Sea waves represents a very promising energy source. Pioneered by Wiercigroch a pendulum system would be feasible for such an energy harvesting purpose. These devices consist of a pendulum with a vertical motion induced by the sea waves. As it is well known, pendulum's stable rotations generate enough energy able to be extracted by an electrical generator attached to its axis. In this paper, using a brushless dc motor as an input torque u(t) to maintain stable pendulum rotations, a singular optimal control formalism provides a very simple control law using a mechanical model.Besides the main objective in this paper: stable controlled rotations, in order to control every motion possibility: rotation, stability or even chaos, a singular optimal control policy is defined A salient property of the the resulting control law lies on its very simple hardware  implementation bang-bang control: only sign is needed using Arduino and operational amplifiers (see Figure 1). While Matlab/Simulink simulations will be presented, the possibility of being tuned for stable/unstable rotations or even asymptotic stability is also an interesting analysis. Conclusions and future work are also depicted.

In this paper, general one-dimensional difference equations (DE) are studied. In fact, the asymptotic behaviour of such DE's will be studied via an algebraic closed-form condition plus a linear DE. Some examples are presented including the Collatz sequence, proving the existence of at most one invariant set surrounding the number 1: {1, 2, 4}. Some conclusions will be also depicted.

This paper formalize the existence’s proof of first-integrals for any second order ODE, allowing to discriminate periodic orbits. Up to the author’s knowledge, such a powerful result is not available in the literature providing a tool to determine periodic orbits/limit cycles in the most general scenario.