Proof of The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation

By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; it is the interval $I$ of Rules for Derivatives of Vector- and Matrix-Valued Maps, Rules for the Riemann Integral of Vector- and Matrix-Valued Maps and The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps throughout, derivatives are taken relative to it as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii), and the functions below whose values are sums, differences, negatives, products or transposes of values of given functions (such as $W$, $K$, $\Delta C$, $D$, $G$ and the integrands) are formed pointwise by Maps Formed Pointwise from Vector and Matrix Operations (i). Products of three matrices, such as $\Phi(t,r)C(r)\Phi(t,r)^{\top}$, are unambiguous by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i). Matrix products, sums and transposes are rearranged by 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 (iv) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vii) (in particular $(XY)^{\top}=Y^{\top}X^{\top}$, $(X^{\top})^{\top}=X$, $I_{n}^{\top}=I_{n}$, and $(-X)^{\top}=-X^{\top}$, as $-X=(-1)X$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv)), products with negatives by Products of Real Matrices with Zero Matrices, Negatives and Differences (ii), and sums and differences by 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).

Preliminaries. (a) Continuity. For every $s\in[a,b]$, the functions $t\mapsto\Phi(t,s)$ and $t\mapsto\Phi(s,t)$ are differentiable, by The State-Transition Matrix of a Linear System (ii) and Basic Properties of the State-Transition Matrix (v), hence continuous on $[a,b]$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (i) and Continuous Map Between Metric Spaces (ii). 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), with $(\mathbb{R},d_{\mathbb{R}})$ for $(X,d)$ and $[a,b]$ for $S$, constant functions, sums, matrix products and transposes of functions $[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ continuous on $[a,b]$ are continuous on $[a,b]$, and by Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii), with the same $(X,d)$ and $S$ and with $n$ for $p$ and $q$, so are their negatives and differences; this gives the continuity of every function below built from $\Phi(\cdot,s)$, $\Phi(s,\cdot)$, $A$, $C$, $\tilde{C}$ and $Q$, and in particular of $A^{\top}:t\mapsto A(t)^{\top}$ and of the functions in (ii) and (vii).

(b) Composition. By Basic Properties of the State-Transition Matrix (ii) and The State-Transition Matrix of a Linear System (iii), $\Phi(t,r)\Phi(r,s)=\Phi(t,s)$ and $\Phi(s,s)=I_{n}$ for all $r,s,t\in[a,b]$; in particular $\Phi(t,s)\Phi(s,t)=I_{n}$.

(c) Congruence. For $X\in\mathcal{S}(n)$ and $Y\in\mathcal{M}_{n}(\mathbb{R})$, $Y^{\top}XY\in\mathcal{S}(n)$, and $Y^{\top}XY$ is positive semidefinite if $X$ is, by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (v) with $n$ for $m$ and $Y$ for $C$.

Proof of (i). The function $A^{\top}$ is continuous on $[a,b]$ by (a). By 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), with $n$ for $q$, $A^{\top}$ for $B$ and $C$ for $G$, the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) with right-hand side $(t,X)\mapsto A(t)X+XA(t)^{\top}+C(t)$ and initial value $P_{0}$, which by The Lyapunov Differential Equation (i) is the Lyapunov differential equation with data $A$, $C$ and $P_{0}$, has exactly one solution $P$. By Solution of an Initial Value Problem on $[a,c]$ (iii) and Solution of an Initial Value Problem on $[a,c]$ (ii), with $b$ for $c$, this means:

($\ast$) for every function $C_{1}:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ continuous on $[a,b]$ and every $P_{1}\in\mathcal{M}_{n}(\mathbb{R})$ (the argument above applies to them in place of $C$ and $P_{0}$), every differentiable $Y:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ with $Y'(t)=A(t)Y(t)+Y(t)A(t)^{\top}+C_{1}(t)$ for every $t\in[a,b]$ and $Y(a)=P_{1}$ is the solution of the Lyapunov differential equation of The Lyapunov Differential Equation (i) with data $A$, $C_{1}$ and $P_{1}$.

