On exact controllability of variational discrete systems

Let X , U be two Banach spaces, let Θ be a metric space and let σ be a flow on Θ . For A ∈ ℓ ∞ ( Θ , B ( X ) ) and B ∈ ℓ ∞ ( Θ , B ( U , X ) ) , we consider the variational discrete system with control ( A , B ) x ( θ ) ( n + 1 ) = A ( σ ( θ , n ) ) x ( θ ) ( n ) + B ( σ ( θ , n ) ) u ( n ) , ∀ ( θ , n ) ∈ Θ × N , where x : Θ → S ( X ) and u ∈ ℓ p ( N , U ) . We prove that if the discrete cocycle associated with the system ( A , B ) is surjective and the variational discrete system ( A , B ) is completely stabilizable, then ( A , B ) is exactly controllable. By illustrative examples we show that our hypotheses cannot be dropped and also we study the validity of the converse implication.