Lyapunov equation

Remarks

The continuous equation is the Sylvester equation $AX + XA^* = -Q$. Many texts in control write it as $A^* P + PA + Q = 0$ instead, which is the same equation with $A^*$ in place of $A$; for real $A$ this reads $A^T P + PA + Q = 0$. If every eigenvalue of $A$ has negative real part, the continuous equation has the unique solution $X = \int_0^\infty e^{At} Q e^{A^* t} \, dt$, and $X$ is positive definite when $Q$ is. If every eigenvalue of $A$ has absolute value less than $1$, the discrete equation has the unique solution $X = \sum_{k=0}^{\infty} A^k Q (A^*)^k$. Example: for $n = 1$, $A = -1$ and $Q = 2$, the continuous equation $-2X + 2 = 0$ gives $X = 1$. Without the eigenvalue condition a solution may fail to exist: for $n = 1$, $A = 0$ and $Q = 1$, the continuous equation reads $1 = 0$.