Proof of Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem

Computations with real numbers are those of The Real Numbers as a Type. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$. As in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (vi), $d_{E}(u,v)=\lVert u-v\rVert$ for $u,v\in\mathbb{R}^{n}$ and $d_{p\times r}(M,N)=\lVert M-N\rVert$ for $M,N\in\mathcal{M}_{p\times r}(\mathbb{R})$.

Proof of (i). Let $A$ and $g$ be as there. For $t\in[a,b]$ and $x\in\mathbb{R}^{n}$, the matrix-vector product $A(t)x$ lies in $\mathbb{R}^{n}$ (Euclidean Space and Real Matrices: Standing Notation (iii)), and so does the sum $A(t)x+g(t)$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv)). So the map $[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$, $(t,x)\mapsto A(t)x+g(t)$, given by this expression, exists (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)); two maps $[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ with these values agree at every point, and so are equal by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv). Likewise, for $A$, $B$ and $G$ as there, $t\in[a,b]$ and $X\in\mathcal{M}_{n\times q}(\mathbb{R})$, the products $A(t)X$ and $XB(t)$ lie in $\mathcal{M}_{n\times q}(\mathbb{R})$ (Euclidean Space and Real Matrices: Standing Notation (iii)), and so does $A(t)X+XB(t)+G(t)$ (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv)); so the map $(t,X)\mapsto A(t)X+XB(t)+G(t)$ from $[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$ to $\mathcal{M}_{n\times q}(\mathbb{R})$ exists (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), and it is the only map with these values by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv). This proves (i).

Step 1: a bound for a continuous matrix-valued map. Let $p\in\mathbb{N}$ and let $C:[a,b]\to\mathcal{M}_{p}(\mathbb{R})$ be continuous on $[a,b]$. We show that there is $K\in\mathbb{R}$ with $K\ge0$ and $\lVert C(t)\rVert\le K$ for every $t\in[a,b]$. The map $\nu:[a,b]\to\mathbb{R}$, $t\mapsto\lVert C(t)\rVert$ (the composite of $C$ with the operator norm, a map $\mathcal{M}_{p}(\mathbb{R})\to\mathbb{R}$ by The Operator Norm of a Real Matrix (i)), 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), with $(\mathbb{R},d_{\mathbb{R}})$ for its $(X,d)$ and $[a,b]$ for its $S$. By Closed Interval $[a,b]$ is Compact in $\mathbb{R}$ (i), $[a,b]$ is compact in $(\mathbb{R},\mathcal{T}_{\mathbb{R}})$, where $\mathcal{T}_{\mathbb{R}}$ is the set of open subsets of $(\mathbb{R},d_{\mathbb{R}})$, that is, the topology $\mathcal{T}_{d}$ of Extreme Value Theorem on a Compact Subset of a Metric Space for $(\mathbb{R},d_{\mathbb{R}})$ in place of $(X,d)$; and $[a,b]$ is nonempty. By Extreme Value Theorem on a Compact Subset of a Metric Space (i), with $[a,b]$ for its $K$ and $\nu$ for its $f$, there is $t_{1}\in[a,b]$ with $\nu(t)\le\nu(t_{1})$ for every $t\in[a,b]$. So $K=\lVert C(t_{1})\rVert$ satisfies $\lVert C(t)\rVert\le K$ for every $t\in[a,b]$, and $K\ge0$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i).

Proof of (ii). Let $A$, $g$ and $F$ be as there. By Step 1, with $n$ for $p$ and $A$ for $C$, there is $K_{0}\in\mathbb{R}$ with $K_{0}\ge0$ and $\lVert A(t)\rVert\le K_{0}$ for every $t\in[a,b]$.

