Let $K$ be a real stable matrix; more specifically, $$ K=\left(\begin{array}{rrrrr} 0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\\ -k_0&-k_1&-k_2&\ldots&-k_{n-1} \end{array}\right) $$
where $k_0,\ldots,k_{n-1}>0$. Is there a (non-numerical) algorithm for obtaining the real symmetric positive definite $P$ that maximizes the value
$$ \sigma=\frac{-\lambda_{\max}\left( K^T P+ PK \right)}{\lambda_{\max}(P)}? $$
Or, at least, is it possible to estimate $\sigma$ for a given $K$? I understand that I can solve the Lyapunov equation $K^T P+ PK=Q$ for some $Q<0$ and use the obtained $P$ to calculate the estimation. But, maybe, there is a better way.
