Proof of The Control Riccati Equation with Cross Term: Global Existence and Uniqueness, Positive Semidefiniteness, and the Lyapunov Bound

Real arithmetic is that of The Real Numbers as a Type (iv). Functions formed pointwise from functions on a set, such as $u\mapsto\tilde A(\psi(u))^{\top}$ and $\Delta=Y-Y^{\top}$ below, are those of Maps Formed Pointwise from Vector and Matrix Operations (i), and pointwise inverses, such as $(t,X)\mapsto R(t)^{-1}$ in the functions $G$ and $K$ of Step 2, are those of Maps Formed Pointwise from Vector and Matrix Operations (ii), applied to the composite of $R$ with the first projection $[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to[a,b]$ of Projection, Evaluation and Currying Maps (i) (a function by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)). 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) 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: the reversed data. Let $R^{-1}:[a,b]\to\mathcal{M}_{m}(\mathbb{R})$ be the function $t\mapsto R(t)^{-1}$ of Maps Formed Pointwise from Vector and Matrix Operations (ii). It is continuous on $[a,b]$ by Continuity and Derivative of the Inverse of a Matrix-Valued Map (i), and every $R(t)^{-1}$ lies in $\mathcal{S}(m)$ and is positive definite by Invertibility of Positive Definite Matrices (ii) and Invertibility of Positive Definite Matrices (iii). Hence $\tilde A$ and $\tilde Q$ are continuous on $[a,b]$ 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). Let $\psi:[a,b]\to[a,b]$ be the reflection $\psi(u)=a+b-u$ of The Reflection of a Closed Interval (i); it satisfies $\psi(\psi(u))=u$ for every $u\in[a,b]$, by The Reflection of a Closed Interval (ii), and is continuous on $[a,b]$, by The Reflection of a Closed Interval (iv). Define $\hat A,\hat C,\hat D:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ by

$$\hat A(u)=\tilde A(\psi(u))^{\top},\qquad\hat C(u)=\tilde Q(\psi(u)),\qquad\hat D(u)=B(\psi(u))R(\psi(u))^{-1}B(\psi(u))^{\top}.$$

They are continuous on $[a,b]$: $\tilde A$ and $\tilde Q$ are, as just shown, and so is $u\mapsto B(u)R(u)^{-1}B(u)^{\top}$, 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); their compositions with $\psi$ are continuous by the composition rule of The Real Line: Standing Notation and Background for Calculus on Intervals (vi), and transposes of continuous functions are continuous by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv). For every $u\in[a,b]$, with $s=\psi(u)$: $\hat C(u)=\tilde Q(s)$ lies in $\mathcal{S}(n)$ and is positive semidefinite, by the statement and the hypothesis; and $\hat D(u)=\bigl(B(s)^{\top}\bigr)^{\top}R(s)^{-1}B(s)^{\top}$ 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), applied with the matrix $B(s)^{\top}\in\mathcal{M}_{m\times n}(\mathbb{R})$ and with $R(s)^{-1}$, which is positive semidefinite by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (x). Also $F\in\mathcal{S}(n)$ is positive semidefinite. Hence The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound (i) and The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound (ii) apply with $\hat A$, $\hat C$, $\hat D$ in place of $A$, $C$, $D$ and $F$ in place of $P_{0}$. Let $\tilde H:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the right-hand side of the initial value problem of The Riccati Differential Equations of Filtering and Control (i) for these data,

$$\tilde H(u,Y)=\hat A(u)Y+Y\hat A(u)^{\top}-Y\hat D(u)Y+\hat C(u).$$

By The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound (i), this initial value problem, with initial value $F$, has exactly one solution $P$; by The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound (ii), $P(u)\in\mathcal{S}(n)$ and $0_{n}\preceq P(u)\preceq\Lambda(u)$ for every $u\in[a,b]$, where $\Lambda$ is the solution of the initial value problem of The Lyapunov Differential Equation (i) with data $\hat A$, $\hat C$ and $F$.

