Reduced-Order Control of Transient Instabilities in High-Dimensional Dynamical Systems

Identification and control of transient instabilities in high-dimensional dynamical systems remain a challenge because transient (non-normal) growth cannot be captured by low-dimensional modal analysis. Here, we leverage the power of the optimally time-dependent (OTD) modes, a time-evolving set of orthonormal modes that capture directions in phase space associated with transient and persistent instabilities, to formulate a control law capable of suppressing transient and asymptotic growth around any fixed point of the governing equations. The control law is derived from a reduced-order system resulting from projecting the linearized dynamics onto the OTD modes, and enforces that the instantaneous growth of perturbations in the OTD-reduced tangent space be nil. We apply the proposed reduced-order control algorithm to infinite-dimensional fluid flows dominated by strong transient instabilities, and demonstrate unequivocal superiority of OTD control over conventional modal control.