Multitime maximum principle for curvilinear integral cost

Recently we have created a multitime maximum principle gathering together some concepts in Mechanics, Field Theory, Differential Geometry, and Control Theory. The basic tools of our theory are variational PDE systems, adjoint PDE systems, Hamiltonian PDE systems, duality, multitime maximum principle, incavity on manifolds etc. Now we justify the multitime maximum principle for curvilinear integral cost using the m-needle variations. Section 1 recalls the multitime control theory and proves the equivalence between curvilinear integral costs and multiple integral costs. Section 2 formulates variants of multitime maximum principle using control Hamiltonian 1-forms produced by a curvilinear integral cost and a controlled m-flow evolution. Section 3 refers to original proofs of the multitime maximum principle using simple and multiple multitime m-needle control variations. The key is to use completely integrable first order PDEs (controlled evolution and variational PDEs) and their adjoint PDEs. Section 4 formulates and proves sufficient conditions that the multitime maximum principle be true. M.S.C. 2010: 49J20, 49K20, 93C20.