Further generalizations of Farkas' theorem and their applications in optimal control
A set-valued generalization is given for Farkas' theorem in Banach spaces. If \(S\) and \(K\) are convex cones, \(A\) linear, and \(T\) a convex process with \(T(S)\) closed, then \[ [(\forall x\in K)(\exists z\in S) Ax\in T(z)]\Leftrightarrow [T^*(y^*)\cap S^*\neq\emptyset\Rightarrow A^*y^*\in K^*]. \] A generalization is given without the closedness assumption on \((T(S))\). The generalized Gale alternative theorem in nonlinear programming is a corollary. The results are applied to constrained controllability theory in Banach spaces, as well as to some multiobjective optimization problems.
scientific article; zbMATH DE number 1115520
scientific article