We consider the ensemble controllability problem for a linear time-invariant system ẋ(t, θ) = A(θ)x(t, θ) + B(θ)u(t), where A and B are continuous matrices with respect to the parameter θ, which belongs to some compact set Θ ⊂ R. Given any continuous initial state datum θ → x⁰(θ) and any continuous target state θ → x¹(θ), we investigate the numerical computation of a θ-independent open loop control u such that x⁰ is steered, in a given time T > 0, at a distance ε > 0 of x¹ in the uniform norm (with respect to the parameter). We approach the problem both theoretically and numerically. Using the Fenchel-Rockafellar duality, we first prove the existence and uniqueness of the ensemble control of a minimal L² norm. The numerical recovery of the optimal control is obtained by solving the dual problem, which consists in the unconstrained minimization of a non-differentiable functional in the space of Radon measures.