Let $\alpha$ be a positive real number and $A\in\mathbb{R}^{n\times n}$ be a stable matrix (that is the eigenvalues of $A$ have negative real part). Consider the following optimization problem $$ \max_{X\in\mathbb{R}^{n\times n}, \, \mathrm{tr}(X)=1,\, X\ge 0} \mathrm{tr}((P+\alpha I)X), $$ where $X\ge 0$ means that $X$ is positive semi-definite and $P$ is the solution of the following Lyapunov equation $$ AP+PA^\top =-X. $$

Question: Does the above-formulated problem admit a closed-form solution?