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Center for Nonlinear Studies |
Many natural and engineered systems consist of interacting nonlinear dynamical components that exhibit complexities and span scales that challenge our ability to model, optimize, and control them. Their dynamics, parameters, and interconnections may be problematic to infer, may be subject to intrinsic uncertainty and external noise, and could vary in time on multiple scales. Periodic dynamics in coupled multi-scale networks motivate compelling mathematical problems with applications to biological rhythms and energy infrastructures, where systems must be designed or controlled in optimal ways that are robust to variability, uncertainty, and disturbances. I will describe a control problem involving oscillations that appear in chronobiology and neuroscience. Inputs can be applied globally to establish and maintain resilient dynamic patterns in a collection of heterogeneous nonlinear oscillators with unobservable state and unknown initial conditions. I will then discuss optimization of dynamic flows in large-scale natural gas pipeline networks in order to compensate for variation in fuel consumption of gas-fired power plants. Here, the mathematical properties of monotonicity and stability enable compact optimization formulations for solutions that are robust to uncertainty in magnitude and timing of loads on these systems.