Step 2: reversing the control problem. Let $G:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the function defined by the right-hand side displayed in The Riccati Differential Equations of Filtering and Control (ii) for $A$, $B$, $Q$, $R$, $V$,

$$G(t,X)=-\Bigl(A(t)^{\top}X+XA(t)-K(t,X)+Q(t)\Bigr),\qquad K(t,X)=\bigl(XB(t)+V(t)\bigr)R(t)^{-1}\bigl(XB(t)+V(t)\bigr)^{\top}.$$

By Time Reversal: Terminal Value Problems as Initial Value Problems (ii), the function $H:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$, $H(u,Y)=-G(\psi(u),Y)$, exists, and a function $Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ is a solution of the terminal value problem with right-hand side $G$ and terminal value $F$ if and only if $Z\circ\psi:u\mapsto Z(\psi(u))$ is a solution of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) with right-hand side $H$ and initial value $F$. With $s=\psi(u)$ and $-(-M)=M$,

$$H(u,Y)=A(s)^{\top}Y+YA(s)-K(s,Y)+Q(s).$$

Step 3: every solution of the reversed problem is symmetric. For every $s\in[a,b]$ and $Y\in\mathcal{M}_{n}(\mathbb{R})$, $K(s,Y)=\bigl((YB(s)+V(s))^{\top}\bigr)^{\top}R(s)^{-1}\bigl(YB(s)+V(s)\bigr)^{\top}$ lies in $\mathcal{S}(n)$ by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (v), and $Q(s)^{\top}=Q(s)$; hence

$$H(u,Y)-H(u,Y)^{\top}=A(s)^{\top}\bigl(Y-Y^{\top}\bigr)+\bigl(Y-Y^{\top}\bigr)A(s).$$

Let $Y$ be a solution of the initial value problem with right-hand side $H$ and initial value $F$. The function $\Delta=Y-Y^{\top}$ is differentiable with $\Delta'=Y'-(Y')^{\top}$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (viii) and Rules for Derivatives of Vector- and Matrix-Valued Maps (iv), so

$$\Delta'(u)=A(\psi(u))^{\top}\Delta(u)+\Delta(u)A(\psi(u))+0_{n},\qquad\Delta(a)=F-F^{\top}=0_{n}.$$

The functions $u\mapsto A(\psi(u))^{\top}$ and $u\mapsto A(\psi(u))$ are continuous on $[a,b]$ (composition rule of The Real Line: Standing Notation and Background for Calculus on Intervals (vi) and Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv)), and so is the constant function $z:u\mapsto0_{n}$. Thus $\Delta$ and $z$ (whose derivative is $0_{n}$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (ix)) are both solutions of the initial value problem with the right-hand side of Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (i) for these two maps and $z$, and initial value $0_{n}$. By Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (v), $\Delta=z$, so $Y(u)=Y(u)^{\top}$, that is $Y(u)\in\mathcal{S}(n)$, for every $u\in[a,b]$.

Step 4: on symmetric matrices, $H=\tilde H$. Let $u\in[a,b]$, $s=\psi(u)$ and $Y\in\mathcal{S}(n)$. Since $R(s)^{-1}$ and $Y$ are symmetric, $(YB(s)+V(s))^{\top}=B(s)^{\top}Y+V(s)^{\top}$ and $\bigl(B(s)R(s)^{-1}V(s)^{\top}\bigr)^{\top}=V(s)R(s)^{-1}B(s)^{\top}$, so, omitting the argument $s$,

$$K(s,Y)=YBR^{-1}B^{\top}Y+YBR^{-1}V^{\top}+VR^{-1}B^{\top}Y+VR^{-1}V^{\top}, \qquad \tilde A^{\top}Y+Y\tilde A=A^{\top}Y-VR^{-1}B^{\top}Y+YA-YBR^{-1}V^{\top}.$$

