KT‐pseudoinvex multidimensional control problem

Abstract

Summary In this paper, we define a Kuhn‐Tucker (KT)–pseudoinvex multidimensional control problem. More exactly, we introduce a new condition on the functions, which are involved in a multidimensional control problem, and we prove that a KT‐pseudoinvex multidimensional control problem is characterized such that a KT point is an optimal solution. Thus, we generalize optimality results in known mathematical programming problems. These theoretical results are illustrated with an application.

Necessary Optimality Conditions in Isoperimetric Constrained Optimal Control Problems

“…In the following, let us consider the relations formulated in (4). Now, we fix the control variable u(t) and the associated solution y(t) of (4).…”

“…Currently, the optimal control theory is under continuous development. This theory is based on the optimization of some functionals with ordinary differential equations/partial differential equations (in short, ODE/PDE) constraints, all depending on the control functions (for variational control problems with first-order PDE constraints, the reader is directed to Mititelu and Treanţȃ [1], Treanţȃ [2,3], and Treanţȃ and Arana-Jiménez [4,5]). There are three approaches: variational calculus, the maximum principle, and dynamic programming.…”

Characterization of efficient solutions for a class of PDE-constrained vector control problems

“…The condition of invexity formulated in Mond and Smart [13] is not a necessary condition in order that all Kuhn-Tucker critical points to be optimal solutions of a scalar control problem (see, also, Arana-Jiménez et al [2], Treanţȃ and Arana-Jiménez [25,26]). In this section, in accordance to Mond and Smart [13], Martin [10], following Treanţȃ and Arana-Jiménez [25,26], we contribute an invexity condition that is characterized such that all Kuhn-Tucker points are efficient solutions in (M V CP ). More exactly, we introduce the notion of V-KT-pseudoinvexity associated with the multidimensional vector control problem (M V CP ).…”

“…Further, in accordance to Mititelu and Treanţȃ [11], following Treanţȃ and Arana-Jiménez [25,26], we introduce the notion of Kuhn-Tucker point associated with (M V CP ).…”

“…The finding of solutions in scalar and vector control problems means the study of efficiency conditions and of functionals (functions) that are involved. Quite recently, Treanţȃ and Arana-Jiménez [25,26] have improved the result formulated by Mond and Smart [13], in accordance to Martin [10], following Arana-Jiménez et al [2], by introducing the notion of KT-pseudoinvexity (associated with a scalar control problem of minimizing a multiple integral cost functional subject to nonlinear equality and inequality constraints involving first-order partial derivatives) as a necessary and sufficient condition in order that all Kuhn-Tucker points to be optimal solutions. In addition, as a consequence, they have proved that the class of invex functions or functionals is equivalent to the class of pseudoinvex functions or functionals.…”