The differential systems that are encountered in applications are fundamentally analytic and therefore share a common geometric base. At the same time, such systems have extra-mathematical properties that differentiate one system from another and account for the particular features of their solutions. The theory of such systems is a blend of the general, shared by all Lie-determined systems, and the particular, due to other mathematical structures, and this recognition provides much insight and understanding, even for systems motivated by narrow practical considerations.

In this chapter we shall expand the theory of linear systems for systems with single inputs initiated in Chapter 4. In contrast to the traditional presentation of this theory, which relies entirely on the use of linear algebra and functional analysis, we shall develop the basic theory using the geometric tools introduced in Chapter 3. The geometric point of view will reveal that much of the theory of linear systems follows from considerations independent of the linear properties of the system and therefore extends to larger classes of systems.

Our selection of topics in this theory is motivated partly by the striking nature of the mathematical results and partly by their relevance to the second part of this book, dealing with optimality. Our treatment of linear systems begins with their controllability properties. We shall show that the main theorem has natural Lie-theory interpretations in terms of the Lie saturate of the system. Second, we shall show that a linear controllable system can be decoupled by means of linear feedback into a finite number of independent scalar linear differential equations. This result is known as Brunovsky's normal form.