Control systems are developed in order to manage or control the output of a given process. Feedback control systems continually monitor the final output and automatically control the process to fall within a given or set range. For example, the thermostat in a home not only controls the on-off settings but monitors the room temperature and applies heat or cooling to maintain a balance. Such a system would be known as a linear control system. In reality there are few such systems - most have numerous inputs and outputs that need to be taken into account if the controls are to work properly or in an optimal range. This book focuses on these nonlinear systems and provides an introduction to analysis techniques used in the design of nonlinear and optimal feedback control systems. The emphasis is on the fundamental topics of stability, controllability, and optimality and on the corresponding geometry associated with these topics.