The importance of utilizing nonlinear models for control systems design has become more prevalent as systems become complex, and simplifying linear model assumptions are inadequate for purposes of control. An inherent problem in using nonlinear models is that properties such as stability are not so easily characterized. Even more difficult is the question of stabilizability of nonlinear systems. Here an investigation of the problem of local smooth stabilization of smooth continuous time nonlinear systems is proposed. The main objective of the proposed research is to obtain a general theory which may apply toward the local smooth stabilization (or testing the local smooth stabilizability) of all smooth nonlinear systems. This research will culminate in the construction of algorithms which will systematically attack the stabilization problem, depending on the structure of the system involved. Further extensions of this research could involve the use of these algorithms in adaptive control of smooth nonlinear systems. The main tools involved will be the construction of Lyapunov functions which test the local smooth stabilizability of smooth nonlinear systems through the use of multivariate power series and the blow-up method of bifurcation analysis. This research will make a significant contribution to the theory of local stability and stabilizability of nonlinear systems. The impact of this research will be far reaching for applications for applications of control of nonlinear systems. Examples of potential applications of this theory can be found in control of mechanical systems such as aircraft or robotic applications.
