We study properties of ω-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains. First, we prove that under certain conditions the ω-limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side. Second, we give conditions ensuring that the ω-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the ω-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.