On exact controllability of variational discrete systems

Let X,UX,U be two Banach spaces, let ΘΘ be a metric space and let σσ be a flow on ΘΘ. For A∈ℓ∞(Θ,B(X))A∈ℓ∞(Θ,B(X)) and B∈ℓ∞(Θ,B(U,X))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)x:Θ→S(X) and u∈ℓp(N,U)u∈ℓp(N,U). We prove that if the discrete cocycle associated with the system (A,B)(A,B) is surjective and the variational discrete system (A,B)(A,B) is completely stabilizable, then (A,B)(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.