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

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 $H_{Q}$ be the quadratic control Hamiltonian, $\rho,\beta$ the constants, $f$ the source, and $K$ a trilinear constant. Suppose that there is a real $L_{f}\ge0$ with

$$|f(\mu)-f(\nu)|\le L_{f}\,d_{\mathcal{L}}(\mu,\nu)\quad\text{for all }\mu,\nu\in\mathcal{L}_{d},\qquad\text{and}\qquad L_{f}\le\frac{\rho\,\beta^{2}}{4K}.$$

Then there is a map $V:\mathcal{L}_{d}\to\mathbb{R}$ with the following properties.

1. (Existence)¶ $V$ is a viscosity solution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$.

2. (Uniqueness)¶ Every viscosity solution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H_{Q}$ equals $V$.

3. (Lipschitz bound)¶ $|V(\mu)-V(\nu)|\le(L_{f}/\rho)\,d_{\mathcal{L}}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{L}_{d}$.

4. (Bound)¶ For every real $M$ with $|f(\lambda)|\le M$ for all $\lambda\in\mathcal{L}_{d}$, $|V(\lambda)|\le M/\rho$ for every $\lambda\in\mathcal{L}_{d}$.