On Coupled Two-Level Variational Problem in Sobolev-Orlicz Space

Abstract

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.