2.11 STABILITY OF HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS

Before investigating the stability of nonlinear systems, it is important to first understand the dynamics of linear homogeneous systems with constant coefficients. Thus, let us consider

with ICs

where A is a real constant square matrix of order n, and y _s is an n-column vector.

If the eigenvalues of A are distinct, from eq. (1.17.13) we note that the solution y is represented by

where ? _j are the eigenvalues of A, and z _j and W ^T _j are the corresponding eigenvectors and eigenrows, respectively. It is then apparent from eq. (2.11.3) that if all eigenvalues ? _j are negative (if real) or have negative real parts (if complex), then

Further, if A has at least one real eigenvalue that is positive or if it has at least one pair of complex eigenvalues with positive real parts, then due to the fact that for such eigenvalues the term e ^{? _j t} increases without limit, the solution y will become unbounded as t ? ?. In this case, we have

Finally, if all eigenvalues of A, except for one, are negative (if real) or have negative real parts (if complex), but there is either a real eigenvalue ? = 0, or there is a pair of purely imaginary eigenvalues ? = i ?, then

or

where