Proof of Global Existence and Uniqueness for the Kalman Covariance Riccati Equation

Conventions. Identify $k\times k$ matrices with $\mathbb{R}^{k^{2}}$. From claims 1 and 2 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals: for matrices $U,V$, $|UV|_{e}\le k\,|U|_{e}|V|_{e}$ where $|\cdot|_{e}$ is the largest absolute entry, and $|X|_{e}\le|X|\le k^{2}|X|_{e}$ for the Euclidean norm $|X|$ on entries; moreover $|X|\le k\,|X|_{e}$, since $|X|^{2}=\sum_{ij}X_{ij}^{2}\le k^{2}|X|_{e}^{2}$ and the nonnegative square root is monotone. Every closed subinterval $[p,q]$ of $[a,b]$ with $p<q$ is nonempty and is compact by Closed Interval $[a,b]$ is Compact in $\mathbb{R}$, so by Extreme Value Theorem on a Compact Subset of a Metric Space every real-valued continuous function on $[p,q]$ attains a maximum and a minimum and is in particular bounded; we use this repeatedly below. In particular, fix reals $\alpha,\delta\ge0$ bounding all entries of $A(t)$ and $D(t)$ on $[a,b]$. Transpose manipulations use claim 3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals. Set $F(t,X)=A(t)X+XA(t)^{\top}-XD(t)X+C(t)$.

Hypotheses of the a priori theorem. Composition continuity: entries of $F(t,h(t))$ are polynomial combinations of continuous real-valued functions (Continuity of Sums and Products of Real-Valued Functions on a Metric Space). Local Lipschitz: for $|X|,|Y|\le R$, write $XDX-YDY=(X-Y)DX+YD(X-Y)$; every entry of the right side is bounded by $2k^{2}\delta R\,|X-Y|_{e}$ (two products of three factors, entries of $X,Y$ bounded by $R$), and the linear terms contribute at most $2k\alpha\,|X-Y|_{e}$ per entry, so $|F(t,X)-F(t,Y)|\le k^{2}\,(2k\alpha+2k^{2}\delta R)\,|X-Y|=:L_R\,|X-Y|$, using $|X-Y|_{e}\le|X-Y|$.

Properties of solutions on subintervals. Let $a<c\le b$ and let $h$ be any continuous solution of the Riccati integral equation on $[a,c]$, in the sense of the solution notion of A Priori Bounded Solutions of Locally Lipschitz Ordinary Differential Equations Exist Globally (whose statement records that the integrand is continuous on $[a,c]$).

(a) Uniqueness on $[a,c]$. If $h,\widetilde h$ are two solutions on $[a,c]$, choose $R$ with $|h(t)|\le R$ and $|\widetilde h(t)|\le R$ on $[a,c]$: each entry is bounded by Extreme Value Theorem on a Compact Subset of a Metric Space, and $|X|\le k\,|X|_{e}$ converts entry bounds to Euclidean bounds. Subtracting the two integral equations and applying claim 5 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals (with $p=k^{2}$) and the Lipschitz estimate, $u(t):=|h(t)-\widetilde h(t)|$ satisfies $u(t)\le k^{2}L_R\int_a^tu(r)\,dr$ on $[a,c]$, and $u$ is continuous. By claim 7 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals (translation), $\bar u(\tau):=u(a+\tau)$ is continuous on $[0,c-a]$ and satisfies $\bar u(\tau)\le k^{2}L_R\int_0^{\tau}\bar u(\sigma)\,d\sigma$. The interval $[0,c-a]$ is nonempty, since $a<c$, and compact by Closed Interval $[a,b]$ is Compact in $\mathbb{R}$. The function $\bar u$ is continuous, hence bounded by Extreme Value Theorem on a Compact Subset of a Metric Space and measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions, and its Riemann integral over a subinterval of $[0,c-a]$ agrees with the corresponding Lebesgue integral by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. Applying Gronwall's Lemma for Bounded Measurable Functions on $[0,c-a]$ with its constants taken to be $0$ and $k^{2}L_R$ gives $\bar u\le0$; since $u\ge0$, $u\equiv0$ and $h=\widetilde h$.

(b) Symmetry. Since $C(t)$, $D(t)$, $P_0$ are symmetric, transposing the equation shows $t\mapsto h(t)^{\top}$ is also a solution on $[a,c]$: $(A h)^{\top}=h^{\top}A^{\top}$, $(hA^{\top})^{\top}=Ah^{\top}$, and $(hDh)^{\top}=h^{\top}D^{\top}h^{\top}=h^{\top}Dh^{\top}$ (claim 3 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals). By (a), $h(t)^{\top}=h(t)$: every $h(t)$ is symmetric.

