Mathematical Sciences: Control Structures

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9409037 Gardner This project is funded jointly by the Applied Mathematics Program and the Geometric Analysis Program. Its goal is to construct and apply new invariants, structure theorems and normal forms of control systems under time-independent, time-dependent, and dynamic feedback. The tools involve the method of equivalence as refined by the principal investigator, exterior differential systems, and the theory of infinite pseudogroups. An important principle is the enlargement of the notion of feedback equivalence to the fully nonlinear category. This program has resulted in major improvements in exact linearization algorithms. Similar ideas will be used to investigate algorithms for other applicable normal forms such as standard bilinear systems. A new concept of control structure, a more global viewpoint of nonsingular control systems, will also be investigated. The theory of feedback control systems, originally motivated by problems in engineering, occupies a central place in applied mathematics and is fundamental to control engineering applications. This project will develop the geometric foundations that underlie the dynamics of high-performance automatic control systems. It focuses primarily on systems that have highly nonlinear dynamics and which may 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 employing techniques from various areas of pure mathematics, such as differential geometry, major improvements in exact linearization algorithms, with applications to nonlinear autopilot design, for example, are expected.