Five: In this chapter a certain constrained dynamic game is shown to be equivalent to a pair of symmetric dual variational problems which have more general formulation than those already existing in the literature. Various duality results are proved under convexity and generalized convexity assumptions on the appropriate functional. The dynamic game is also viewed as equivalent to a pair of dual variational problems without the condition of fixed points. It is also indicated that our equivalent formulation of a pair of symmetric dual variational problems as dynamic generalization of those already studied in the literature. Chapter Six: In this chapter a mixed type second-order dual to a variational problem is formulated as a unification of Wolfe and Mond-Weir type dual problems already treated in the literature and various duality results are validated under generalized second order invexity. Problems with natural boundary values are formulated and it also is pointed out that our duality results can be regarded as dynamic generalizations of those of (static) nonlinear programming. The subject matter of the present research thesis is fully published in the form of the following research papers written by the author: (1) Sufficiency and Duality In Control Problems with Generalized Invexity, Journal of Applied Analysis,Vol, 14 No. 1 (2008),pp.27-42. (2) Mixed Type Duality for Control Problems with Generalized Invexity, Journal of Applied Mathematics and Informatics,Vol. 26(2008), No.5-6 , pp. 819-837. (3) On Multiobjective Duality for Variational Problems, The Open Operational Research Journal,2012, 6, 1-8. (4) Constrained Dynamic Game and Symmetric Duality For Variational Problems, Journal of Mathematics and System Science 2(2012), 171-178. (5) Mixed Type Second – Order Dulaity For Variational Problems, Journal of Informatics and Mathematical Sciences , Vol5,No.1, pp.1-13,(2013)