In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.

Buoy systems are an alternative for micropowering small devices in remote locations. Portal frames are very useful to harvest the energy of the waves into usable energy. Thus, using the current models for a portal frame in the literature and the spectrum of available energy in sea waves, a nonlinear mathematical model accounting for the coupling of a nonlinear piezoelectric material is considered. The neighbour of selected variables is analyzed and then optimized by a process utilizing the particle swarm optimization (PSO) algorithm. Furthermore, an optimal control using the linear‐quadratic regulator (LQR) controller is applied to control the load resistance of the piezoelectric circuit. The optimization process and the LQR show to be effective. The results show a general gain due to optimization and a relatively small gain using the controller.