Mathematical Sciences: Nonlinear Differential Equations, Vector Field Approximations and Control

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The focus of this research is the development of a coherent program to determine the stabilizability of nonlinear control systems. Understanding stabilizability is important; almost all issues in both systems theory and systems practice eventually reduce to some version of the stabilization problem. The importance of linear systems theory rests more on the ease of testing a linear systems for stabilization than on its linearity per se. Earlier work concentrated on the question of short time local controllability, a necessary condition for stabilizability. viewed as stabilization by open loop controls while the latter is stabilization by closed loop controls. As a result of the prior work, one now has computational criteria for deciding STLC. Work has turned to approximations of nonlinear systems by simpler ones such as nilpotent systems and systems which are homogeneous with respect to a dilation. This work holds promise for a relatively comprehensive theory of nonlinear stabilizability. In the present project, a new concept of vector fields which are stable with respect to measurement are introduced. The concept is closely related to the shadowing ideas of nonlinear dynamics. It has been shown already that if Brockett's locally onto conditions not satisfied then a nonlinear system is not locally stabilizable to a vector field which is stable with respect to measurement. Work will also be done using homogeneous approximations and homogeneous Lagrangians to construct dynamic asymptotically stabilizing feedbacks for systems which don't satisfy Brockett's locally onto condition using approximate feedback linearization and optimal control. The combination of mathematical power and engineering applications has long made control theory one of the healthiest sources of new mathematical themes. The bridges built between mathematicians and engineers have provided for dynamic interchanges leading to the advancement of both fields. This project takes up the problem of what one can do when some of the traditional tests for stabilizing state feedback control are not present. In many important problems, including mechanical systems with nonholonomic constraints and the inverted pendulum in a gravity free environment, efforts are to be made to construct discontinuous state feedback controls.