Consider the matrix-valued differential Riccati equation (DRE): $$ \dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G, $$ where all coefficients are continuous.
It is well-known that if $Q-S^\top S\ge 0$ and $G\ge 0$, then the DRE admits a non-negative definite solution for all $T>0$.
My question is whether there exists any known condition such that the DRE admits a non-positive definite solution for all $T>0$? I am particularly interested in the case with $Q=G=0$. Numerical experiments show that for some choices of $A$ and $B$, the DRE seems to have non-positive definite solution for all $T>0$. However, I am not sure whether it is just a coincidence, since in this case, the existence of solutions to DRE does not follow from standard results.
