Multi-point penalty-based optimization for optimal control of chaotic turbulent flow

Gradient-based optimization can augment the utility of a simulation for both scientific and engineering applications, but obtaining useful gradient for a chaotic turbulent flow is challenging because any quantity-of-interest $\mathcal{J}$ to become significantly non-convex over time. As such, even an exact gradient is restricted in finding a useful optimum. We introduce an optimization framework to circumvent this challenge for optimal control of turbulent flow. It splits the simulation time into intervals with discontinuities that are increasingly penalized. The challenge and method are illustrated with a logistics map and the Lorenz system before being successfully applied to a three-dimensional turbulent Kolmogorov flow. For the turbulence case, the method suppresses large-scale pressure fluctuations without laminarization, effectively targeting a controllable component of the flow amidst the more chaotic turbulence. The utility of such an optimization, which might be questioned because of the restricted controllability of chaotic systems, is discussed.