Proof of The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound

Real arithmetic is that of The Real Numbers as a Type (iv) and The Real Numbers as a Type (vi). Let $F:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the right-hand side of the initial value problem of The Riccati Differential Equations of Filtering and Control (i) for $A$, $C$, $D$,

$$F(t,X)=A(t)X+XA(t)^{\top}-XD(t)X+C(t)\qquad\bigl(t\in[a,b],\ X\in\mathcal{M}_{n}(\mathbb{R})\bigr),$$

that is, $F$ is the function $H$ of The Riccati Differential Equations of Filtering and Control (i). For $a<c\le b$, a solution on $[a,c]$ below is a solution on $[a,c]$, in the sense of Solution of an Initial Value Problem on $[a,c]$ (ii), of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) with right-hand side $F$ and initial value $P_{0}$; for $c=b$ this is exactly a solution of The Riccati Differential Equations of Filtering and Control (i) with data $A$, $C$, $D$ and $P_{0}$, since $F$ is its function $H$. Functions formed pointwise from functions on a set, such as $\bar A=A-hD$, $h^{\top}$, $hDh$ and $\Lambda-h$ below, are those of Maps Formed Pointwise from Vector and Matrix Operations (i). Matrix computations below use Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (ii), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vii), together with Products of Real Matrices with Zero Matrices, Negatives and Differences (i), Products of Real Matrices with Zero Matrices, Negatives and Differences (ii), Products of Real Matrices with Zero Matrices, Negatives and Differences (iii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i) and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii); in particular $(-M)^{\top}=-M^{\top}$, the case $s=-1$ of $(sM)^{\top}=sM^{\top}$, since $(-1)M=-M$ entrywise by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv).

Step 1: bounds on the coefficients. The functions $t\mapsto\lVert A(t)\rVert$ and $t\mapsto\lVert D(t)\rVert$ are continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), hence bounded on $[a,b]$ by the extreme value theorem of The Real Line: Standing Notation and Background for Calculus on Intervals (x). Fix $\alpha,\delta\in\mathbb{R}$ with $\lVert A(t)\rVert\le\alpha$ and $\lVert D(t)\rVert\le\delta$ for every $t\in[a,b]$; then $\alpha,\delta\ge0$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i).

Step 2: continuity of $F$. Let $d$ be the product metric of $d_{\mathbb{R}}$ and $d_{n}$ on $\mathbb{R}\times\mathcal{M}_{n}(\mathbb{R})$, a metric by The Product Metric is a Metric (i), and let $S=[a,b]\times\mathcal{M}_{n}(\mathbb{R})$. By Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (i), the function $S\to\mathcal{M}_{n}(\mathbb{R})$, $(t,X)\mapsto X$, the restriction to $S$ of the second projection, is continuous on $S$, and, $A$, $C$ and $D$ being continuous on $[a,b]$, so are the functions $(t,X)\mapsto A(t)$, $(t,X)\mapsto C(t)$ and $(t,X)\mapsto D(t)$ on $S$. Since $F(t,X)=\bigl(A(t)X+XA(t)^{\top}+C(t)\bigr)-XD(t)X$, Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v) (products, a transpose and sums of continuous functions) and Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii) (the difference), applied to the metric space $(\mathbb{R}\times\mathcal{M}_{n}(\mathbb{R}),d)$ and its subset $S$, show that $F$ is continuous on $S$.

Step 3: the local Lipschitz bound. Let $\varrho\in\mathbb{R}$ with $\varrho>0$, let $t\in[a,b]$ and let $X,Y\in\mathcal{M}_{n}(\mathbb{R})$ with $\lVert X\rVert\le\varrho$ and $\lVert Y\rVert\le\varrho$. Since $XD(t)X-YD(t)Y=\bigl(XD(t)X-YD(t)X\bigr)+\bigl(YD(t)X-YD(t)Y\bigr)=(X-Y)D(t)X+YD(t)(X-Y)$,

$$F(t,X)-F(t,Y)=A(t)(X-Y)+(X-Y)A(t)^{\top}-\bigl((X-Y)D(t)X+YD(t)(X-Y)\bigr).$$

By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vi), Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (v) (so $\lVert-M\rVert=\lVert M\rVert$), Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vii) and The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (v),

$$\lVert F(t,X)-F(t,Y)\rVert\le2\alpha\lVert X-Y\rVert+2\delta\varrho\lVert X-Y\rVert=L_{\varrho}\lVert X-Y\rVert,\qquad L_{\varrho}=2\alpha+2\delta\varrho\ge0.$$

