A Short Course in Control Theory

Mathematics, as well as several areas of application, abounds with situations where it is desired to control the behavior of the trajectories of a given dynamical system. The goal can be either geometric (keep the state of the system in a given set, or bring it toward the set), or functional (find the trajectory that is optimal relative to a given criterion). More specific issues arise subsequently, such as the construction of feedback control mechanisms achieving the aims we have in mind. In this chapter we will identify a complex of such fundamental and related issues, as they arise in connection with the control of ordinary differential equations in a deterministic setting. The first section sets the scene and develops a technical base for the entire chapter.