PDI&PDE-constrained optimization problems with curvilinear functional quotients as objective vectors

In this work we introduce and perform a study on the multitime multi-objective fractional variational problem of minimizing a vector of quotients of path independent curvilinear integral functionals (MFP ) subject to certain partial differential equations (PDE) and/or partial differential inequations (PDI), using a geometrical language. The paper is organized as follows: §1 formulates a PDI&PDE-constrained optimization problem. §2 states and proves necessary conditions for the optimality of the problem (MP ) of minimizing a vector of path independent curvilinear integral functionals constrained by PDIs and PDEs. §3 analyzes necessary efficiency conditions for the problem (MFP ), and §4 studies different types of dualities. M.S.C. 2000: 49J40, 49K20, 58E17, 65K10, 53C65.