Mathematical Sciences: Nonlinear Differential Equations and Control; High Order Homogeneous Approximations

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In this project the principal investigator will continue his study of vector fields that are homogeneous with respect to a dilation in order to apply these results to the approximation of solutions of nonlinear differential equations and control systems. In particular, he will study the existence of asymptotically stabilizing feedback controls for nonlinear systems via homogeneous approximations, and then he will attempt to construct such feedback controls for systems that are small- time locally controllable about their uncontrolled rest state. The construction of asymptotically stabilizing feedback controls remains one of the fundamental problems of control theory. Control systems are a common feature of modern everyday life. Think of the thermostat in your home that controls the furnace or the air-conditioner. Many of these control systems are governed by linear equations, and for them there is now an extensive and nearly complete theory. On the other hand, many technologically important control systems are nonlinear, that is, the process to be controlled is governed by a nonlinear equation or system of equations. For nonlinear control systems the theory is far from complete. In this project the principal investigator will seek approximate solutions of nonlinear control problems by means of sophisticated methods based upon linear system theory.