Proof of (ii). Let $V:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\Phi(a,r)$; by Basic Properties of the State-Transition Matrix (v), with $a$ for $t$, $V$ is differentiable with $V'(r)=-\bigl(V(r)A(r)\bigr)$. Let $W:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto V(r)C(r)V(r)^{\top}$, continuous on $[a,b]$ by (a), and let $K:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto V(r)P(r)V(r)^{\top}$. As $P$ is differentiable with $P'(r)=A(r)P(r)+P(r)A(r)^{\top}+C(r)$, Rules for Derivatives of Vector- and Matrix-Valued Maps (v), applied twice, and Rules for Derivatives of Vector- and Matrix-Valued Maps (viii) show that $K$ is differentiable, with, writing $V,A,P,C$ for their values at $r$,

$$K'(r)=\bigl(-VAP+V(AP+PA^{\top}+C)\bigr)V^{\top}+VP\bigl(-(VA)\bigr)^{\top}=-VAPV^{\top}+VAPV^{\top}+VPA^{\top}V^{\top}+VCV^{\top}-VPA^{\top}V^{\top}=W(r).$$

So $K'=W$ is continuous on $[a,b]$, and The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (iii), with $a$ for $u$ and $t$ for $v$, shows that $\int_{a}^{t}W$ is defined and equals $K(t)-K(a)$. By (b), $K(a)=I_{n}P_{0}I_{n}^{\top}=P_{0}$, so $K(t)=P_{0}+\int_{a}^{t}W$. By (b), $\Phi(t,a)V(t)=\Phi(t,a)\Phi(a,t)=I_{n}$, hence

$$P(t)=\bigl(\Phi(t,a)V(t)\bigr)P(t)\bigl(\Phi(t,a)V(t)\bigr)^{\top}=\Phi(t,a)K(t)\Phi(t,a)^{\top}=\Phi(t,a)P_{0}\Phi(t,a)^{\top}+\Bigl(\Phi(t,a)\int_{a}^{t}W\Bigr)\Phi(t,a)^{\top}.$$

By Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (vi), with $\Phi(t,a)$ for $M$, $\Phi(t,a)^{\top}$ for $N$ and $W$ for $A$, the last term is the integral from $a$ to $t$ of $r\mapsto\Phi(t,a)W(r)\Phi(t,a)^{\top}$, and by (b) this value is $\bigl(\Phi(t,a)\Phi(a,r)\bigr)C(r)\bigl(\Phi(t,a)\Phi(a,r)\bigr)^{\top}=\Phi(t,r)C(r)\Phi(t,r)^{\top}$. So the function $r\mapsto\Phi(t,r)C(r)\Phi(t,r)^{\top}$, continuous on $[a,b]$ by (a), has a defined integral from $a$ to $t$, and the formula of (ii) holds.

Proof of (iii) and (iv). Let $t\in[a,b]$, and assume the hypothesis of (iii), which that of (iv) includes. By (c) with $\Phi(t,a)^{\top}$ for $Y$, $\Phi(t,a)P_{0}\Phi(t,a)^{\top}=\bigl(\Phi(t,a)^{\top}\bigr)^{\top}P_{0}\Phi(t,a)^{\top}$ is symmetric, and positive semidefinite if $P_{0}$ is. Likewise, by (c) with $\Phi(t,r)^{\top}$ for $Y$, the function $M_{t}:r\mapsto\Phi(t,r)C(r)\Phi(t,r)^{\top}$ of (ii), continuous on $[a,b]$, takes values in $\mathcal{S}(n)$, positive semidefinite ones if every $C(r)$ is. So, by Integrals Preserve Order, Symmetry and Positive Semidefiniteness (ii) with $M_{t}$ for $M$ and $a$, $t$ for $u$, $v$, $\int_{a}^{t}M_{t}\in\mathcal{S}(n)$, and, as $a\le t$, $\int_{a}^{t}M_{t}$ is positive semidefinite under the hypothesis of (iv) by Integrals Preserve Order, Symmetry and Positive Semidefiniteness (iii). By (ii), $P(t)$ is the sum of the two, hence symmetric by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i), and positive semidefinite under the hypothesis of (iv) by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (vii).

