130 A solution of Lyapunov’s equation

Suppose that $A$ is a real square matrix such all its eigenvalues are in the open left half plane . Then for all positive definite $P$, there exists a positive definite $K$ such that

$$AK+K{A}^{†}=-P.$$

Note that $\exp (tA)P\exp (t{A}^{†})$ is positive definite by exercise A3.2. Indeed $⟨\exp (tA)P\exp (t{A}^{†})Y,Y⟩=⟨P\exp (t{A}^{†})Y,\exp (t{A}^{†})Y⟩$ is positive and continuous for $t>0$ and $Y\ne 0$. Hence

$$K={\int }_0^{\mathrm{\infty }}\exp (tA)P\exp (t{A}^{†})𝑑t$$

is also positive definite. [MATLAB gives $K=lyap(A,P)$.]