Step 4: solutions on initial subintervals. Let $\Lambda$ be as in (ii); every $\Lambda(t)$ lies in $\mathcal{S}(n)$ by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (iii). Fix $c\in\mathbb{R}$ with $a<c\le b$ and a solution $h:[a,c]\to\mathcal{M}_{n}(\mathbb{R})$ on $[a,c]$. The restrictions of $A$, $C$ and $D$ to $[a,c]$ are continuous on $[a,c]$ by Restricting a Continuous Map to a Smaller Subset Preserves Continuity (ii); the restriction of $\Lambda$ to $[a,c]$ is differentiable relative to $[a,c]$, with the restriction of $\Lambda'$ as its derivative, by Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (iii) (with $a$ and $c$ for $c$ and $d$); and $h$ is continuous on $[a,c]$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (i). Below, $A$, $C$, $D$ and $\Lambda$ denote these restrictions, $t$ ranges over $[a,c]$, and the argument $t$ is often omitted. The Lyapunov Differential Equation (i), The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (i), The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (iv), Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (i) and Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (v) are applied below with $c$ in place of $b$, which is allowed since $a<c$.

(a) $h(t)\in\mathcal{S}(n)$. The function $h^{\top}:t\mapsto h(t)^{\top}$ is differentiable with $(h^{\top})'=(h')^{\top}$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (viii), so $\Delta=h-h^{\top}$ is differentiable with $\Delta'=h'-(h')^{\top}$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (iv). As $C^{\top}=C$ and $D^{\top}=D$,

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

and $hDh-h^{\top}Dh^{\top}=\bigl(hDh-hDh^{\top}\bigr)+\bigl(hDh^{\top}-h^{\top}Dh^{\top}\bigr)=hD\Delta+\Delta Dh^{\top}$. Hence, with the function $\bar A=A-hD:[a,c]\to\mathcal{M}_{n}(\mathbb{R})$, whose transpose is $\bar A^{\top}=A^{\top}-Dh^{\top}$,

$$\Delta'=A\Delta+\Delta A^{\top}-hD\Delta-\Delta Dh^{\top}=\bar A\Delta+\Delta\bar A^{\top}+0_{n},\qquad\Delta(a)=P_{0}-P_{0}^{\top}=0_{n}.$$

The functions $\bar A$ and $\bar A^{\top}$, and the constant function $z:t\mapsto0_{n}$, are continuous on $[a,c]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v) and Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii). Thus $\Delta$ is a solution of the initial value problem with right-hand side $(t,X)\mapsto\bar A(t)X+X\bar A(t)^{\top}+0_{n}$ (the function of Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (i) for $\bar A$, $\bar A^{\top}$ and $z$) and initial value $0_{n}$; so is $z$, whose derivative is $0_{n}$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (ix), while $\bar A(t)0_{n}+0_{n}\bar A(t)^{\top}+0_{n}=0_{n}$. By Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (v), $\Delta=z$, so $h(t)-h(t)^{\top}=0_{n}$ and $h(t)=h(t)^{\top}$ for every $t$.

