128 An integral solution of Sylvester’s equation

Suppose that all the eigenvalues of $A$ and $B$ are in the open left half plane . Then

$$Y={\int }_0^{\mathrm{\infty }}\exp (tA)C\exp (tB)𝑑t$$

gives the unique solution to

$$AY+YB=-C.$$

MATLAB gives $Y=lyap(A,B,C)$, so $lyap(A,B,C)={\int }_0^{\mathrm{\infty }}\exp (tA)C\exp (tB)𝑑t$.