Proof of (v). By (iii), applied as stated and with $\tilde{C}$, $\tilde{P}_{0}$ for $C$, $P_{0}$, $P(t)$ and $\tilde{P}(t)$ are symmetric for every $t\in[a,b]$. Let $\Delta C:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\tilde{C}(r)-C(r)$, continuous on $[a,b]$ by Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii), with $(\mathbb{R},d_{\mathbb{R}})$ for $(X,d)$, $[a,b]$ for $S$, $n$ for $p$ and $q$, and $\tilde{C}$, $C$ for $A$, $B$, and let $D:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\tilde{P}(r)-P(r)$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (iv), $D$ is differentiable, and, by 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 (ii) and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i),

$$D'(r)=\bigl(A\tilde{P}+\tilde{P}A^{\top}+\tilde{C}\bigr)-\bigl(AP+PA^{\top}+C\bigr)=A(r)D(r)+D(r)A(r)^{\top}+\Delta C(r),$$

with $D(a)=\tilde{P}_{0}-P_{0}$. By ($\ast$), $D$ is the solution of the Lyapunov differential equation of The Lyapunov Differential Equation (i) with data $A$, $\Delta C$ and $\tilde{P}_{0}-P_{0}$. These data are symmetric by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i) and positive semidefinite by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vi), since $P_{0}\preceq\tilde{P}_{0}$ and $C(r)\preceq\tilde{C}(r)$. By (iv), applied with $\Delta C$ for $C$ and $\tilde{P}_{0}-P_{0}$ for $P_{0}$, $D(t)=\tilde{P}(t)-P(t)$ is positive semidefinite for every $t\in[a,b]$, so $P(t)\preceq\tilde{P}(t)$ by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vi).

Proof of (vi) and (vii). Let $U:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\Phi(r,b)$, differentiable with $U'(r)=A(r)U(r)$ by The State-Transition Matrix of a Linear System (ii); let $V:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\Phi(b,r)$, differentiable with $V'(r)=-\bigl(V(r)A(r)\bigr)$ by Basic Properties of the State-Transition Matrix (v); and let $G:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto U(r)^{\top}Q(r)U(r)$, continuous on $[a,b]$ by (a). By (b), $U(t)V(t)=I_{n}$ and $U(r)V(t)=\Phi(r,t)$ for all $r,t\in[a,b]$, so

$$V(t)^{\top}G(r)V(t)=\bigl(U(r)V(t)\bigr)^{\top}Q(r)\bigl(U(r)V(t)\bigr)=\Phi(r,t)^{\top}Q(r)\Phi(r,t),\qquad V(t)^{\top}G(t)V(t)=Q(t).$$

By The Lyapunov Differential Equation (ii), a solution is a differentiable function $Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ with $Z'(t)=-\bigl(A(t)^{\top}Z(t)+Z(t)A(t)+Q(t)\bigr)$ for every $t\in[a,b]$ and $Z(b)=F$.

Every solution has the stated form. Let $Z$ be a solution, and let $K:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto U(r)^{\top}Z(r)U(r)$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (v), applied twice, and Rules for Derivatives of Vector- and Matrix-Valued Maps (viii), $K$ is differentiable with, writing $U,A,Z,Q$ for their values at $r$,

$$K'(r)=U^{\top}A^{\top}ZU-U^{\top}\bigl(A^{\top}Z+ZA+Q\bigr)U+U^{\top}ZAU=-U^{\top}QU=-G(r).$$