Hence, substituting the expansion of $K(s,Y)$ into the formula for $H$ of Step 2 and regrouping the four terms of $-K(s,Y)$,

$$H(u,Y)=\bigl(A^{\top}Y+YA-VR^{-1}B^{\top}Y-YBR^{-1}V^{\top}\bigr)-YBR^{-1}B^{\top}Y+\bigl(Q-VR^{-1}V^{\top}\bigr):$$

the cross terms $-VR^{-1}B^{\top}Y$ and $-YBR^{-1}V^{\top}$ of $-K(s,Y)$ join $A^{\top}Y+YA$ to give $\tilde A^{\top}Y+Y\tilde A$, and $-VR^{-1}V^{\top}$ joins $Q$ to give $\tilde Q$. Therefore

$$H(u,Y)=\tilde A(s)^{\top}Y+Y\tilde A(s)-Y\bigl(B(s)R(s)^{-1}B(s)^{\top}\bigr)Y+\tilde Q(s)=\hat A(u)Y+Y\hat A(u)^{\top}-Y\hat D(u)Y+\hat C(u)=\tilde H(u,Y),$$

using $\hat A(u)^{\top}=\bigl(\tilde A(s)^{\top}\bigr)^{\top}=\tilde A(s)$.

Step 5: proof of (i). The solution $P$ of Step 1 has symmetric values, so by Step 4 $P'(u)=\tilde H(u,P(u))=H(u,P(u))$ for every $u$, and $P(a)=F$: $P$ is a solution of the initial value problem with right-hand side $H$ and initial value $F$. Conversely, let $Y$ be any solution of this problem; by Step 3 its values are symmetric, so by Step 4 $Y'(u)=H(u,Y(u))=\tilde H(u,Y(u))$, and $Y$ is a solution of the filter problem of Step 1, whence $Y=P$ by The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound (i). So $P$ is the only solution of the reversed problem. Let $Z=P\circ\psi$, the function $t\mapsto P(a+b-t)$ of Time Reversal: Terminal Value Problems as Initial Value Problems (ii). Then $Z\circ\psi=P$, since $\psi(\psi(u))=u$, so $Z$ is a solution of the terminal value problem by Step 2. If $Z_{1}$ is any solution of the terminal value problem, then $Z_{1}\circ\psi$ is a solution of the reversed problem by Step 2, so $Z_{1}\circ\psi=P$ and $Z_{1}=Z_{1}\circ\psi\circ\psi=P\circ\psi=Z$. This proves (i).

Step 6: proof of (ii). For $t\in[a,b]$, $Z(t)=P(\psi(t))$ lies in $\mathcal{S}(n)$ and satisfies $0_{n}\preceq Z(t)$ by Step 1, so $Z(t)$ is positive semidefinite by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vii).

Step 7: proof of (iii). Let $\hat Z$ be as in (iii), and let $G_{L}:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the function defined by the right-hand side displayed in The Lyapunov Differential Equation (ii) for $\tilde A$ and $\tilde Q$, $G_{L}(t,X)=-\bigl(\tilde A(t)^{\top}X+X\tilde A(t)+\tilde Q(t)\bigr)$. By Time Reversal: Terminal Value Problems as Initial Value Problems (ii), $\hat Y=\hat Z\circ\psi$ is a solution of the initial value problem with right-hand side $(u,Y)\mapsto-G_{L}(\psi(u),Y)$ and initial value $F$. As

$$-G_{L}(\psi(u),Y)=\tilde A(\psi(u))^{\top}Y+Y\tilde A(\psi(u))+\tilde Q(\psi(u))=\hat A(u)Y+Y\hat A(u)^{\top}+\hat C(u),$$

$\hat Y$ is a solution of the initial value problem of The Lyapunov Differential Equation (i) with data $\hat A$, $\hat C$ and $F$, so $\hat Y=\Lambda$ by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (i). Hence, for $t\in[a,b]$, Step 1 with $u=\psi(t)$ gives

$$Z(t)=P(\psi(t))\preceq\Lambda(\psi(t))=\hat Z(\psi(\psi(t)))=\hat Z(t).$$