(c) Positive semidefiniteness. Define $\bar A(r)=A(r)-h(r)D(r)$ on $[a,c]$; its entries are continuous, being sums and products of continuous real-valued functions (Continuity of Sums and Products of Real-Valued Functions on a Metric Space). Using symmetry of $h(r)$ and $D(r)$,

$$\bar A h+h\bar A^{\top}+\bigl(C+hDh\bigr)=Ah-hDh+hA^{\top}-hDh+C+hDh=Ah+hA^{\top}-hDh+C ,$$

since $h\bar A^{\top}=h(A^{\top}-Dh)=hA^{\top}-hDh$. Hence $h$ satisfies on $[a,c]$ the linear matrix equation of Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations, applied with $b$ replaced by $c$ and data $(\bar A,\bar C,P_0)$ where $\bar C:=C+hDh$; $\bar C$ has continuous entries, is symmetric ($(hDh)^{\top}=hDh$ by (b)), and is positive semidefinite: $x\cdot(h(r)D(r)h(r)x)=(h(r)x)\cdot\bigl(D(r)(h(r)x)\bigr)\ge0$ by the transpose-dot identity and symmetry of $h(r)$. By the uniqueness in claim 1 of Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations, $h$ is the solution treated there, and claims 2-3 there give: every $h(t)$ is positive semidefinite.

(d) Comparison with $\Lambda$. Let $\Lambda$ be as in the statement (existing on $[a,b]$ by Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations; symmetric positive semidefinite by claims 2-3 there). On $[a,c]$, $\Delta:=\Lambda-h$ satisfies, subtracting the two integral equations,

$$\Delta(t)=\int_a^t\bigl(A(r)\Delta(r)+\Delta(r)A(r)^{\top}+h(r)D(r)h(r)\bigr)\,dr ,$$

an equation of the form treated in Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations (with $b$ replaced by $c$), with coefficient $A$, inhomogeneity $hDh$ (continuous, symmetric, positive semidefinite by (c)'s computation), and initial value $0$ (positive semidefinite). By claims 1-3 there, $\Delta(t)$ is positive semidefinite: $h(t)\preceq\Lambda(t)$ in the semidefinite order.

(e) Entry bound. By (c), (d), and Entry Bounds for Positive Semidefinite Matrices, $|h_{ij}(t)|\le\max_{l}\Lambda_{ll}(t)\le\lambda^{*}$, where $\lambda^{*}\ge0$ is the maximum over $[a,b]$ of the continuous function $t\mapsto\max_l\Lambda_{ll}(t)$ (Extreme Value Theorem on a Compact Subset of a Metric Space; a maximum of finitely many continuous functions is continuous, since $\max(u,v)=\tfrac12(u+v+|u-v|)$, and it is nonnegative since the diagonal entries of the positive semidefinite $\Lambda(t)$ are nonnegative by Entry Bounds for Positive Semidefinite Matrices). Hence $|h(t)|\le k\,|h(t)|_{e}\le k\lambda^{*}=:\rho$. Also $|P_0|\le\rho$: at $t=a$ the Lyapunov integral equation gives $\Lambda(a)=P_0$ (degenerate interval, convention of Mean-Square Riemann Integral of a Family of Random Variables), so $0\preceq P_0=\Lambda(a)$ and $|(P_0)_{ij}|\le\max_l\Lambda_{ll}(a)\le\lambda^{*}$ by Entry Bounds for Positive Semidefinite Matrices, whence $|P_0|\le k\lambda^{*}=\rho$.

Conclusion. The hypotheses (i)-(iii) of A Priori Bounded Solutions of Locally Lipschitz Ordinary Differential Equations Exist Globally hold with this $\rho$: (i) and (ii) were checked above, and (iii) holds by (e), every solution on every $[a,c]$ being bounded by $\rho$ in the Euclidean norm on entries, with $\rho\ge|P_0|$. Hence there is exactly one continuous solution $P$ on $[a,b]$. Applying (b), (c), (d) with $c=b$ to $P$ gives symmetry and $0\preceq P(t)\preceq\Lambda(t)$, and the final entry bound is Entry Bounds for Positive Semidefinite Matrices applied to $0\preceq P(t)\preceq\Lambda(t)$. $\blacksquare$