Robust passivity-based boundary control of the 2-D Navier-Stokes equation with chaotic vortex

This work introduced a novel control strategy for the chaos suppression in a numerical wave tank with chaotic vortices. The control strategy is based on designing a robust passivity-based boundary control for the uncertain Navier-Stokes equation interacting with a wave energy converter. First, the dynamic analysis of the uncertain Navier-Stokes equation was presented by determining the eigenfunction and eigenvalues along with computing the phase portraits, bifurcation diagrams, and Lyapunov exponents. Additionally, the proposed boundary controller was derived by selecting an appropriate Lyapunov functional, aiming to suppress the chaotic vortices present in the uncertain Navier-Stokes equation. In addition to the theoretical and numerical results, we also numerically evaluated the interactions between an energy wave converter generator and the chaotic vortices, first as an open-loop problem, and next, the proposed boundary control strategy was tested to suppress the chaotic behavior. Finally, the discussion and conclusion of this research study were presented.