Positive Definite just on a Cone

Given $A \in \mathbb{R}^{n \times n}$, $C \in \mathbb{R}^{m \times n}$, and the cone $\mathcal{C}:=\{x \in \mathbb{R}^n \mid C x \geq 0\}$,

find necessary and sufficient conditions on $(A,C)$ such that the following property holds.

There exists $P \stackrel{\mathcal{C}}{\succ} 0$, $P \in \mathbb{R}^{n \times n}$, such that: $-(A^\top P + P A) \stackrel{\mathcal{C}}{\succ} 0$.

As first step, it would be interesting to solve the question for just $\mathcal{C}=(\mathbb{R}_{\geq 0})^n.$



Definition:

A square matrix $P$ is positive definite on a cone $\mathcal{C}$ (notation $P \stackrel{\mathcal{C}}{\succ} 0$) if $x^\top P x >0$ for all $x \in \mathcal{C}\setminus\{0\}$.

Known results:

(1. Sufficient condition) If all eigenvalues of $A$ have negative real part, then the property holds with $\mathcal{C} = \mathbb{R}^{n}$.

(2. Necessary condition) If the property holds, then $A$ has no eigenvectors in $\mathcal{C}$ associated to positive eigenvalues.

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