Stability describes the tendency of a dynamical system to return to its desired performance after being disturbed by external forces. For systems with controllable input channels stabilizability indicates that inputs can be chosen in such a way that a (possibly unstable) system becomes stable. For systems with linear dynamics the basic questions of stability and stabilizability are settled, while for nonlinear systems a variety of sufficient criteria is known, without a general theory emerging. This project is to study the precise exponential growth behavior of control systems, based on the theory of Lyapunov exponents, geometric control theory and infinite time optional (periodic) control. The results will enable one to - characterize stabilizable and destabilizable systems, - determine a measure for the stability or instability margin of a system, - compute exact stabilizability diagrams depending on the system parameters, - pinpoint the design parameters that are crucial for stabilizability of a system, - design stabilizing controllers. Bilinear systems will be analyzed as prototypes of general nonlinear systems, concentrating on developing a general theory for necessary and sufficient conditions along with numerical procedures necessary to compute exact exponential stability and stabilizability diagrams.
