A solution of the continuous Lyapunov equation by means of power series

(2) ^ © = A' X(t) + X\t) A + B, te(0,co) dt for continuous systems. Here A is a given stable matrix, i.e. all its eigenvalues a, satisfy |a,| < i or Re a,< 0, respectively, and B is a given symmetric matrix. This difference/differential equation is to be solved with a given initial condition X(0). Of special interest is a limit for / -» co; it is a solution of the stationary Lyapunov equation (3) A'XA ~ X + B = 0 or (4) A'X + XA + B = 0 . The latter equation is investigated in matrix algebra books [1], [2] as a special case of the non-symmetric equation (5) AVX + XA2 + B = 0. The structure of solution of (5) is exhibited using elementary divisors or Jordan