Equivalence of multitime optimal control problems

Many science and engineering problems can be formulated as optimization problems that are governed by m-flow type PDEs (multitime evolution systems) and by cost functionals expressed as curvilinear integrals or multiple integrals. Though these functionals are mathematically equivalent on m-intervals, their meaning is totally different in real life problems. Our paper discusses the m-flow type PDE-constrained optimization problems of Mayer, Lagrange and Bolza, focussing on their equivalence. Section 1 formulates the Mayer problem with a terminal cost functional. In Section 2, the idea of equivalence is motivated for the Mayer, Lagrange and Bolza problems, based on curvilinear integral cost, using the curvilinear primitive. In Section 3, similar results are proved for the Mayer, Lagrange and Bolza problems, based on multiple integral cost, using both the curvilinear primitive and the hyperbolic primitive. Section 4 shows that curvilinear integral functionals and multiple integral functionals are equivalent on m-intervals. M.S.C. 2000: 93C20, 93C35, 49K20, 49J20.