New Representation Theorems for Consistent Flows

The notion of a generalized flow with a finite boundary is introduced. It is shown that the generalized flows which may be represented as mixtures of generalized curves are precisely those with finite boundaries. In the case of a unidirectional flow with finite boundary, the representation may be given in terms of generalized curves with specified endpoints. These results have significant applications in the calculus of variations and optimal control theory as regards problem reformulation and development of necessary conditions. 0. Comments on notation Let S be a compact topological space. C(S) will denote the Banach space of continuous real‐valued functions on S with sup norm. When S is a cube in R n , C 1 (S) is the subset of C(S) comprised of restrictions to s of continuously differentiable functions on R n . The (normed) dual of C(s) , C * (S) , is known to be isometrically isomorphic to the space of Radon measures on s (finite, regular, signed Borel measures), denoted frm( s ) and normed by total variation. We shall not distinguish between bounded linear functionals and the measures which represent them, writing interchangeably μ( f ) and fd μ for the action of μ upon f . Various norms on C * (s) will be considered, the strong norm always being denoted by |.| for any μ ɛ C * ( S ),‖ μ ‖ = sup { | ∫ g d μ | : g ɛ ( S ) , | g | ⩽ 1 } ,where |.| is the sup norm in C(S) . P(S) will denote the set of non‐negatively valued functions in C(S) , with P⊕( S ) its positive polar cone in C * (S) , P ⊕ ( S ) = { μ ɛ C * ( S ) : ∫ g d μ ⩾ 0 f o r a l l g ɛ P ( S ) } .For any measure μ ɛ C * ( S ), the support of μ is defined to be the complement of the largest open set S 1 ⊂ S such that μ( S 1 ) = 0, that is, supp{μ} = S i , in the usual notation. For x, y ɛ R n , x′y denotes the euclidean inner product of x and y and x | = ( x′x )½.