How to force matlab output any solution when the Lyapunov solution is not unique?

Question

I want to solve the Lyapunov equation of the following form

and I use this code in Matlab: Lambda = lyap(Cbar,Cbar,-2*Cbarprime); However, the Matlab displays the error message: The solution of this Sylvester equation does not exist or is not unique. How can I force it to output any one solution?

Answer 1

We can use the following code which is amended from the official function sylvester by changing some of the elements(actually zero elements) in matrix dA+dB' to be inf.

function X = mysylvester(A,B,C)
%         Solve the equation A*X + X*B = C 
A = (A + A') / 2;
B = (B + B') / 2;
[QA, dA] = eig(A, 'vector');
[QB, dB] = eig(B, 'vector');

CC = QA'*C*QB;
tmp = dA + dB';
idx = find(abs(tmp) <= 0.00001);
tmp(idx) = inf;
X = CC ./ tmp;
X = QA*X*QB';
end