I would like to solve a coupled matrix differential equation.
All are $2\times 2$ matrices. Then, I have
\begin{align} &\dot{X}=-i (A X - X A)-\eta (B Y - Y B);\\ &\dot{Y}=-\kappa Y +(B X - X B) \end{align} with initial conditions $Y(0)=0$ and $X(0)=\mathcal{I}$ and where all the constants $\kappa, \eta$ are positive.
While this seems like a complicated system to solve, I am interested in some particular limits that I think could make it solvable. The first one is that $\kappa\ll 1$, which when simplifies one of the equations to $\dot{Y}\simeq \eta (B X - X B)$.
The second one is that I am interested in the steady-state solution, that is, the limit when $t\rightarrow \infty$. However, I assumed that I could just take $\dot{X}=0$ and $\dot{Y}=0$ and solve the remaining system. And doing so gives me the wrong result! ($\dot{Y}=0$ would just give me $XB=BX$ and no information about $Y$ which I have to plug in into the 1st equation).
If I don't take the limit of $\kappa\ll 1$ and solve for the steady-state $\dot{X}=\dot{Y}=0$, I then get
\begin{equation} -i (AX-XA)=\frac{\eta}{\kappa}(BBX-2BXB-XBB) \end{equation}
And I am not sure how to solve this equation.
So my question is two-fold:
- Are my approximations for the steady-state correct? $\dot{X}=\dot{Y}=0$ even if the system is coupled? Why is it not compatible with taking $\kappa\ll 1$?
- If so, is there a way to solve for $X$? While it reminds me of the Sylvester equation ($AX+XA=C$), it is not true and it is just more complicated.
$ \def\LR#1{\left(#1\right)} \def\vc#1{\operatorname{vec}\LR{#1}} \def\qiq{\quad\implies\quad} \def\c#1{\color{red}{#1}} \def\CLR#1{\c{\LR{#1}}} $Rewrite the steady-state equation in the following form $$\eqalign{ C \;=\; DX \;+\; EXF \;+\; XG \\ }$$ then vectorize it $$\eqalign{ &\vc{C} \;=\; \LR{I\otimes D \;+\; F^T\otimes E \;+\; G^T\otimes I}\vc{X} \\ &c \;=\; {M}x \\ }$$ This admits a least-squares solution in terms of the pseudoinverse $M^+$ $$\eqalign{ x = M^+c + \LR{I-M^+M}w \\ }$$ where $w$ is an arbitrary vector that can be used to satisfy additional constraints, e.g. symmetry.
If there are no further constraints, then set to it zero.