Transversality conditions and a strong maximum principle for systems of differential inclusions

An extension of the author's previous (1998) result on the finite-dimensional Pontryagin maximum principle (PMP), in which the classical transversality condition is replaced by an improved form, involving "weakly approximating cones". Our version of the PMP is similar in spirit to the classical statement, though much more general, and is proved using the same strategy of (a) reducing the optimal control problem to a geometric separation problem, (b) constructing needle variations, (c) using a topological argument, based on a result closely related to the Brouwer fixed-point theorem, to construct an "adjoint vector" that satisfies a finite subset of the collection of inequalities that occur in the Hamiltonian maximisation condition of the PMP, and (d) concluding with a compactness argument to end up with an adjoint vector that satisfies all the inequalities of the Hamiltonian maximization condition.