Lipschitz bound. Let $K\in\mathbb{R}$ with $K\ge0$ and $\lVert A(t)\rVert\le K$ for every $t\in[a,b]$, and let $t\in[a,b]$ and $x,y\in\mathbb{R}^{n}$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (v) (and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)), $F(t,x)-F(t,y)=\bigl(A(t)x-A(t)y\bigr)+\bigl(g(t)-g(t)\bigr)=A(t)x-A(t)y=A(t)(x-y)$. By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii), and as $\lVert x-y\rVert\ge0$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i),

$$\lVert F(t,x)-F(t,y)\rVert\le\lVert A(t)\rVert\,\lVert x-y\rVert\le K\,\lVert x-y\rVert .$$

Continuity. Let $d$ be the product metric on $\mathbb{R}\times\mathbb{R}^{n}$ and let $(t_{0},x_{0})\in[a,b]\times\mathbb{R}^{n}$. The choices are made in this order. First a real $\varepsilon>0$ is given; put $\mu=\varepsilon/\bigl(3(1+\lVert x_{0}\rVert)\bigr)$, a positive real number as $\lVert x_{0}\rVert\ge0$. Next, $A$ is continuous at $t_{0}$ relative to $[a,b]$ by Continuous Map Between Metric Spaces (ii), and Continuous Map Between Metric Spaces (i) gives a real $\delta_{1}>0$ such that $\lVert A(t)-A(t_{0})\rVert<\mu$ for every $t\in[a,b]$ with $d_{\mathbb{R}}(t_{0},t)<\delta_{1}$; likewise it gives, for $g$, a real $\delta_{2}>0$ such that $\lVert g(t)-g(t_{0})\rVert<\varepsilon/3$ for every $t\in[a,b]$ with $d_{\mathbb{R}}(t_{0},t)<\delta_{2}$. Let $\delta>0$ be the smallest of $\delta_{1}$, $\delta_{2}$ and $\varepsilon/\bigl(3(1+K_{0})\bigr)$. Now let $(t,x)\in[a,b]\times\mathbb{R}^{n}$ with $d\bigl((t_{0},x_{0}),(t,x)\bigr)<\delta$. By The Product Metric is a Metric (iii), $d_{\mathbb{R}}(t_{0},t)<\delta$ and $\lVert x_{0}-x\rVert=d_{E}(x_{0},x)<\delta$, and $\lVert x-x_{0}\rVert=\lVert x_{0}-x\rVert$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v). By Products of Real Matrices with Zero Matrices, Negatives and Differences (iii) (and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)),

$$F(t,x)-F(t_{0},x_{0})=A(t)(x-x_{0})+\bigl(A(t)-A(t_{0})\bigr)x_{0}+\bigl(g(t)-g(t_{0})\bigr),$$

so, by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (vi) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (ii),

$$\lVert F(t,x)-F(t_{0},x_{0})\rVert\le K_{0}\,\lVert x-x_{0}\rVert+\mu\,\lVert x_{0}\rVert+\lVert g(t)-g(t_{0})\rVert\le\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{3}+\lVert g(t)-g(t_{0})\rVert<\varepsilon .$$

Thus $d_{E}\bigl(F(t,x),F(t_{0},x_{0})\bigr)<\varepsilon$. So $F$ is continuous at $(t_{0},x_{0})$ relative to $[a,b]\times\mathbb{R}^{n}$ by Continuous Map Between Metric Spaces (i), and, the point being arbitrary, continuous on $[a,b]\times\mathbb{R}^{n}$ by Continuous Map Between Metric Spaces (ii). This proves (ii).

Proof of (iii). Let $A$, $g$ and $F$ be as there. By (ii), there is $K_{0}\in\mathbb{R}$ with $K_{0}\ge0$ and $\lVert A(t)\rVert\le K_{0}$ for every $t\in[a,b]$, and $F$ is continuous on $[a,b]\times\mathbb{R}^{n}$ and satisfies the Lipschitz bound with the constant $K_{0}$. By The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (i), with $K_{0}$ for $L$, for every $\xi\in\mathbb{R}^{n}$ the initial value problem with right-hand side $F$ and initial value $\xi$ has exactly one solution. This proves (iii).

Proof of (iv). Let $A$, $B$, $G$ and $F$ be as there. Step 1, with $n$ and $A$, and with $q$ and $B$, for $p$ and $C$, gives the bounds $K_{A}$ and $K_{B}$; fix such $K_{A},K_{B}\ge0$.

