Further Generalizations of Farkas’ Theorem and Their Applications in Optimal Control*

This paper deals with some new generalizations of Farkas’ theorem for a class of set-valued mappings with arbitrary convex cones in infinite-dimensional Banach spaces. A modified Farkas’ theorem with no closedness assumption is given. The generalized Gale alternative theorem in nonlinear programming is derived as an easy consequence. The results are applied to constrained controllability theory in Banach spaces as well as to some multiobjective optimization problem