The question is in the context of linear control systems. Let $A(t)$, $R(t)$, and $Q(t)$ be time-varing square $n\times n$ matrices of reals. For all $t$, $R(t)\ge 0$ and $Q(t)\ge 0$ are semi positive-definite. Consider the differential matrix Riccati equation $$\dot{X}(t) = A(t) X(t) + X(t) A^\top(t) - X(t) R(t) X(t) + Q(t)$$ with a positive-definite initial condition $X(0)>0$.
My question are
- When $X(t)$ is bounded for all $t$? Is it enough to say that $R(t)> 0$ or it should be $R(t) \ge R_0 > 0$?
- When $X(t)$ is non-singular for all $t$? Is it enough to say that $Q(t)> 0$ or it should be $Q(t) \ge Q_0 > 0$?
- When (necessary, sufficient, iff) does there exist $X_0>0$ such that $X(t) \ge X_0$ for all $t$?
Usually, control books consider the algebraic equation, i.e. for $\dot{X}(t)=0$, or start with assumptions on uniform controllability and observability of $R$ and $Q$ implying they are positive definite. Are you aware of books/papers that coniser semi definite cases?
I'm likewise currently doing some research along the lines of your question. I was looking at the chapter on Hermitian matrix Riccati DEs in the Abou-Kandil et. al. book, but I stumbled across this neat report which is pretty straightforward https://ntrs.nasa.gov/api/citations/19650020082/downloads/19650020082.pdf.
I'm almost sure that these aren't the only circumstance when this is true, but it seems like if $A(t)$, $R(t)$, and $Q(t)$ are locally integrable, the unstable eigenvectors of $A(t)$ are not in the nullspace of $R(t)$ (maybe this need only hold almost everywhere), unstable eigenvectors of $A(t)^{\top}$ are not in the nullspace of $Q(t)$ (again, maybe this need only hold almost everywhere, I need to understand more concretely how these assumptions are used) then:
If $X(0) \succeq 0$ and $R(t), Q(t) \succeq 0$ almost everywhere, then $X(t)$ exists and is not infinity for all $t\in [0,\infty)$. (Theorem 2-2 in the reference)
If $X(t)$ satisfies $'()=()()+()^{\top}()−()()()+()$ on $[0,a)$, and both $X(0) \succeq 0$, $Q(t)\succeq 0 $ for almost every $t$ on $[0,a)$, then $X(t)\succeq 0$ for $[0,a)$. If additionally $X(0)\succ 0$ or $Q(t) \succ 0$ for almost every $t\in[0,a)$, then $X(t)\succ 0$ for $t\in[0,a)$.(Theorem 2-1 in the reference)
The proofs in that report are easy to follow, and these hypotheses were good enough for what I was trying to do. I haven't played around with the assumptions at all. It seems like the results here might be due to an (even older) paper by Kalman (reference 3 in the report), but I have not tried to dig in there.