Lipschitz bound. Let $K_{A}',K_{B}'\in\mathbb{R}$ with $K_{A}',K_{B}'\ge0$, $\lVert A(t)\rVert\le K_{A}'$ and $\lVert B(t)\rVert\le K_{B}'$ for every $t\in[a,b]$, and let $t\in[a,b]$ and $X,Y\in\mathcal{M}_{n\times q}(\mathbb{R})$. By Products of Real Matrices with Zero Matrices, Negatives and Differences (iii) (and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)), $F(t,X)-F(t,Y)=A(t)(X-Y)+(X-Y)B(t)$. 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 (vii) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i),

$$\lVert F(t,X)-F(t,Y)\rVert\le\lVert A(t)\rVert\,\lVert X-Y\rVert+\lVert X-Y\rVert\,\lVert B(t)\rVert\le(K_{A}'+K_{B}')\,\lVert X-Y\rVert .$$

Continuity. Let $d$ be the product metric on $\mathbb{R}\times\mathcal{M}_{n\times q}(\mathbb{R})$ and let $(t_{0},X_{0})\in[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$. The choices are made in this order. First a real $\varepsilon>0$ is given; put $\mu=\varepsilon/\bigl(5(1+\lVert X_{0}\rVert)\bigr)>0$, using Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (i). Next, by Continuous Map Between Metric Spaces (ii) and Continuous Map Between Metric Spaces (i), there are real $\delta_{A},\delta_{B},\delta_{G}>0$ such that, for every $t\in[a,b]$, $d_{\mathbb{R}}(t_{0},t)<\delta_{A}$ implies $\lVert A(t)-A(t_{0})\rVert<\mu$, $d_{\mathbb{R}}(t_{0},t)<\delta_{B}$ implies $\lVert B(t)-B(t_{0})\rVert<\mu$, and $d_{\mathbb{R}}(t_{0},t)<\delta_{G}$ implies $\lVert G(t)-G(t_{0})\rVert<\varepsilon/5$. Let $\delta>0$ be the smallest of $\delta_{A}$, $\delta_{B}$, $\delta_{G}$ and $\varepsilon/\bigl(5(1+K_{A}+K_{B})\bigr)$. Now let $(t,X)\in[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$ with $d\bigl((t_{0},X_{0}),(t,X)\bigr)<\delta$. By The Product Metric is a Metric (iii), $d_{\mathbb{R}}(t_{0},t)<\delta$ and $\lVert X_{0}-X\rVert<\delta$, and $\lVert X-X_{0}\rVert=\lVert X_{0}-X\rVert$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (v). By Products of Real Matrices with Zero Matrices, Negatives and Differences (iii) (and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i)),

$$F(t,X)-F(t_{0},X_{0})=A(t)(X-X_{0})+\bigl(A(t)-A(t_{0})\bigr)X_{0}+(X-X_{0})B(t)+X_{0}\bigl(B(t)-B(t_{0})\bigr)+\bigl(G(t)-G(t_{0})\bigr),$$

so, by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vi) and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (vii),

$$\lVert F(t,X)-F(t_{0},X_{0})\rVert\le(K_{A}+K_{B})\,\lVert X-X_{0}\rVert+2\mu\,\lVert X_{0}\rVert+\lVert G(t)-G(t_{0})\rVert\le\tfrac{\varepsilon}{5}+\tfrac{2\varepsilon}{5}+\lVert G(t)-G(t_{0})\rVert<\tfrac{3\varepsilon}{5}+\tfrac{\varepsilon}{5}=\tfrac{4\varepsilon}{5}<\varepsilon ,$$

since $\lVert G(t)-G(t_{0})\rVert<\varepsilon/5$ as $d_{\mathbb{R}}(t_{0},t)<\delta\le\delta_{G}$; that is, $d_{n\times q}\bigl(F(t,X),F(t_{0},X_{0})\bigr)<\varepsilon$. As in the proof of (ii), $F$ is continuous on $[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$. This proves (iv).

Proof of (v). Let $A$, $B$, $G$ and $F$ be as there. By (iv), there are $K_{A},K_{B}\in\mathbb{R}$ with $K_{A},K_{B}\ge0$, $\lVert A(t)\rVert\le K_{A}$ and $\lVert B(t)\rVert\le K_{B}$ for every $t\in[a,b]$, and $F$ is continuous on $[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$ and satisfies the Lipschitz bound with the constant $K_{A}+K_{B}\ge0$. By The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (iii), with $n$ for $p$ and $K_{A}+K_{B}$ for $L$, for every $\Xi\in\mathcal{M}_{n\times q}(\mathbb{R})$ the initial value problem with right-hand side $F$ and initial value $\Xi$ has exactly one solution. This proves (v).