In this work, we study a technique of exploiting open-loop generated trajectories for a constrained control problem, using them to shape suitable non-quadratic Control Lyapunov Functions. These trajectories, generated off-line, allow detecting a suitable domain of attraction in which a candidate Lyapunov function has a negative derivative. Given a suitably constructed basis function, our working machinery is based on linear programming, hence, the technique can be applied to problems of non-trivial size in terms of the number of basis functions and points in the state space. For linear systems, we seek convex Lyapunov functions, which are homogeneous polynomials. Simulation and experimental results for drone control are given.