Title: Balanced truncation model reduction and control of high-dimensional nonlinear systems

Abstract: Optimal control of high-dimensional nonlinear systems is challenging from a computational perspective, as it requires solving nonlinear optimality systems in high dimensions. Nonlinear model reduction for control systems is one avenue to this problem, yet the model reduction problem in turn requires solving high-dimensional control problems in the first place to find the proper projections and manifolds where reduced trajectories evolve. Thus, the nonlinear control and model reduction problem are intricately related. In this talk, we build on the theoretically rigorous framework of nonlinear balanced truncation model reduction, a system-theoretic method that is built on the notion of controllability and observability of a nonlinear system. The framework requires solving well-behaved high-dimensional Hamilton-Jacobi-Bellman partial differential equations, which we do so with Taylor-series-based techniques to produce scalable algorithms for systems with 1,000s of state variables. We then describe ways to use these solutions to control nonlinear systems, even going to 100,000s of dimension through some additional approximations, as well as how to derive nonlinear balanced truncation ROMs. We illustrate the methods on a variety of problems in aerospace, mechanical engineering, and fluids.