Robust & Linear Control Systems The characteristic polynomial of a control system is the following uncertain polynomial:
$$s^3 + a_2 s^2 + a_1 s + 3.5 $$
Where $a_1 \in  [1.5,4.2],a_2 ∈ [1.2,4.25]$ and $ 4.2 ≤ a_1 + a_2 ≤ 6.3$
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Is this uncertain polynomial stable?
How do I solve this problem, when I have got two values from $a_1$ and $a_2$.

The Hurwitz matrix of the polynomial is:
$$ \begin{bmatrix} a_2 & 3.5 & 0 \\ 1 & a_1 & 0 \\ 0 & a_2 & 3.5 \end{bmatrix}$$
To stability this matrix must be definite-positive, this is, all subdeterminants must be positive. The conditions are:
$$a_2 >0$$
$$a_2 a_1 >3.5 $$
If the second subdeterminant is positive, it follows that the third one also is (because $3.5 \gt 0$ ). You can check the polynomial will always be stable for the values you have.
