This paper focuses on designing a versatile state-space representation for parallel shear flows to facilitate both modal analysis and closed-loop control. Our objective in designing a control law is to regulate shear flows around a stationary desired state—a laminar flow—despite disturbances while ensuring the physical interpretability of the decision-making process. Our closed-loop control implementation involves linearizing the Navier-Stokes equations around the desired state, spatially discretizing the resulting linear system, and computing the feedback gain using an optimal control formulation. We assume that the actuators operate at the boundaries and model their effects as a forcing term using the elimination method rather than the lifting method, the former being standard in partial differential equations solvers but less common in flow control. Furthermore, we assume that the flow state is reconstructed from image sensors. To illustrate the efficiency of our approach, we derive a state-space representation for both the 2D periodic channel flow and the 2D spatially developing mixing layer flow. Hydrodynamic stability analysis confirms the representativeness of the state matrix and identifies the primary natural modes with exponential growth. The resolvent analysis highlights forcing modes, analogous to Orr structures in 2D shear flows, that induce misaligned response modes, analogous to natural modes, with the most significant transient energy growth. These observations provide insights for designing controllers that mitigate disturbances and enhance flow stability. Balanced truncation corroborates these observations and suggests an opposing control mechanism in which the control signal is derived by projecting the flow state onto the adjoint balanced modes. This approach targets dominant dynamic features, particularly the Orr structures, for effective control. For the two flows, linear quadratic regulator controllers are deduced from the full linear state-space model and implemented within a nonlinear Navier-Stokes solver. Additionally, for the periodic channel flow case, pole placement controllers are deduced from the reduced-order state-space model obtained via balanced truncation, as this reduced model naturally lends itself to effective pole placement. All simulation results confirm that these control laws stabilize the 2D periodic channel flow, preventing the development of unstable modes. Furthermore, they delay the onset of large eddies in the 2D spatially developing mixing layer by attenuating relatively slow endogenous disturbances in the control signal.

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• Flow control
• Shear flows