Question

Explain why a continuous time system is stable if the real part of its eigenvalues are negative.

Answer 1

Answer:3)

We know that the general solution is x(t) = eAtx0

thus, x(t) 0 if and only if as

we will now show that this happens if and only if all the eigenvalues of A have negative real parts

let be the jordan canonical form for A

then,

let be the eigenvalue of a associated with Ji then e Jit will tend to 0

if and only if has negative real parts

therefore, tend to 0 if and only i all the eigenvalues of A have negative real parts