Ordinary Differential Equations

We shall speak of ordinary differential equation if an equation contains time-dependent (or more generally, scalar-dependent) variables as well as their derivatives with respect to time (or another scalar). Since we shall always consider ordinary differential equations in this book, we shall drop the adjective ordinary. We shall see the deep similarity between the theorems on differential equations and those on difference equations. To avoid lengthy references to Chapters 1 and 3, we shall only ‘repeat’ the most important definitions and theorems with the necessary changes. But we will refrain from displaying all the parallels. In Section 5.1 we shall introduce the basic concepts of differential equations. In Section 5.2 we outline the theory concerning linear systems. In Section 5.3 we shall return to nonlinear systems and extend stability theorems obtained for linear systems to nonlinear ones. In Section 5.4 we just outline the basic elements of control in continuous time. Useful information is to be found in Samuelson (1947), Coddington and Levinson (1955), Pontryagin (1962), Gandolfo (1971), Arnold (1973) and Martos (1981).