Proof of Well-Posedness for Polynomial Sources in the Quadratic Control Problem on Unitary Laws

Each result cited is universally quantified over the data in its own statement.

Conventions. The conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces are in force. Let $W^{\circ}_{d}$ be the set of cyclically reduced words, $W^{\circ}_{d,k}$ the set of those of length $k\in\mathbb{N}$, and $\theta_{d}$ and the weights $c_{w}$ as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights. Fix $\mu,\nu\in\mathcal{L}_{d}$ and put $x=\iota_{d}(\mu)-\iota_{d}(\nu)\in E_{d}$. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev, $x(u)=\mu(u)-\nu(u)$ for every $u\in W^{\circ}_{d}$, and $d_{\mathcal{L}}(\mu,\nu)=\lVert x\rVert_{d}$ by The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance. Square roots are the nonnegative ones of Existence and Uniqueness of the Nonnegative Square Root, and comparisons of nonnegative reals through their squares use claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Natural numbers are read in $\mathbb{R}$ through the canonical map (The Real Numbers: Standing Notation and Background §numbers). Put $c=6144\,d$. Then $1\le c$: $1\le6144$ and $1\le d$ by claim 2 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and for reals $a,b$ with $1\le a$ and $1\le b$ one has $1\le a=a\cdot1\le ab$, by claim 5 of Elementary Arithmetic in an Ordered Field applied to $1\le b$ and $0\le1\le a$ (claim 1 of that lemma). Also $\theta_{d}\,c=1$.

Step 1 (One coefficient is controlled by the gauge norm). For $k\in\mathbb{N}$ let $X_{k}=\sum_{w\in W^{\circ}_{d,k}}c_{w}|x(w)|^{2}$ be the $k$-th block sum of $w\mapsto c_{w}|x(w)|^{2}$ and $\Sigma_{n}=\sum_{k=1}^{n}X_{k}$ its partial sums (Series of Real Numbers §partial-sums). Each term $c_{w}|x(w)|^{2}$ is nonnegative, $c_{w}$ being positive (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §weights), so $X_{k}\ge0$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. As $x\in E_{d}$, the series $\sum_{k=1}^{\infty}X_{k}$ converges (The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space), and by The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §sums and The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §norm,

$$\lVert x\rVert_{d}^{2}=c_{\varnothing}|x(\varnothing)|^{2}+\sum_{k=1}^{\infty}X_{k},\qquad c_{\varnothing}=1 .$$

By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, $0\le\Sigma_{n}\le\sum_{k=1}^{\infty}X_{k}$ for every $n\in\mathbb{N}$.

Let $u\in W^{\circ}_{d}$. If $u=\varnothing$, then $|x(u)|^{2}\le\lVert x\rVert_{d}^{2}$, the series having a nonnegative sum; hence $|x(u)|\le\lVert x\rVert_{d}$. If $u$ has length $j\in\mathbb{N}$, then $u\in W^{\circ}_{d,j}$, and Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone with $E=\{u\}$ gives $c_{u}|x(u)|^{2}\le X_{j}$, the sum over the one-element set $\{u\}$ being its single term (Sum over a Finite Index Set with $n=1$). By claim 1 of Properties of a Sum over a Finite Index Set, $\Sigma_{j}$ is the sum of $k\mapsto X_{k}$ over $[j]$, so Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §monotone with $E=\{j\}$ gives $X_{j}\le\Sigma_{j}$. As $c_{\varnothing}|x(\varnothing)|^{2}\ge0$, we obtain $c_{u}|x(u)|^{2}\le\Sigma_{j}\le\sum_{k=1}^{\infty}X_{k}\le\lVert x\rVert_{d}^{2}$. Now $c_{u}=\theta_{d}^{\,j}/(j+1)^{4}$, and $\theta_{d}^{\,j}c^{\,j}=(\theta_{d}c)^{j}=1^{j}=1$ by claims 3 and 2 of Properties of Natural Number Powers in a Field, so $1/c_{u}=c^{\,j}(j+1)^{4}=\bigl(\sqrt{c^{\,j}}\,(j+1)^{2}\bigr)^{2}$. The last equality combines three elementary identities: $(\sqrt{c^{\,j}})^{2}=c^{\,j}$ (Existence and Uniqueness of the Nonnegative Square Root, as $0\le c^{\,j}$ by claim 5 of Properties of Natural Number Powers in a Field); $(ab)^{2}=a^{2}b^{2}$ for reals $a,b$ (claim 3 of that lemma); and $((j+1)^{2})^{2}=(j+1)^{4}$ (claim 1 of that lemma, both sides unfolding to the product of four factors $j+1$). Multiplying by the positive number $1/c_{u}$,

$$|x(u)|^{2}\le\bigl(\sqrt{c^{\,j}}\,(j+1)^{2}\,\lVert x\rVert_{d}\bigr)^{2},\qquad\text{hence}\qquad|x(u)|\le\sqrt{c^{\,j}}\,(j+1)^{2}\,\lVert x\rVert_{d}.\tag{1}$$

