Relationship with trace and asymptotic stability in control theory

Question

What is the relationship between $\mathrm{tr}(\exp(tA) \exp(tA^\ast))$ and asymptotic stability in control theory ?

Answer 1

The expression is the square of the Frobenius matrix norm. If the norm tends to zero, the system $\dot{x} = A x$ is asymptotically stable.

This is more a definition than a practical test, because it requires the solution to the ordinary differential equations.