4.1 Introduction

Systems of first order ordinary differential equations (ODEs) play a vital role in control systems theory. In this chapter, results from the theory of ODEs , limited to those necessary for developing subsequent chapters, are presented.

In Section 4.2, the definition and classification of ODEs of order n are given. In Section 4.3, solutions to first order ODEs and initial value problems (IVPs) are discussed. Moreover, in Theorem 4.3.8, a sufficient condition for the existence of solutions to first order ODEs is given. The existence and uniqueness of solutions to IVPs is proved in Section 4.4 using the Banach fixed point theorem. In Sections 4.5 and 4.6, system of ODEs and its solution in series form are discussed. In Section 4.7, solutions of time-invariant linear systems in terms of eigenpairs of the coefficient matrix is presented. Section 4.8 contains some results on fundamental matrices and the state transition matrix. In Section 4.9, solution to nonhomogeneous linear system is discussed. Gronwall's inequality is briefly discussed in Section 4.10. Theorem 4.5.1 on the existence and uniqueness of solutions of IVPs, Section 4.6 on systems of linear ODEs, Section 4.8 on fundamental matrices and the state transition matrix, and Theorem 4.10.1 on Gronwall's inequality are important focal points for a better understanding of the contents of Chapters 5–8.