Dynamic optimization theory with multiple objectives

Let V(t) be a vector-valued function for t belonging to closed interval a,b open interval, a real interval. The main purpose of this paper is to establish the existence of a closed interval alpha,beta contained in closed interval a,b for which there exists a t(sub O) belonging to closed interval alpha,beta contained in closed interval a,b such that V(t(sub O)) = 0, the zero vector. Use of such information in the dynamic optimization theory with multiple objectives present is needed. Examples of such systems will be given.