Introduction The transition from laminar flow to turbulence remains one of the most important and challenging problems in fluid mechanics due to its strong nonlinear behavior, multiscale interactions, and sensitivity to perturbations. In many practical engineering systems, including bluff-body wakes, boundary layers, thermoacoustic combustors, rotating flows, and interfacial transport systems, the onset of oscillatory instability is frequently associated with a Hopf bifurcation. At the Hopf point, a pair of complex conjugate eigenvalues crosses the imaginary axis, destabilizing the steady equilibrium state and generating self-sustained oscillatory dynamics. These oscillations may subsequently evolve into increasingly complex nonlinear flow structures and ultimately transition toward turbulence. Consequently, understanding and controlling Hopf bifurcation behavior plays a critical role in the suppression of oscillatory instability and the delay of turbulence transition. Traditional turbulence-control strategies are often based on perturbation-energy minimization, empirical feedback laws, reduced drag objectives, or linearized stability analysis around a fixed operating point. Although these methods have achieved success in many applications, they frequently neglect the underlying nonlinear bifurcation structure governing the flow dynamics. As a result, optimal-control trajectories may inadvertently drive the system toward unstable operating regimes where oscillatory instabilities become amplified. In highly nonlinear fluid systems, this limitation becomes particularly important because the transition to turbulence is fundamentally governed by changes in the stability structure of the nonlinear dynamical system. Incorporating bifurcation information directly into the optimal-control formulation therefore provides a promising strategy for improving turbulence suppression and stability management. Recent advances in reduced-order modeling and nonlinear dynamical systems theory have enabled the representation of complex fluid flows using low-dimensional Galerkin models derived from the incompressible Navier–Stokes equations. Proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and Galerkin projection techniques have been widely used to construct reduced-order models capable of capturing dominant coherent structures and oscillatory flow dynamics. Such models provide a computationally efficient framework for continuation analysis and bifurcation tracking using software tools such as MATCONT. In particular, reduced-order models are well suited for identifying Hopf bifurcation points, limit cycles, and stability transitions associated with vortex shedding and transitional flow behavior. Despite substantial progress in nonlinear flow control, directly incorporating bifurcation stability information into optimal-control formulations remains computationally challenging. Eigenvalue computations embedded within dynamic optimization are expensive and may introduce numerical non-smoothness near eigenvalue crossings, making gradient-based optimization difficult. This challenge becomes even more significant in large-scale PDE-constrained optimization problems where repeated Jacobian evaluation and eigenvalue analysis may become prohibitively expensive. Consequently, there is a strong need for computationally efficient surrogate models capable of representing the local stability structure of nonlinear fluid systems while preserving differentiability required by gradient-based solvers. Machine-learning approaches, particularly neural-network surrogate models, provide a promising mechanism for overcoming these limitations. Feedforward neural networks with smooth activation functions can approximate highly nonlinear mappings while maintaining differentiability. By training a neural network to predict the maximum real eigenvalue of the system Jacobian as a function of the reduced-order state variables and bifurcation parameter, stability information can be embedded directly into the optimal-control problem without repeated eigenvalue computations. This enables the construction of differentiable Hopf-avoidance penalties suitable for integration into nonlinear programming solvers such as IPOPT within Pyomo. The primary objective of the present work is therefore to develop a bifurcation-aware optimal-control framework for turbulence suppression in nonlinear fluid-mechanics systems. The proposed methodology combines Galerkin reduced-order modeling, Hopf bifurcation analysis, neural-network stability prediction, and optimal control within a unified computational framework. A three-mode reduced-order turbulence model is first constructed from the incompressible Navier–Stokes equations and analyzed using continuation techniques to identify Hopf bifurcation points and limit cycles. A neural-network surrogate is then trained to approximate the spectral abscissa of the reduced-order system, enablin