As $G$ is continuous on $[a,b]$, so is $K'=-G$, by Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii) with $(\mathbb{R},d_{\mathbb{R}})$ for $(X,d)$, $[a,b]$ for $S$ and $n$ for $p$ and $q$; and, for $t\in[a,b]$, The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (iii), with $t$ for $u$ and $b$ for $v$, gives $K(b)-K(t)=\int_{t}^{b}K'=-\int_{t}^{b}G$, by Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (iii) with $-1$ for $c$, the integral of $G$ from $t$ to $b$ being defined by Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (i). As $K(b)=I_{n}^{\top}FI_{n}=F$ by (b), $K(t)=F+\int_{t}^{b}G$. Since $U(t)V(t)=I_{n}$, and by Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (vi) with $V(t)^{\top}$ for $M$, $V(t)$ for $N$ and $G$ for $A$,

$$Z(t)=\bigl(U(t)V(t)\bigr)^{\top}Z(t)\bigl(U(t)V(t)\bigr)=V(t)^{\top}K(t)V(t)=\Phi(b,t)^{\top}F\,\Phi(b,t)+\int_{t}^{b}V(t)^{\top}G(r)V(t)\,dr ,$$

and the integrand is $r\mapsto\Phi(r,t)^{\top}Q(r)\Phi(r,t)$, continuous on $[a,b]$ by (a). So every solution satisfies the formula of (vii); in particular there is at most one solution.

A solution exists. By The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (i), with $a$ for $c$ and $G$ for $A$, the function $H:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto\int_{a}^{t}G$ (Maps Given by Integrals of Vector- and Matrix-Valued Maps (ii)), is differentiable with $H'=G$. Let $\bar{K}:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto\bigl(F+H(b)\bigr)-H(t)$; by Rules for Derivatives of Vector- and Matrix-Valued Maps (ix) and Rules for Derivatives of Vector- and Matrix-Valued Maps (iv) it is differentiable with $\bar{K}'(t)=-G(t)$, and $\bar{K}(b)=F$. Let $\bar{Z}:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto V(t)^{\top}\bar{K}(t)V(t)$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (v), applied twice, and Rules for Derivatives of Vector- and Matrix-Valued Maps (viii), $\bar{Z}$ is differentiable with, writing $V,A,\bar{K},G,Q$ for their values at $t$,

$$\bar{Z}'(t)=\bigl(-(VA)\bigr)^{\top}\bar{K}V-V^{\top}GV+V^{\top}\bar{K}\bigl(-(VA)\bigr)=-A^{\top}\bar{Z}(t)-Q-\bar{Z}(t)A=-\bigl(A(t)^{\top}\bar{Z}(t)+\bar{Z}(t)A(t)+Q(t)\bigr),$$

and $\bar{Z}(b)=I_{n}^{\top}FI_{n}=F$ by (b). So $\bar{Z}$ is a solution, and by the previous step it is the only one, which proves (vi), and it satisfies the formula of (vii).

Proof of (viii) and (ix). Let $t\in[a,b]$, and assume the hypothesis of (viii), which that of (ix) includes. By (c) with $\Phi(b,t)$ for $Y$, $\Phi(b,t)^{\top}F\,\Phi(b,t)$ is symmetric, and positive semidefinite if $F$ is; by (c) with $\Phi(r,t)$ for $Y$, the integrand $N_{t}:r\mapsto\Phi(r,t)^{\top}Q(r)\Phi(r,t)$ of (vii), continuous on $[a,b]$, takes values in $\mathcal{S}(n)$, positive semidefinite ones if every $Q(r)$ is. So, by Integrals Preserve Order, Symmetry and Positive Semidefiniteness (ii) with $N_{t}$ for $M$ and $t$, $b$ for $u$, $v$, $\int_{t}^{b}N_{t}\in\mathcal{S}(n)$, and, as $t\le b$, $\int_{t}^{b}N_{t}$ is positive semidefinite under the hypothesis of (ix) by Integrals Preserve Order, Symmetry and Positive Semidefiniteness (iii). By (vii), $Z(t)$ is the sum of the two, hence symmetric by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i), and positive semidefinite under the hypothesis of (ix) by Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (vii).