Proof of Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise

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 and Unitary Laws with Free Unitary Noise: Standing Data §test-functions are in force, and viscosity sub- and supersolutions are those of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$, in the sense of Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws; $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric, nonempty by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §nonempty and compact in the topology induced by $d_{\mathcal{L}}$ by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §compact. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, $E_{d}$ is a real inner product space with the zero map as zero vector $0$, so Elementary Identities in a Real Inner Product Space applies to it; every member of $C^{1}(E_{d})$ is differentiable at every point of $E_{d}$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1). By The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility §admissible, $H_{Q}$ is an admissible Hamiltonian. A constant function $c$ on $\mathcal{L}_{d}$ is bounded (with bound $|c|$), and it is upper and lower semicontinuous, since $c-\varepsilon<c<c+\varepsilon$ for every real $\varepsilon>0$ (Upper Semicontinuous Function on a Subset of a Metric Space, Lower Semicontinuous Function on a Subset of a Metric Space). By Unitary Laws with Free Unitary Noise: Standing Data §constants, $\rho>0$ and $\beta\ge0$. The hypotheses on $L_{f}$ and $K$ are those of Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise, which we therefore may apply, with the same $L_{f}$ and $K$.

Step 0 ($H_{Q}$ at extrema of test functions). Let $F\in C^{1}(E_{d})$ and $\lambda\in\mathcal{L}_{d}$ be such that $F\circ\iota_{d}$ has a local minimum or a local maximum at $\lambda$ relative to $\mathcal{L}_{d}$, and put $p=DF(\iota_{d}(\lambda))$. We claim $H_{Q}(\lambda,p)=0$, and moreover $\langle p,\widehat{\Theta}\lambda\rangle_{d}\ge0$ in the case of a minimum and $\langle p,\widehat{\Theta}\lambda\rangle_{d}\le0$ in the case of a maximum. Indeed, $F$ is differentiable at $\iota_{d}(\lambda)$, so Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones §minimum, respectively Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones §maximum, gives $\langle p,v\rangle_{d}=0$ for every $v\in T^{\pm}_{\lambda}$, together with the sign of $\langle p,v\rangle_{d}$ on $T^{+}_{\lambda}$; the sign claim follows because $\widehat{\Theta}\lambda\in T^{+}_{\lambda}$ by The Free Unitary Heat Generator Points into the Space of Unitary Laws §inward. For $v\in T^{\pm}_{\lambda}$, $\langle p-0,v\rangle_{d}=\langle p,v\rangle_{d}-\langle0,v\rangle_{d}=\langle p,v\rangle_{d}=0$ by Elementary Identities in a Real Inner Product Space §bilinear and Elementary Identities in a Real Inner Product Space §zero. So Admissible Hamiltonians on Unitary Laws §tangential, with $p'=0$, gives $H_{Q}(\lambda,p)=H_{Q}(\lambda,0)$. By The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility §bounds, $0\le H_{Q}(\lambda,0)\le\frac12\kappa_{d}^{2}\lVert0\rVert_{d}^{2}=0$, with $\kappa_{d}$ the constant of that lemma, since $\lVert0\rVert_{d}=0$ by Elementary Identities in a Real Inner Product Space §zero. Hence $H_{Q}(\lambda,p)=0$.

Step 1 (Constant barriers). Let $M$ be a real number with $|f(\lambda)|\le M$ for every $\lambda\in\mathcal{L}_{d}$. We claim that the constant function $-M/\rho$ is a viscosity subsolution and the constant function $M/\rho$ a viscosity supersolution. Both are bounded and upper and lower semicontinuous (Conventions).

Subsolution. Let $F\in C^{1}(E_{d})$ and $\lambda\in\mathcal{L}_{d}$ be such that $-M/\rho-F\circ\iota_{d}$ has a local maximum at $\lambda$ relative to $\mathcal{L}_{d}$, with radius $t>0$. For $d_{\mathcal{L}}(\lambda,\lambda')<t$, $-M/\rho-F(\iota_{d}(\lambda'))\le-M/\rho-F(\iota_{d}(\lambda))$, that is $F(\iota_{d}(\lambda))\le F(\iota_{d}(\lambda'))$; so $F\circ\iota_{d}$ has a local minimum at $\lambda$ (Local Minimum of a Function Relative to a Subset of a Metric Space). With $p=DF(\iota_{d}(\lambda))$, Step 0 gives $H_{Q}(\lambda,p)=0$ and $\langle p,\widehat{\Theta}\lambda\rangle_{d}\ge0$. As $\beta^{2}/2\ge0$ and $-M\le-|f(\lambda)|\le f(\lambda)$,

$$\rho\Bigl(-\frac{M}{\rho}\Bigr)+H_{Q}(\lambda,p)-\frac{\beta^{2}}{2}\langle p,\widehat{\Theta}\lambda\rangle_{d}\le-M\le f(\lambda),$$

which is the inequality of Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution.

