I want to prove/disprove the following claim:
Let $P>0$, $P\in\mathbb{R}^{n\times n}$. For every $A\in\mathbb{R}^{n\times n}$ there exists $c>0$: $A^\top P+PA\leq cP$.
This is a follow up to Let $P\geq0$, $P\in\mathbb{R}^{n\times n}$. For every $A\in\mathbb{R}^{n\times n}$ there exists $c\geq0$: $A^\top P+PA\leq cP$.
This is true. You may simply pick any positive number $c\ge\rho\left(P^{-1/2}A^TP^{1/2}+P^{1/2}AP^{-1/2}\right)$, so that $cI\succeq P^{-1/2}A^TP^{1/2}+P^{1/2}AP^{-1/2}$ and in turn $$ cP \succeq P^{1/2}\left(P^{-1/2}A^TP^{1/2}+P^{1/2}AP^{-1/2}\right)P^{1/2} =A^TP+PA. $$