Control Data and the Control Hamiltonian on the Torus Wasserstein Space

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space: Standing Notation, with $T_{\mu}$ the tangent space of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §calculus and $d_{\mathbb{T}}$ the flat torus distance.

1. (Control datum)¶ A control datum is a pair $(b,f)$ of maps $b:\mathbb{R}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}^{d}$ and $f:\mathbb{R}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ for which there is a real number $L\ge0$ such that, for all $x,y\in\mathbb{R}^{d}$ and $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$,

$$\lVert b(x,\mu)\rVert\le L,\qquad|f(x,\mu)|\le L,$$ $$\lVert b(x,\mu)-b(y,\nu)\rVert\le L\bigl(d_{\mathbb{T}}(x,y)+W_{\mathbb{T}}(\mu,\nu)\bigr),\qquad|f(x,\mu)-f(y,\nu)|\le L\bigl(d_{\mathbb{T}}(x,y)+W_{\mathbb{T}}(\mu,\nu)\bigr).$$

Such an $L$ is a constant of the datum.

2. (Control Hamiltonian)¶ Let $(b,f)$ be a control datum with constant $L$. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ the maps $b(\cdot,\mu)$ and $f(\cdot,\mu)$ are continuous, since $d_{\mathbb{T}}(x,y)\le\lVert y-x\rVert$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal, hence Borel, and they are bounded by $L$. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$, $p\in T_{\mu}$ and a Borel representative of $p$, the function $x\mapsto\tfrac12\lVert p(x)\rVert^{2}-b(x,\mu)\cdot p(x)-f(x,\mu)$ is therefore Borel and integrable with respect to $\mu$, its absolute value being at most $\tfrac12\lVert p(x)\rVert^{2}+L\lVert p(x)\rVert+L$ by the Cauchy-Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product, and $\int\lVert p\rVert\,d\mu\le\lVert p\rVert_{\mu}$ by Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm; its integral does not depend on the representative, by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. The control Hamiltonian of $(b,f)$ is the map $H_{b,f}$ assigning to every $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $p\in T_{\mu}$ the real number

$$H_{b,f}(\mu,p)=\int\Bigl(\tfrac12\lVert p(x)\rVert^{2}-b(x,\mu)\cdot p(x)-f(x,\mu)\Bigr)\,\mu(dx).$$