Bifurcation-Aware Reduced-Order Modeling and Optimal Control of Orr-Sommerfeld Instabilities in Shear Flows Exhibiting Hopf Bifurcation

Abstract

This study presents a reduced-order modeling and optimal control framework for a shear-flow instability system derived from the incompressible Navier-Stokes equations via linear stability analysis of the Orr-Sommerfeld equation and subsequent center-manifold reduction. The resulting four-dimensional nonlinear dynamical system captures the essential interactions among the dominant Tollmien-Schlichting disturbance mode, the mean-flow correction induced by nonlinear Reynolds-stress feedback, and the actuator dynamics driven by an external control input. The model exhibits Hopf bifurcation, marking the transition from a stable equilibrium to sustained oscillations in the form of a stable limit cycle. Bifurcation analysis using numerical continuation reveals the existence of a Hopf point at (−0.000147, 0.052789, −0.001858, 0.105579, 0.105579), with a negative first Lyapunov coefficient indicating a supercritical bifurcation. Eigenvalue analysis of the Jacobian matrix at the bifurcation point confirms the presence of a conjugate imaginary pair crossing the stability boundary, validating the onset of oscillatory instability. An optimal control problem is formulated to minimize the energy of the dominant instability mode while penalizing control effort, with the control input simultaneously acting as a bifurcation parameter. The problem is solved using PYOMO.DAE with IPOPT under both unconstrained and Hopf-bifurcation-aware formulations. Results show that incorporating a Hopf constraint significantly reduces the objective function value and suppresses oscillatory control behavior. The study demonstrates that integrating bifurcation information into optimal control design enhances stability, reduces disturbance energy, and improves overall control efficiency in nonlinear fluid systems.