Proof of Variation of Constants for Linear Systems

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, and derivatives are taken relative to it. By The State-Transition Matrix of a Linear System (ii), the map $U=\Phi(\cdot,a)$ is differentiable with $U'(t)=A(t)\Phi(t,a)$ for every $t\in[a,b]$, and by Basic Properties of the State-Transition Matrix (v), with $a$ for $t$, the map $V:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $r\mapsto\Phi(a,r)$, is differentiable; by Rules for Derivatives of Vector- and Matrix-Valued Maps (i) and Continuous Map Between Metric Spaces (ii), $V$ is continuous on $[a,b]$. By Basic Properties of the State-Transition Matrix (ii) and The State-Transition Matrix of a Linear System (iii), $\Phi(t,a)\Phi(a,t)=\Phi(t,t)=I_{n}$ and $\Phi(t,a)\Phi(a,r)=\Phi(t,r)$ for all $r,t\in[a,b]$.

Proof of (i). Exactly one solution. By Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (iii), with $F(t,x)=A(t)x+g(t)$, the initial value problem has exactly one solution $x$; by Solution of an Initial Value Problem on $[a,c]$ (iii) and Solution of an Initial Value Problem on $[a,c]$ (i), with $b$ for $c$, every differentiable $y:[a,b]\to\mathbb{R}^{n}$ with $y'(t)=A(t)y(t)+g(t)$ for every $t\in[a,b]$ and $y(a)=\xi$ is a solution, and so equals $x$.

A candidate. Let $h:[a,b]\to\mathbb{R}^{n}$, $r\mapsto\Phi(a,r)g(r)=V(r)g(r)$; it is continuous on $[a,b]$ 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 its $(X,d)$ and $[a,b]$ for its $S$. By The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (i), with $n$ for $q$, $h$ for $x$ and $a$ for $c$, $\int_{a}^{t}h$ is defined for every $t\in[a,b]$, and the map $H:[a,b]\to\mathbb{R}^{n}$, $t\mapsto\int_{a}^{t}h$ (Maps Given by Integrals of Vector- and Matrix-Valued Maps (i)), is differentiable with $H'(t)=h(t)$. For every $k\in[n]$, $\int_{a}^{a}h_{k}=0$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii), so $H(a)=0$ by Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii). The constant map $t\mapsto\xi$ is differentiable with derivative $0$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (ix), so $z:t\mapsto\xi+H(t)$ is differentiable with $z'(t)=0+h(t)=h(t)$ by Rules for Derivatives of Vector- and Matrix-Valued Maps (ii) (and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)). Let $y:[a,b]\to\mathbb{R}^{n}$, $t\mapsto U(t)z(t)$. By Rules for Derivatives of Vector- and Matrix-Valued Maps (vi), $y$ is differentiable at every $t\in[a,b]$, hence differentiable, and, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v),

$$y'(t)=\bigl(A(t)\Phi(t,a)\bigr)z(t)+\Phi(t,a)\bigl(\Phi(a,t)g(t)\bigr)=A(t)y(t)+\bigl(\Phi(t,a)\Phi(a,t)\bigr)g(t)=A(t)y(t)+g(t).$$

Moreover $y(a)=\Phi(a,a)(\xi+0)=I_{n}\xi=\xi$ by The State-Transition Matrix of a Linear System (iii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v). Hence $y=x$.

The formula. Let $t\in[a,b]$. The map $r\mapsto\Phi(t,r)g(r)$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)) has the values $\bigl(\Phi(t,a)\Phi(a,r)\bigr)g(r)=\Phi(t,a)h(r)$, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v). As $\int_{a}^{t}h$ is defined, Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (vi), with $\Phi(t,a)$ for $L$, $a$ and $t$ for $u$ and $v$, and $h$ for $x$, shows that $\int_{a}^{t}\Phi(t,r)g(r)\,dr$ is defined and equals $\Phi(t,a)H(t)$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v),

$$x(t)=y(t)=\Phi(t,a)\xi+\Phi(t,a)H(t)=\Phi(t,a)\xi+\int_{a}^{t}\Phi(t,r)g(r)\,dr .$$

This proves (i).

Proof of (ii). Let $O:[a,b]\to\mathcal{M}_{q}(\mathbb{R})$ be the constant map with value $0_{q}$, continuous on $[a,b]$ 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). For $Y\in\mathcal{M}_{n\times q}(\mathbb{R})$, $Y0_{q}=0_{n\times q}$ by Products of Real Matrices with Zero Matrices, Negatives and Differences (i), so the right-hand side of Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem (v), with $A$, $O$ and $G$ for $A$, $B$ and $G$, is $(t,Y)\mapsto A(t)Y+G(t)$ (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)), and the initial value problem has exactly one solution $X$; as in the proof of (i), every differentiable $Y:[a,b]\to\mathcal{M}_{n\times q}(\mathbb{R})$ with $Y'(t)=A(t)Y(t)+G(t)$ for every $t\in[a,b]$ and $Y(a)=\Xi$ equals $X$, by Solution of an Initial Value Problem on $[a,c]$ (iii) and Solution of an Initial Value Problem on $[a,c]$ (ii). The rest of the proof of (i) applies word for word, with $G$, $\Xi$, $\mathcal{M}_{n\times q}(\mathbb{R})$, the zero matrix $0_{n\times q}$, the map $h:r\mapsto\Phi(a,r)G(r)$ into $\mathcal{M}_{n\times q}(\mathbb{R})$ and its entry maps $h_{ij}$ ($i\in[n]$, $j\in[q]$) in place of $g$, $\xi$, $\mathbb{R}^{n}$, $0$, the map $h$ into $\mathbb{R}^{n}$ and its component maps $h_{k}$ ($k\in[n]$); using the matrix halves of The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps (i), Rules for Derivatives of Vector- and Matrix-Valued Maps (ix) and Rules for Derivatives of Vector- and Matrix-Valued Maps (ii), Maps Given by Integrals of Vector- and Matrix-Valued Maps (ii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv) in place of Maps Given by Integrals of Vector- and Matrix-Valued Maps (i) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii), Rules for Derivatives of Vector- and Matrix-Valued Maps (v) in place of Rules for Derivatives of Vector- and Matrix-Valued Maps (vi), the rules 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 (iv) in place of Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v), and the identity $\int_{a}^{t}MA(r)\,dr=M\int_{a}^{t}A$ of Rules for the Riemann Integral of Vector- and Matrix-Valued Maps (vi), with $\Phi(t,a)$ for $M$ and $h$ for the map $A$ of that lemma. This proves (ii).