Global existence for differential riccati equation?

Question

Consider an autonomous Riccati Differential Equation $$ \dot{P} + A^\top P + P\,A − P\,B\,B^\top P + Q = 0,\quad P(0) = 0 $$

all matrices are real and continuous. You can find in many textbooks existence and uniqueness on $(-\infty,0]$ under suitable conditions. Moreover $P(t)\rightarrow P_{\infty}$ for $t\rightarrow -\infty$ in a monotonic way, where $P_{\infty}$ is the unique symmetric positive semidefinite solution of $A^\top P + P\, A − P\, B\, B^\top P + Q = 0$.

What can be said about existence on $(-\infty,\infty)$. Or has somebody an example for a finite-time blow up for $t\rightarrow +\infty$?

Answer 1

$\dot y = y^2$ is an instance of this problem class with blow-up in finite positive time for $y(0)>0$, as its well-known solution $$ y(t)=\frac{y(0)}{1-y(0)·t} $$ has a pole at $t=\frac1{y(0)}$.

Answer 2

One-dimensional counter-example: $$ A=0,\,B=1,\, Q=-1. $$ Our initial value problem becomes $$ p'-p^2-1=0,\quad p(0)=0, $$ with UNIQUE solution $$ p(t)=\tan t. $$ Maximal domain of the solution $I=(\pi/2,\pi/2)$.