I am considering a matrix differential Riccati equation
$$\dot{X}(t) = -AX(t) - X(t)A^T + XDX - Q$$
where it is known that D is a symmetric positive semi-definite matrix and Q is a symmetric matrix and we have initial condition $X(0) = 0$.
Suppose I know that a particular solution $X_*$ to the associated algebraic Riccati equation
$$-AX_* -X_*A^{T} + X_*DX_* - Q = 0$$
is the unique stabilizing solution, such that all eigenvalues of $A - DX_*$ have negative real part. In other words, for the particular system under consideration I have already found a symmetric matrix $X_*$ that satisfies the condition that the eigenvalues of $A - DX_*$ have negative real part, and so existence of a unique stabilizing solution is not an issue.
Given all this, is it known that $X(t) \rightarrow X_*$ as $t \rightarrow \infty$? If not, what is the relevance or importance of the unique stabilizing solution?