Step 2 (Monotonicity in the length). Let $w\in S$ have length $k\in\mathbb{N}$, so that $s_{w}=\sqrt{c^{\,k}}$. For every $m\in\mathbb{N}$, $0\le c^{\,m}$ by claim 5 of Properties of Natural Number Powers in a Field, so multiplying $1\le c$ by $c^{\,m}$ (claim 5 of Elementary Arithmetic in an Ordered Field) and using claim 1 of Properties of Natural Number Powers in a Field gives $c^{\,m}\le c^{\,m}c=c^{\,m+1}$. Now fix $j\in\mathbb{N}$ and let $A$ be the set of $m\in\mathbb{N}$ with $c^{\,j}\le c^{\,j+m}$. Then $1\in A$, by the preceding inequality for $j$; and if $m\in A$, then $c^{\,j}\le c^{\,j+m}\le c^{\,(j+m)+1}=c^{\,j+(m+1)}$, so $m+1\in A$. By Principle of Induction for the Natural Numbers, $A=\mathbb{N}$. Let $j\le k$ in $\mathbb{N}$. If $j=k$, then $c^{\,j}=c^{\,k}$; if $j<k$, then $k=j+m$ for some $m\in\mathbb{N}$ (Order on the Natural Numbers), and $c^{\,j}\le c^{\,k}$ as $m\in A$. Moreover $1=1^{k}\le c^{\,k}$ by claims 2 and 5 of Properties of Natural Number Powers in a Field. Taking square roots, $\sqrt{c^{\,j}}\le s_{w}$ for $j\le k$, and $1\le s_{w}$. Next, for $j\le k$ in $\mathbb{N}$ we have $1\le j+1\le k+1$ in $\mathbb{R}$: the real number $j+1$ is the image of the natural number $j+1$ (claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), so $1\le j+1$ by claim 2 of that lemma; if $j=k$ then $j+1=k+1$, and if $j<k$ then $j<k$ in $\mathbb{R}$ by claim 6 of that lemma, so $0\le k-j=(k+1)-(j+1)$ and $j+1\le k+1$ by claim 3 of Elementary Arithmetic in an Ordered Field. Hence $(j+1)^{2}\le(k+1)^{2}$ and $1=1^{2}\le(k+1)^{2}$, by claims 2 and 5 of Properties of Natural Number Powers in a Field. Products of nonnegative reals being monotone in each factor,

$$\sqrt{c^{\,j}}\,(j+1)^{2}\le s_{w}(|w|+1)^{2}\ \ (j\in\mathbb{N},\ j\le k),\qquad 1\le s_{w}(|w|+1)^{2}.\tag{2}$$

Step 3 (Clause 1). Let $w\in S$, of length $k\in\mathbb{N}$. By Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §reduction there is $u\in W^{\circ}_{d}$ with $|u|\le|w|$ and $\lambda(w)=\lambda(u)$ for every $\lambda\in\mathcal{L}_{d}$, so $\mu(w)-\nu(w)=x(u)$. If $u=\varnothing$, Step 1 and (2) give $|\mu(w)-\nu(w)|\le\lVert x\rVert_{d}\le s_{w}(|w|+1)^{2}\lVert x\rVert_{d}$. If $u$ has length $j\in\mathbb{N}$, then $j\le k$ in $\mathbb{N}$ (otherwise $k<j$, and claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field would give $|w|<|u|$), and (1) and (2) give the same bound. Thus

$$|\mu(w)-\nu(w)|\le s_{w}(|w|+1)^{2}\,\lVert x\rVert_{d}\qquad(w\in S).\tag{3}$$

For complex $z,z'$, $\operatorname{Re}(z+z')=\operatorname{Re}z+\operatorname{Re}z'$ and $\operatorname{Re}(-z)=-\operatorname{Re}z$: by claims 1 and 2 of Properties of Complex Conjugation and Modulus, $2\operatorname{Re}(z+z')=(z+z')+\overline{z+z'}=(z+\overline{z})+(z'+\overline{z'})=2\operatorname{Re}z+2\operatorname{Re}z'$, and similarly for $-z$. With claims 3 and 4 of Properties of a Sum over a Finite Index Set (the latter with the scalar $-1$),

$$f(\mu)-f(\nu)=\operatorname{Re}Z,\qquad Z=\sum_{w\in S}\gamma_{w}\bigl(\mu(w)-\nu(w)\bigr).$$

By claim 6 of Properties of Complex Conjugation and Modulus, $\operatorname{Re}Z\le|Z|$ and $-\operatorname{Re}Z\le|Z|$, so $|f(\mu)-f(\nu)|\le|Z|$, the absolute value of the real number $\operatorname{Re}Z$ being $\operatorname{Re}Z$ or $-\operatorname{Re}Z$. By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus, claim 4 of Properties of Complex Conjugation and Modulus, (3) with $|\gamma_{w}|\ge0$, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison and claim 4 of Properties of a Sum over a Finite Index Set,

$$|Z|\le\sum_{w\in S}|\gamma_{w}|\,|\mu(w)-\nu(w)|\le\sum_{w\in S}|\gamma_{w}|\,(|w|+1)^{2}s_{w}\,\lVert x\rVert_{d}=\Gamma\,\lVert x\rVert_{d}=\Gamma\,d_{\mathcal{L}}(\mu,\nu).$$

As $\mu,\nu$ were arbitrary, this proves clause 1.

Step 4 (Clause 2). Each summand $|\gamma_{w}|(|w|+1)^{2}s_{w}$ of $\Gamma$ is nonnegative, so $\Gamma\ge0$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. Suppose $\Gamma\le\rho\beta^{2}/(4K)$. By clause 1, the source $f$ of Unitary Laws with Free Unitary Noise: Standing Data §source satisfies the hypotheses of Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise with $L_{f}=\Gamma$ and the same trilinear constant $K$. That theorem provides $V:\mathcal{L}_{d}\to\mathbb{R}$ which is a viscosity solution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$ (Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §existence), such that every viscosity solution equals $V$ (Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §uniqueness), so that the equation has exactly one viscosity solution, and such that $|V(\mu)-V(\nu)|\le(\Gamma/\rho)\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$ (Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §lipschitz). This proves clause 2.