Pontryagin’s Maximum Principle for Multidimensional Control Problems

Abstract

A weak maximum principle is shown for general problems

$${\text{minimize}}\,f\left( {x,{\text{ }}w} \right)\,\,{\text{on }}{X_0} \times {X_{\text{1}}}\,{\text{with respect to}}\,linear\,{\text{state constraints}}\,{A_0}x = {A_{\text{1}}}w$$

in Banach spaces X ₀ and local convex topological vector spaces X ₁, where f(x, •) is a convex functional on X ₁ and X _j are linear and continuous operators from X _j to a Hilbert space X (j = 0,1). The proved theorem is applied to Dieudonné-Rashevsky-type and relaxed control problems.