Supersolution. Let $F\in C^{1}(E_{d})$ and $\lambda\in\mathcal{L}_{d}$ be such that $M/\rho-F\circ\iota_{d}$ has a local minimum at $\lambda$ relative to $\mathcal{L}_{d}$. As above, $F\circ\iota_{d}$ has a local maximum at $\lambda$ (Local Maximum of a Function Relative to a Subset of a Metric Space); with $p=DF(\iota_{d}(\lambda))$, Step 0 gives $H_{Q}(\lambda,p)=0$ and $\langle p,\widehat{\Theta}\lambda\rangle_{d}\le0$. Since $f(\lambda)\le|f(\lambda)|\le M$,

$$\rho\,\frac{M}{\rho}+H_{Q}(\lambda,p)-\frac{\beta^{2}}{2}\langle p,\widehat{\Theta}\lambda\rangle_{d}\ge M\ge f(\lambda),$$

which is the inequality of Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §supersolution.

Step 2 (A bound for the source). The source $f$ is continuous on $(\mathcal{L}_{d},d_{\mathcal{L}})$ by Unitary Laws with Free Unitary Noise: Standing Data §source; by Continuous Map Between Metric Spaces, with the metric $d_{\mathbb{R}}(s,t)=|s-t|$ of The Real Numbers: Standing Notation and Background §numbers, it has the continuity property of Extreme Value Theorem on a Compact Subset of a Metric Space. By that theorem, applied with the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ and the nonempty compact set $\mathcal{L}_{d}$, there are $\lambda_{\min},\lambda_{\max}\in\mathcal{L}_{d}$ with $f(\lambda_{\min})\le f(\lambda)\le f(\lambda_{\max})$ for every $\lambda\in\mathcal{L}_{d}$. Put $M_{0}=|f(\lambda_{\min})|+|f(\lambda_{\max})|\ge0$. Then $-M_{0}\le-|f(\lambda_{\min})|\le f(\lambda_{\min})\le f(\lambda)\le f(\lambda_{\max})\le|f(\lambda_{\max})|\le M_{0}$, so $|f(\lambda)|\le M_{0}$ for every $\lambda\in\mathcal{L}_{d}$. Put $C=M_{0}/\rho\ge0$. By Step 1 with $M=M_{0}$, $-C$ is a viscosity subsolution and $C$ a viscosity supersolution.

Step 3 (The Perron family). Let $\mathcal{S}$ be the set of viscosity subsolutions $u$ with $u(\lambda)\le C$ for every $\lambda\in\mathcal{L}_{d}$. By Step 2, $-C\in\mathcal{S}$, as $-C\le C$. By Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution, applied with the admissible Hamiltonian $H_{Q}$, with $\mathcal{S}$ and with $M=C$, the function $w(\lambda)=\sup\{u(\lambda):u\in\mathcal{S}\}$ is bounded (Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution §bound) and its upper semicontinuous envelope $W=w^{*}$ is a viscosity subsolution (Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution §subsolution). As $w$ is bounded, there is a real $M'\ge0$ with $|w(\lambda)|\le M'$ for every $\lambda\in\mathcal{L}_{d}$ (Bounded Real-Valued Function on a Set); since $w(\lambda')\le|w(\lambda')|\le M'$ for every $\lambda'$, $M'\in A_{w}(\lambda)$ (with $r=1$) for every $\lambda\in\mathcal{L}_{d}$, so $w$ is bounded above near each point (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds) and its envelope $W$ is defined. We apply Properties of the Upper Semicontinuous Envelope and Properties of the Lower Semicontinuous Envelope, by Duality with the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ and $S=\mathcal{L}_{d}$. Since $C$ is an upper bound of each set $\{u(\lambda):u\in\mathcal{S}\}$, $w\le C$; the constant $C$ being upper semicontinuous, Properties of the Upper Semicontinuous Envelope §least gives $W\le C$. So $W\in\mathcal{S}$, whence $W\le w$; with $w\le W$ (Properties of the Upper Semicontinuous Envelope §bounds) we get $W=w$. As $-C\in\mathcal{S}$, $-C\le w=W$. Thus

$$-C\le W\le C\quad\text{on }\mathcal{L}_{d},\qquad\text{and}\qquad u\le W\ \text{ for every }u\in\mathcal{S}.\tag{P1}$$

