An Exact Control for Stabilizing a Flying Sailing Boat with Foils

In this text, we introduce a new optimal control strategy for linear dynamic mechanical systems. The main goal is to focus on reducing the control oscillations. We aim at minimizing the energy consumption spent and the mechanical fatigue phenomenon of actuators. Therefore, we suggest a non-standard cost functional that incorporates a non-differentiable term proportional to the L1–norm of the control time derivative. This additional term penalizes high-frequency variations in the control and aims at reducing the oscillations of the system. Several tests are suggested for a simplified model of a flying boat with foils. The inclusion of this non-differentiable term induces significant mathematical difficulties, rendering the optimization problem non-smooth. In a first step, we adapt a duality regularization technic pioneered by R. Glowinski for thixotropic fluids. The model obtained is a non-linear variational inequation following the ideas of G. Stampacchia. In a second step, we develop an asymptotic analysis versus the marginal cost ε of the control. It is rigorously proved that the optimal control converges to an exact control when ε → 0. In other words, this limit control drives the system’s final state exactly to the desired position, while simultaneously minimizing the control cost among all such exact controls. Keywords exact control; optimal control; total variations; stabilization; flying boats; foils