ODE in symmetric definite positive matrices

Question

It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt a-\sqrt bx} + \frac {dx}{\sqrt a+\sqrt bx})$ and integrate both parts.

I am interested in the same equation with $x$ taking its values in symmetric positive matrices, ie $\frac {dx}{dt} = a - x b x$, with $a$ and $b$ symmetric positive. In that case, because the involved matrices might not commute, I cannot use the same trick to get a closed form solution.

Would you know of any techniques I can use to solve such a problem?

Answer 1

Consider a solution of the linear system $$ \dot x=by$$ $$\dot y =ax$$ Then $u:=yx^{-1}$ solves your Riccati equation $$\dot u= a- ubu\, .$$

rmk. Note that the solution of the linear system is an exponential, defined for all $t\in\mathbb{R}$; while the solution of the Riccati equation is bounded to the interval where $x$ is invertible (possibly not the whole $\mathbb{R}$) Also, if you have an initial data $u(0)=u_0$, you can take $x(0)=I$ and $y(0)=u_0$ as initial data for the linear system.