(b) $0_{n}\preceq h(t)$. The function $G=hDh:[a,c]\to\mathcal{M}_{n}(\mathbb{R})$ is continuous on $[a,c]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), and by (a) $G(t)=h(t)^{\top}D(t)h(t)$, which lies in $\mathcal{S}(n)$ and is positive semidefinite by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (v). Hence the function $C+G$ is continuous on $[a,c]$ by the same two clauses, and every $C(t)+G(t)$ lies in $\mathcal{S}(n)$ by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i) and is positive semidefinite by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (vii). Let $\bar A=A-hD$ be as in (a), continuous on $[a,c]$. By (a) and $D^{\top}=D$, $\bar A^{\top}=A^{\top}-Dh$, so $\bar Ah=Ah-hDh$ and $h\bar A^{\top}=hA^{\top}-hDh$, and

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

Thus $h$ is a solution of the initial value problem of The Lyapunov Differential Equation (i) with data $\bar A$, $C+G$ and $P_{0}$; it is the only one by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (i), and since $P_{0}$ and every $C(t)+G(t)$ lie in $\mathcal{S}(n)$ and are positive semidefinite, every $h(t)$ is positive semidefinite by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (iv), that is, $0_{n}\preceq h(t)$ by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vii).

(c) $h(t)\preceq\Lambda(t)$ and $\lVert h(t)\rVert\le\lVert\Lambda(t)\rVert$. The function $E=\Lambda-h$ is differentiable, by Rules for Derivatives of Vector- and Matrix-Valued Maps (iv), with

$$E'=\bigl(A\Lambda+\Lambda A^{\top}+C\bigr)-\bigl(Ah+hA^{\top}-hDh+C\bigr)=AE+EA^{\top}+G,\qquad E(a)=P_{0}-P_{0}=0_{n},$$

with $G=hDh$ as in (b), continuous on $[a,c]$ with every $G(t)$ in $\mathcal{S}(n)$ and positive semidefinite. Also $0_{n}$ is positive semidefinite: for every $x\in\mathbb{R}^{n}$, $0_{n}x=0$ by Products of Real Matrices with Zero Matrices, Negatives and Differences (i), so $x\cdot(0_{n}x)=x\cdot0=0\ge0$ by the dot product formula, which is the condition of Positive Semidefinite and Positive Definite Real Matrices (i). So $E$ is a solution of the initial value problem of The Lyapunov Differential Equation (i) with data $A$, $G$ and $0_{n}$; by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (i) and The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (iv), every $E(t)=\Lambda(t)-h(t)$ is positive semidefinite, so $h(t)\preceq\Lambda(t)$ by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vi). With (b), The Operator Norm of a Symmetric Matrix: Quadratic Forms, Eigenvalues and the Semidefinite Ordering (xi) gives $\lVert h(t)\rVert\le\lVert\Lambda(t)\rVert$.

Step 5: the a priori bound. $\Lambda$ is differentiable on $[a,b]$, hence continuous by Rules for Derivatives of Vector- and Matrix-Valued Maps (i), so $t\mapsto\lVert\Lambda(t)\rVert$ is continuous on $[a,b]$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iii) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v) and has a greatest value $\rho$ on $[a,b]$ by The Real Line: Standing Notation and Background for Calculus on Intervals (x). Then $\lVert P_{0}\rVert=\lVert\Lambda(a)\rVert\le\rho$, and by Step 4(c) every solution $h$ on $[a,c]$, for every $c$ with $a<c\le b$, satisfies $\lVert h(t)\rVert\le\lVert\Lambda(t)\rVert\le\rho$ for every $t\in[a,c]$.

Step 6: conclusion. By Steps 2, 3 and 5, the hypotheses of A Priori Bounded Initial Value Problems with Locally Lipschitz Right-Hand Side Have Exactly One Solution on the Whole Interval (iii) hold with $p=q=n$, right-hand side $F$, $\Xi=P_{0}$, $L_{\varrho}$ for each radius $\varrho>0$, and $\rho$; so the initial value problem has exactly one solution $P$ on $[a,b]$, which is (i). $P$ is a solution on $[a,c]$ with $c=b$, so Step 4 applies to $h=P$: for every $t\in[a,b]$, $P(t)\in\mathcal{S}(n)$ and $0_{n}\preceq P(t)\preceq\Lambda(t)$ by Step 4(a)-(c), which is (ii), and $\lVert P(t)\rVert\le\lVert\Lambda(t)\rVert$ by Step 4(c), which is (iii).