In the paper the following optimization problem is considered: Let X be a Banach space, Q1,...,QnℝX, intQ1,...,intQp=ab42/0ab21/, intQpÅ+1,...,intQn=0ab21/, Ii:X⇒Rµ, i=1,...,s, I(x)=(I1(x),...,Is(x))’ℝµ is a vector performance index. We are looking for a point x0∈Q:¸=ℝnab21kÅ=1Qk such that I(x0)=minxÅ∈QÅℝUÅ(xÅ)I(x) in the sense of Pareto where U(x) denotes some neighbourhood of x. Making use of results of Walczak and Censor we generalize the well-known Dubovicki-Milutin Theorem and then we apply it to the optimization problem obtaining a necessary condition for local Pareto optimum. For convex problems a local Pareto optimum is also a global Pareto one. If additional assumptions are fulfilled for convex problems i.e. Ii, i=1,...,s, are continuous and Ponstein convex and the so-called Slater’s condition is satisfied, then the Euler-Lagrange Equation gives a sufficient condition for global Pareto optimum. In the paper an example of a Pareto optimal control problem is also given. A dynamical cascade system is described by two partial differential equations of parabolic type in a Sobolev space.
