On Efficiency Conditions for Multiobjective Variational Problems Involving Higher Order Derivatives∗

This paper aims to formulate and prove necessary and sufficient conditions of efficiency for a class of multiobjective variational problems involving higher order derivatives. Consider a multiobjective optimization problem of minimizing a vector of simple integral functionals subject to certain higher order differential equations and/or inequations. We establish sufficient efficiency conditions for a feasible solution using the notion of quasiinvexity. Key–Words: efficient solution, quasi-invexity, multiobjective variational problem. 1 Our framework and problem describtion In this work we extend and further develop some optimization results connected to the efficiency of a feasible solution for a class of multiobjective nonfractional programming problems. We introduce and perform a study on the multiobjective variational problem of minimizing a vector of simple integral functionals (MVP) constrained by higher order differential equations and inequations. This paper is strongly motivated by its applications in natural phenomena and in wide areas of research for new technology as well, where there are needed derivatives of order higher than one or two (engineering, chemistry, games theory, etc.) The passing from the first order derivatives to the higher order derivatives is not a facile task because it requests specific techniques, a new quasiinvexity and an appropriate mathematical framework. In time, several authors have been interested in the study of vector programming problems which involve a generalized convexity (see [4]-[6], [8]). Thus, many of them extend this notion and develop a multitime optimization theory, using a geometrical language (see [7]). For other different ideas but connected to this subject, we address the readers to the works [1]-[3] and [9]. Before presenting our results, ∗15th WSEAS International Conference on Automatic Control, Modelling & Simulation (ACMOS-13), Brasov, Romania, June 1-3, 2013. for the completeness of the exposition, we set the following notations. Let consider the real interval I := [t0, t1] ⊆ R and f = (fα) : I ×R → R, α = 1, p, (f1(t, x(t), x (t), ..., x(t)), ..., fp(t, x(t), x (t), ..., x(t))), a C-class function, where x(t) := dk dtk x(t), with k ≥ 1 a fixed natural number. Also, let be given g = (g1, ..., gm) : I × R → R, with m < n, and h = (h1, ..., hr) : I×R → R, with r < n, two C-class functions. Assume that the previous C-class Lagrangians, fα(t, x(t), x (t), ..., x(t)), α = 1, p, generate the simple integral functionals Fα (x(·)) := ∫ t1 t0 fα(t, x(t), x (t), ..., x(t))dt, α = 1, p. Let C∞ ([t0, t1], R) be the space of all functions x : [t0, t1] → R of C∞-class, with the norm