theorem, stability of linear analog system equilibrium state at origin

Theorem

For a linear analog system, the equilibrium state at the origin is

• stable (thus also bounded) if and only if $\lVert \phi(t,t_0) \rVert$ ) is bounded for all $t \geq t_0$, where
  • $\phi(t,t_0)$ is the transition matrix, and
  • $\lVert \cdot \rVert$ is the norm of a matrix defined as $\lVert \mathbf{T} \rVert = \max_{\mathbf{x} \neq \mathbf{0}} \frac{\lVert\mathbf{Tx} \rVert}{\lVert \mathbf{x} \rVert}$
• asymptomatically stable if and only if in addition to the previous condition, $\lVert \phi(t,t_0) \rVert \to 0$ as $t \to \infty$.

Corollary

For a linear analog system, equilibrium state is S if and only if $\lvert \phi_{ij}(t,t_0) \rvert$ is bounded for all $i,j$ and all $t \geq t_0$. It is AS if and only if in addition for all $i$ and $j$, $\phi_{ij}(t,t_0) \to 0$ as $t \to \infty$.

See also

• State equation of finite dimensional linear systems

References

1. P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, p. 186.
   • (Theorem 9.1)