Simplified multitime maximum principle

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 multiple integrals or curvilinear integrals. Our paper discuss the m-flow type PDEconstrained optimization problems, focussing on a simplified multitime maximum principle. This extends the simplified single-time maximum principle of Pontryaguin in the ODEs case (curves) to include the case of PDEs (submanifolds). In Section 1 the idea of multitime is motivated. In Section 2 a multitime maximum principle, for the case of multiple integral functionals, is stated and proved. A version of multitime maximum principle, for the case of curvilinear integral functionals, is formulated in Section 3. Though a multiple integral functional is mathematically equivalent to a curvilinear integral functional (Section 4), their meaning is totally different in real life problems. A multitime maximum principle approach of variational calculus is presented in Section 5. M.S.C. 2000: 93C20, 93C35, 49K20, 49J20, 53C44.