[THEOREMS] Stability of systems

Discrete stability from root moduli

Write a characteristic root in polar form as

r_j=\rho_j e^{i\theta_j}.

Then

|r_j^n|=\rho_j^n.

Thus $\rho_j<1$ gives a decaying mode, $\rho_j>1$ gives an unbounded mode and $\rho_j=1$ gives a non-decaying mode.

For the whole system, every solution decays precisely when all characteristic roots satisfy $|r_j|<1$ . A root with $|r_j|>1$ supplies an unbounded solution. If all roots satisfy $|r_j|\le1$ and a unit-modulus root is present, its multiplicity decides the remaining case: when each such root is simple, every solution is bounded but some do not decay; a repeated unit-modulus root instead produces an unbounded term $nr_j^n$ .

Continuous stability from real parts

For a characteristic root $r=\alpha+i\beta$ ,

|e^{rt}|=e^{\alpha t}.

Therefore $\alpha<0$ gives decay, $\alpha>0$ gives an unbounded mode and, for a simple root, $\alpha=0$ gives a non-decaying oscillatory or constant mode. A repeated root on the imaginary axis introduces a term $t e^{rt}$ , whose magnitude is unbounded.

For the whole system, every solution decays precisely when all characteristic roots have negative real part. A root with positive real part supplies an unbounded solution. If no root has positive real part but a root lies on the imaginary axis and each such root is simple, every solution is bounded but some do not decay. A repeated root there gives an unbounded solution.

Theorems from Stability of systems

Discrete stability from root moduli

Continuous stability from real parts

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