Given the continuous-time differential Lyapunov equation
$$\mathbf{A}(t)^T\mathbf{P}(t)+\mathbf{P}(t)\mathbf{A}(t)+\dot{\mathbf{P}}(t)=-\mathbf{Q}(t)$$
where $\mathbf{A}(t)$ and $\mathbf{Q}(t)$ are both continuous and bounded, there exists a unique solution $\mathbf{P}(t)$. In addition, assume that $\mathbf{Q}(t)$ is positive definite and $\mathbf{A}(t)$ is Hurwitz.
Under what additional assumptions can one guarantee that the solution $\mathbf{P}(t)$ is positive definite? Is this even possible? I am not interested in the solution to the equation.
I would greatly appreciate any comments or pointers where I can find more info about this problem. Thank you.