Suppose we have two monic polynomials with real coefficients
\begin{align*}
& p_1(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0 \\
& p_2(t) = t^n + b_{n-1} t^{n-1} + \dots + b_0.
\end{align*}
Further assume $p_1(t)$ and $p_2(t)$ are both Hurwitz stable, i.e., the roots are all lying on the open left half plane of $\mathbb C$; equivalently, real parts of all roots are $<0$.
I am wondering whether or not there is some $M < 0$ (existence of such $M$ is enough) such that if the real parts of all roots smaller than $M$, then the convex combination of the two polynomials is Hurwitz stable.
Intuitively, I am thinking by making real parts of roots "negative" enough, the convex combination would stay in the left half plane.
p.s. Routh-Hurwitz Theorem gives sufficient condition to determine whether a polynomial is Hurwitz stable. But the rule seems complicated.
Best Answer
As shown in this answer, the sum of two stable polynomials does not have to be stable. The example given there is that, for $a>11$, the polynomial $(x+1)^3 + (x+a)$ has roots in the right half plane, although $(x+1)^3$ and $x+a$ are stable (and even have their roots real!).
This is easily adapted to give a counterexample to your question: For any positive real $M$ and any $a>11$, the polynomials $(x+M)^3$ and $(x+aM)$ have roots with real part $<M$, but their sum has roots with positive real part.