4.2 Basic Definitions of Stability

We consider a class of dynamical systems modeled by the equation:

where is the state vector and is a vector-valued function with the components

We assume that f _i's are continuous and have continuous first partial derivatives. Under these assumptions, the solution to , exists and is unique. We denote this solution as x( t) = x( t; t ₀, x ₀). If the f _i's do not depend explicitly on t, then the system represented by is called autonomous; otherwise it is called nonautonomous. An equilibrium point or state is a constant vector, say x _e, such that

Observe that an equilibrium state x _e is a constant solution to (4.1).

If x _e is an equilibrium point, we can arrange for this equilibrium point to be transferred to the origin of by introducing the new variable:

We assume that this has been done for an equilibrium point under consideration. Therefore, we have

We now introduce basic definitions of stability in the sense of A. M. Lyapunov ^[*] (1857 1918). In the following considerations we use the standard Euclidean norm of a vector. Thus, if , then