  We consider a periodic evolution inclusion defined on an evolution triple of
spaces. The inclusion involves also a subdifferential term. We prove existence
theorems for both the convex and the nonconvex problem, and we also produce
extremal trajectories. Moreover, we show that every solution of the convex
problem can be approximated uniformly by certain extremal trajectories (strong
relaxation). We illustrate our results by examining a nonlinear parabolic
control system.
