Maximize Trace with Lyapunov inequality as constraint

Question

Let $C$ be a given symmetric matrix, $A$ be a given Hurwitz matrix and $X$ be a symmetric positive definite matrix. Is the following problem solvable to give unique, optimal $X$?

max $tr (CX)$ subject to

$AX + XA^T<0$ and $X>0$

Answer 1

Without any extra conditions on $C$, then no. For example, think of $C=I$, $A=-\frac{1}{2}I$. Then your program becomes

$$\max \sum_i x_{ii}$$

$$\text{subject to } X>0$$

which does not have a unique solution.