Stability of a sum of an unstable matrix and a stable matrix

Question

I remember reading somewhere, probably in StackExchange, but now I couldn't find it, about the sum of two real matrices of dimension $n\times n$ $$ A + \theta B $$ where $A$ is non-Hurwitz, $B$ is Hurwitz, and $\theta > 0$. My question is about the claim that

There exists $\phi>0$ such that $A+\theta B$ is Hurwitz for all $\theta \geq \phi$, where $B$ is Hurwitz and $A$ is non-Hurwitz.

Is this true?

Answer 1

I found the answer to the above question. The above claim is true.

To prove, let's consider the Lyapunov theorem:

A matrix $B$ is Hurwitz if and only if there exists a symmetric positive definite $P$ such that $PB + B^T P < 0$, where $<$ denotes negative definiteness.

Now, assume that $A+\mu B$ is Hurwitz. Then, $$ \begin{array}{rcl} P(A+\mu B) + (A+\mu B)^T P &<& 0 \\ (PA + A^T P) + \mu (PB + B^T P) &<& 0. \end{array} $$ Since $PB+B^T P <0$, we can choose $\mu$ to be sufficiently large to satisfy the above inequality. This completes the proof.