The Quadratic Control Hamiltonian 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 $\mathcal{P}_{d}$ be the set of polynomial controls with energies $\lVert a\rVert_{\lambda}^{2}$, let $b_{a}(\lambda)$ be the feedback drift, which lies in $E_{d}$ by Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §bound, and let $\kappa_{d}$ be the constant of Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §constant.

(Quadratic Hamiltonian)¶ For $\lambda\in\mathcal{L}_{d}$ and $p\in E_{d}$,

$$H_{Q}(\lambda,p)=\sup\Bigl\{-\bigl\langle p,b_{a}(\lambda)\bigr\rangle_{d}-\tfrac12\lVert a\rVert_{\lambda}^{2}\ :\ a\in\mathcal{P}_{d}\Bigr\}.$$

The supremum exists by The Real Numbers: Standing Notation and Background §bounds: the set is nonempty, $\mathcal{P}_{d}$ being nonempty by Polynomial Controls on Unitary Laws and Their Energy §controls, and it is bounded above by $\frac12\kappa_{d}^{2}\lVert p\rVert_{d}^{2}$. Indeed, for $a\in\mathcal{P}_{d}$ put $s=\kappa_{d}\lVert p\rVert_{d}$ and $t=\lVert a\rVert_{\lambda}$; by the Cauchy–Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space, applicable since $E_{d}$ is a real inner product space with norm $\lVert\cdot\rVert_{d}$ by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, and by Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency §bound, $-\langle p,b_{a}(\lambda)\rangle_{d}\le\lVert p\rVert_{d}\lVert b_{a}(\lambda)\rVert_{d}\le st$, and $st-\frac12t^{2}\le\frac12s^{2}$ because $\frac12s^{2}-st+\frac12t^{2}=\frac12(s-t)^{2}\ge0$. The map $H_{Q}:\mathcal{L}_{d}\times E_{d}\to\mathbb{R}$ is the quadratic control Hamiltonian.