  We introduce the optimality question to the relaxation in multiple control
problems described by Sobolev type nonlinear fractional differential equations
with nonlocal control conditions in Banach spaces. Moreover, we consider the
minimization problem of multi-integral functionals, with integrands that are
not convex in the controls, of control systems with mixed nonconvex constraints
on the controls. We prove, under appropriate conditions, that the relaxation
problem admits optimal solutions. Furthermore, we show that those optimal
solutions are in fact limits of minimizing sequences of systems with respect to
the trajectory, multi-controls, and the functional in suitable topologies.
