Mathematical Sciences: Nonlinear Control: Feedback Stabilization and Cardiac Arrhythmia Control

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ABSTRACT Proposal: DMS-9530973 PI: Hermes Hermes proposes to continue his research program on the problem of existence and construction of asymptotically stabilizing feedback control (ASFC) for affine control systems, in particular via the use of high order, homogeneous, approximations. The dilations and local coordinates relative to which the approximating systems are constructed can be naturally obtained from the Lie algebra structure of the vector fields defining the original system. Also, an ASFC of appropriate homogeneous order for the approximating system is a local ASFC for the original system. The method will be to extend the classical linear, quadratic, regulator technique to a nonlinear, homogeneous regulator. A first goal is to classify homogeneous, affine, systems which admit a smooth ASFC. Linear, controllable, approximating systems are divided into equivalence classes via the action of the linear feedback group, and canonical representatives of each equivalence class are well known. Hermes proposes tho study the equivalence classes of small time, or large time, locally controllable homogeneous approximations having known smooth ASFC, under the action of a homogeneity preserving group of transformations, and to find canonical representatives of the equivalence classes. The odd power integrators provide an example of such. The above topics have implications to questions of regularity conditions for viscosity solutions of Hamilton-Jacobi-Bellman equations, another topic to be studied. Many mathematical systems, e.g., an inverted compound pendulum, a robotic arm, a high performance aircraft, a communications satellite, either are, or are designed to be, unstable in their uncontrolled mode of operation. Stability of the system is achieved via the introduction of external controls. In past decades, construction of stabilizing controllers was achieved via the use of linear approximations, which had to be "controllable" for the method to work. Many space age prob lems are basically nonlinear, have uncontrollable linear approximations, and their analysis requires high order approximations which retain relevant nonlinear behavior but are still amenable to analysis. The high order, homogeneous approximations are such. The goal of Hermes research is to use these for the construction of stabilizing controls.