I come across a matrix differential equation which is similar to the Lyapunov differential equation, that is
\begin{equation} \frac{d A(t)}{d t} = M(t)A^\top(t) + A^\top(t)M(t) + R(t), \quad A(0) = A_0 \end{equation}
where $A(t), M(t) \in \mathbb{R}^{m \times m}$. I am wondering how to solve this matrix ODE. Can someone give me some hints?
Thanks!
This is just a simple example where $m=2$ but can be generalized to any $m$. I will drop the $(t)$ just to ease notation and use $\dot{a} = \frac{d a}{d t}$. The dynamics when $m=2$ is,
\begin{align} \frac{d A}{dt} = \begin{bmatrix}\dot{a}_{11} & \dot{a}_{12} \\ \dot{a}_{21} & \dot{a}_{22}\end{bmatrix} = \begin{bmatrix}m_{11} & m_{12} \\ m_{21} & m_{22}\end{bmatrix}\begin{bmatrix}a_{11} & a_{21} \\ a_{12} & a_{22}\end{bmatrix} + \begin{bmatrix}a_{11} & a_{21} \\ a_{12} & a_{22}\end{bmatrix}\begin{bmatrix}m_{11} & m_{12} \\ m_{21} & m_{22}\end{bmatrix} + \begin{bmatrix}r_{11} & r_{12} \\ r_{21} & r_{22}\end{bmatrix} \end{align}
Notice that we can convert the matrix $A$ into a vector $a$ with the same dynamics.
Let $a=\begin{bmatrix}a_{11} \\ a_{12} \\ a_{21} \\ a_{22}\end{bmatrix}$ then the matrix dynamics can be written as,
\begin{align} \dot{a} = \begin{bmatrix}\dot{a}_{11} \\ \dot{a}_{12} \\ \dot{a}_{21} \\ \dot{a}_{22}\end{bmatrix} = \begin{bmatrix} 2 m_{11} & m_{12} & m_{21} & 0 \\ m_{12} & 0 & m_{11}+m_{22} & m_{12} \\ m_{21} & m_{22} + m_{11} & 0 & m_{21} \\ 0 & m_{12} & m_{21} & 2m_{22} \\ \end{bmatrix} \begin{bmatrix}a_{11} \\ a_{12} \\ a_{21} \\ a_{22}\end{bmatrix} + \begin{bmatrix}r_{11} \\ r_{12} \\ r_{21} \\ r_{22}\end{bmatrix} \end{align}
which is just a linear time-varying system.