Mathematical Sciences: Geometry of highly nonlinear multi-input systems

点击翻译按钮获取中文摘要

This research program aims at developing the geometric foundations that underlie practically relevant properties of highly nonlinear control systems (that is, systems whose dynamics are inadequately, or not at all, described by linear approximations). Highly oscillatory control variations will be examined with the goal of obtaining better criteria for controllability and optimality than presently exist, and to obtain new algorithms for stabilizing feedback laws. Methods from algebraic combinatorics will be used to characterize the computational complexity of deciding controllability, stabilizability and other relevant control objectives and to develop new path planning algorithms. This project will develop the geometric foundations that underlie the dynamics of high-performance automatic control systems. It focuses primarily on systems that have very highly nonlinear dynamics and that have many interconnected components influencing each other. The engineering demands for ever higher performance, or even optimality, often require completely new control designs since classical linearization techniques for control design is often inadequate in such situations. By bringing together techniques from various areas of pure mathematics, such as differential geometry and algebraic combinatorics, new algorithms for finding optimal solutions to path planning problems (e.g., for robot arms or largely autonomously operating intelligent vehicles) will be developed.