Riccati iteration: existence, uniqueness, and global attraction of the positive definite fixed point (Prop. 4.4.1, parts 1–3)

Let $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times m}$, $Q = C^{\top}C \succeq 0$ with $C \in \mathbb{R}^{q\times n}$, and $R \succ 0$. Assume that $(A,B)$ is controllable and $(A,C)$ is observable in the sense of Definition 4.1.1 of Bertsekas, and let

be the discrete-time Riccati operator. Then there exists a positive definite matrix $P$ such that:

1. $P$ solves the algebraic Riccati equation $F(P) = P$;
2. $P$ is the unique positive semidefinite fixed point: every $P' \succeq 0$ with $F(P') = P'$ equals $P$;
3. the Riccati iteration converges from every positive semidefinite start: $F^{k}(P_0) \to P$ as $k \to \infty$ for every $P_0 \succeq 0$.

Proof sketch (Bertsekas, Vol. I, \S4.4): $F$ is order-preserving on the symmetric positive semidefinite cone; controllability gives a uniform upper bound $\bar P$ for the iterates $F^k(0)$, which are nondecreasing since $F(0) = Q \succeq 0$, hence they converge to some $P \succeq 0$, and continuity of $F$ gives $F(P) = P$. Observability shows $P \succ 0$ (an $x \ne 0$ with $x^{\top}Px = 0$ forces $CA^jx = 0$ for all $j$, contradicting observability). Minimality of $P$ among positive semidefinite fixed points follows from $F^k(0) \le F^k(P') = P'$, and the attraction property from comparing the iterates from $P_0$ with those from $0$ and from any fixed point.