Existence of quadratic forms as \(V\)-function of a stable system (Q1178382)

The author presents (without proofs) two theorems on the solvability of the Lyapunov matrix equation \((1)\;A^ T B+BA=C\) in cases where the matrix \(A\) is stable and \(C\) is positive semidefinite. (If \(B\) is a solution of (1), then \(V(x)=x^ T Bx\) is a Lyapunov function for the linear differential system \(\dot x=Ax\).) Three simple examples are given.