It is known from R. Bellman that the value of the functional $J = \int_{0}^{\infty}xWx \ dt, \ W>0$ along the solution of the linear time-invariant system $\dot x = Ax, \ x(0)=x_0$, with Schur stable matrix $A$ is equal to $J = x_0^\top P_{eq} x_0$, where $P_{eq}>0$ is the solution of the Lyapunov equation $$ A^\top P_{eq} + P_{eq} A = -W. $$
Consider now the Lyapunov inequality $A^\top P + P A \leq -W.$ Then any solution $P$ satisfies $P\geq P_{eq}$. Hence, for every initial condition $x_0$ we have $x_0^\top P_{eq}x_0 \leq x_0^\top Px_0$, so that the value of the functional $J$ can be found as a solution of the following semi-definite program (SDP): $$ \min \ x_0^\top Px_0 \quad \text{s.t. } \ A^\top P + P A \leq -W $$ This problem can then be converted in LMI form. Here we focus solely on the term $x_0^\top Px_0$. Then, by setting $Q = P^{-1}>0$ and a slack scalar $\gamma$, the minimization of $x_0^\top Px_0$ can be equivalently written as $$ \min \gamma \quad \text{s.t. } \ \begin{bmatrix} \gamma & x_0^\top\\ x_0 & Q \end{bmatrix} \geq 0 $$
My question is: how to get rid of the dependence on the initial condition? In the reference [1, section 3.3] they used the fact that since $P\geq P_{eq}>0$, instead of minimizing $x_0^\top P x_0$, (or equivalently, $x_0^\top Q^{-1} x_0$ for $Q=P^{-1}$), it is possible to minimize $$ \min \ \text{trace}(P) \qquad (\text{or } \ \max \ \text{trace}(Q) ) $$
The above fact is not clear to me. I get that $x_0^\top P x_0 = \text{trace}(x_0^\top Px_0)$ but I don't know how to get rid of $x_0$. Note that I do not want to assume that $x_0$ lies inside a unit circle, so the cyclic trick of the trace and $x_0 x_0^\top \leq I$ cannot be used.
[1] Khlebnikov, Mikhail V., et al. "Linear-quadratic regulator. I. a new solution." Automation and Remote Control 76.12 (2015): 2143-2155.
(Note that $\star^\top$ denotes the transpose operator).
You will not be able to fully get rid of the initial conditions. The optimal cost depends on them, this is a fact. Considering the trace is a good way to get rid of the initial conditions assuming that the initial conditions are uncorrelated random variables with zero mean and unit variance. Note that you can use a scaling to consider larger sets but it does not really matter here.