System of nonlinear first-order differential equations (Matrix Riccati system)
Do any of you have an idea how to solve the following system analytically,
A'(t) = a + b * A(t)^T - A(t) * c * A(t)^T ,where ^T means transpose
a, b and c are (2x2)-matrices containing constants and A(t) is also a (2x2)-matrix.
Initial condition is that A(0) = 0
I though about transforming the system in the following way,
Y(t) = (A(t) I) , where I is a (2x2) identity matrix
Y'(t) = d * Y - Y * e * Y ,where d = (b , a ; c ; 0) e = (c , 0) 0 = (2x2)-matrix with zeros
Now the system become homogeneous, but Y is not invertible, so it doesn't get me any further.
If the original system was homogenegous I think I would be able to solve it by transforming it to a linear ODE system and solve that one,
then transform the solution back.
I have already looked at some early litterature on the topic "matrix riccati differential equations", by Reid (1946), Levin (1959) and Coles (1965), but I didn't find it very helpful
Any suggestions on how to solve it or a referrence to some litterature describing this situation?