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

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, let $W_{2d}$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words with lengths $|w|$, let $H_{Q}$ be the quadratic control Hamiltonian, $\rho,\beta$ the constants, and $K$ a trilinear constant. Let $\mathbb{C}$ be the complex numbers with modulus $|z|$ and real part $\operatorname{Re}z$. Suppose that the source is

$$f(\lambda)=\operatorname{Re}\sum_{w\in S}\gamma_{w}\,\lambda(w)\qquad(\lambda\in\mathcal{L}_{d})$$

for a nonempty finite set $S\subseteq W_{2d}$ of nonempty words and complex numbers $\gamma_{w}$, $w\in S$, and put

$$\Gamma=\sum_{w\in S}|\gamma_{w}|\,(|w|+1)^{2}\,s_{w},$$

where, for $w$ of length $k\in\mathbb{N}$, $s_{w}$ is the nonnegative square root, given by Existence and Uniqueness of the Nonnegative Square Root, of the natural power $(6144\,d)^{k}$.

1. (Lipschitz source)¶ $|f(\mu)-f(\nu)|\le\Gamma\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$.

2. (Well-posedness)¶ If $\Gamma\le\rho\beta^{2}/(4K)$, then The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$ has exactly one viscosity solution $V$, and $|V(\mu)-V(\nu)|\le(\Gamma/\rho)\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$.