Step 4 ($W_{*}$ is a viscosity supersolution). By (P1), $-C\in B_{W}(\lambda)$ for every $\lambda$, so $W$ is bounded below near each point (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds) and its lower semicontinuous envelope $W_{*}$ is defined. By Properties of the Lower Semicontinuous Envelope, by Duality §bounds and Properties of the Lower Semicontinuous Envelope, by Duality §lsc, $W_{*}\le W$ and $W_{*}$ is lower semicontinuous; as the constant $-C$ is lower semicontinuous and $-C\le W$, Properties of the Lower Semicontinuous Envelope, by Duality §greatest gives $-C\le W_{*}$. So $-C\le W_{*}\le W\le C$, and $W_{*}$ is bounded. Let $F\in C^{1}(E_{d})$ and $\lambda_{0}\in\mathcal{L}_{d}$ be such that $W_{*}-F\circ\iota_{d}$ has a local minimum at $\lambda_{0}$ relative to $\mathcal{L}_{d}$, with radius $t>0$, and suppose, for a contradiction, that

$$\rho\,W_{*}(\lambda_{0})+H_{Q}\bigl(\lambda_{0},DF(\iota_{d}(\lambda_{0}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda_{0})),\widehat{\Theta}\lambda_{0}\bigr\rangle_{d}<f(\lambda_{0}).\tag{$F_{0}$}$$

Case $W_{*}(\lambda_{0})<C$. Apply Perron's Method on Unitary Laws: the Bump Construction with the admissible Hamiltonian $H_{Q}$, the subsolution $W$ (whose lower semicontinuous envelope is $W_{*}$), the constant function $g=C$, which is lower semicontinuous with $W\le g$, and $F$, $\lambda_{0}$: its hypotheses are the local minimum, $W_{*}(\lambda_{0})<g(\lambda_{0})$ and ($F_{0}$). By Perron's Method on Unitary Laws: the Bump Construction §bump with $r=1$, there are a viscosity subsolution $\widetilde{w}$ with $W\le\widetilde{w}\le C$ and a point $\lambda'$ with $W(\lambda')<\widetilde{w}(\lambda')$. But $\widetilde{w}\in\mathcal{S}$, so $\widetilde{w}(\lambda')\le W(\lambda')$ by (P1), a contradiction.

Case $W_{*}(\lambda_{0})=C$. (As $W_{*}\le C$, these two cases are exhaustive.) For $d_{\mathcal{L}}(\lambda,\lambda_{0})<t$, $C-F(\iota_{d}(\lambda))\ge W_{*}(\lambda)-F(\iota_{d}(\lambda))\ge W_{*}(\lambda_{0})-F(\iota_{d}(\lambda_{0}))=C-F(\iota_{d}(\lambda_{0}))$, so the constant $C$ minus $F\circ\iota_{d}$ has a local minimum at $\lambda_{0}$. Since the constant $C$ is a viscosity supersolution (Step 2), Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §supersolution gives $\rho\,C+H_{Q}(\lambda_{0},DF(\iota_{d}(\lambda_{0})))-\frac{\beta^{2}}{2}\langle DF(\iota_{d}(\lambda_{0})),\widehat{\Theta}\lambda_{0}\rangle_{d}\ge f(\lambda_{0})$, contradicting ($F_{0}$) because $\rho\,C=\rho\,W_{*}(\lambda_{0})$.

Hence ($F_{0}$) fails, that is, the inequality of Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §supersolution holds, and $W_{*}$ is a viscosity supersolution.

Step 5 (Existence). By Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §comparison with $u=W$ and $v=W_{*}$ (its clause 1, Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §cone, applies with $L=L_{f}/\rho$, which satisfies $L_{f}/\rho\le L\le\beta^{2}/(4K)$ by the hypothesis $L_{f}\le\rho\beta^{2}/(4K)$ on $L_{f}$ and $\rho>0$), $W\le W_{*}$; with $W_{*}\le W$, $W=W_{*}$. Put $V=W$, a map $\mathcal{L}_{d}\to\mathbb{R}$; this is the map of the statement. Then $V$ is a viscosity subsolution (Step 3) and, being equal to $W_{*}$, a viscosity supersolution (Step 4); so $V$ is a viscosity solution (Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §solution), which proves clause 1.

Step 6 (Uniqueness). Let $V'$ be a viscosity solution. By Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §uniqueness, which allows at most one viscosity solution, $V'=V$. This proves clause 2.

Step 7 (Lipschitz bound). By Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §lipschitz applied to the viscosity solution $V$, $|V(\mu)-V(\nu)|\le(L_{f}/\rho)\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$. This proves clause 3.

Step 8 (Bound). Let $M$ be real with $|f(\lambda)|\le M$ for every $\lambda\in\mathcal{L}_{d}$. By Step 1, $-M/\rho$ is a viscosity subsolution and $M/\rho$ a viscosity supersolution. By Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise §comparison with $(u,v)=(-M/\rho,V)$ and with $(u,v)=(V,M/\rho)$, $-M/\rho\le V(\lambda)\le M/\rho$, and since $|V(\lambda)|$ is $V(\lambda)$ or $-V(\lambda)$, $|V(\lambda)|\le M/\rho$ for every $\lambda\in\mathcal{L}_{d}$. This proves clause 4.