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Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations

Bettina Kazandjian, Eugenio Pozzoli, Mario Sigalotti

TL;DR

The paper addresses approximate controllability of Liouville transport equations along mechanical Hamiltonians by characterizing the $L^r$-closure of the orbit of a density under DHam$(T^*M)$. It proves that for $M=\\mathbb{R}^d$ or $\\mathbb{T}^d$, the orbit-closure equals the set of densities with identical sub-/super-level-set volumes, extending Moser-type results to Hamiltonian diffeomorphisms. The authors then show small-time approximate reachability of all DHam elements for systems on $\\mathbb{R}^d$ and on $\\mathbb{T}^d$ (and exact controllability for finite ensembles), using a two-step construction: first approximate target rearrangements by mesh permutations, then realize those permutations via localized Hamiltonian diffeomorphisms built from vertical and horizontal shears. The work provides a rigorous bridge between diffeomorphism controllability and Liouville-density control in mechanical settings, with implications for transport control and geometric control of density evolution.

Abstract

We study the approximate controllability problem for Liouville transport equations along a mechanical Hamiltonian vector field. Such PDEs evolve inside the orbit $$\mathcal{O}(ρ_0):=\left\{ρ_0\circ Φ\mid Φ\in {\rm DHam}(T^*M)\right\},\quad ρ_0\in L^r(T^*M,\mathbb{R}), \quad r\in[1,\infty),$$ where $ρ_0$ is the initial density and ${\rm DHam}(T^*M)$ is the group of Hamiltonian diffeomorphisms of the cotangent bundle manifold $T^*M$. The approximately reachable densities from $ρ_0$ are thus contained in $\overline{\mathcal{O}(ρ_0)}$, where the closure is taken with respect to the $L^r$-topology. Our first result is a characterization of $\overline{\mathcal{O}(ρ_0)}$ when the manifold $M$ is the Euclidean space $\mathbb{R}^d$ or the torus $\mathbb{T}^d$ of arbitrary dimension: $\overline{\mathcal{O}(ρ_0)}$ is the set of all the densities whose sub- and super-level sets have the same measure as those of $ρ_0$. This result is an approximate version, in the case of ${\rm DHam}(T^*M)$, of a theorem by J. Moser (Trans. Am. Math. Soc. 120: 286-294, 1965) on the group of diffeomorphisms. We then present two examples of systems, respectively on $M=\mathbb{R}^d$ and $\mathbb{T}^d$, where the small-time approximately attainable diffeomorphisms coincide with ${\rm DHam}(T^*M)$, respectively at the level of the group and at the level of the densities. The proofs are based on the construction of Hamiltonian diffeomorphisms that approximate suitable permutations of finite grids, and Poisson bracket techniques.

Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations

TL;DR

The paper addresses approximate controllability of Liouville transport equations along mechanical Hamiltonians by characterizing the -closure of the orbit of a density under DHam. It proves that for or , the orbit-closure equals the set of densities with identical sub-/super-level-set volumes, extending Moser-type results to Hamiltonian diffeomorphisms. The authors then show small-time approximate reachability of all DHam elements for systems on and on (and exact controllability for finite ensembles), using a two-step construction: first approximate target rearrangements by mesh permutations, then realize those permutations via localized Hamiltonian diffeomorphisms built from vertical and horizontal shears. The work provides a rigorous bridge between diffeomorphism controllability and Liouville-density control in mechanical settings, with implications for transport control and geometric control of density evolution.

Abstract

We study the approximate controllability problem for Liouville transport equations along a mechanical Hamiltonian vector field. Such PDEs evolve inside the orbit where is the initial density and is the group of Hamiltonian diffeomorphisms of the cotangent bundle manifold . The approximately reachable densities from are thus contained in , where the closure is taken with respect to the -topology. Our first result is a characterization of when the manifold is the Euclidean space or the torus of arbitrary dimension: is the set of all the densities whose sub- and super-level sets have the same measure as those of . This result is an approximate version, in the case of , of a theorem by J. Moser (Trans. Am. Math. Soc. 120: 286-294, 1965) on the group of diffeomorphisms. We then present two examples of systems, respectively on and , where the small-time approximately attainable diffeomorphisms coincide with , respectively at the level of the group and at the level of the densities. The proofs are based on the construction of Hamiltonian diffeomorphisms that approximate suitable permutations of finite grids, and Poisson bracket techniques.

Paper Structure

This paper contains 27 sections, 30 theorems, 111 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Approximate orbits of Hamiltonian diffeomorphisms on densities
  4. Examples of approximate controllability in ${\rm DHam}(T^*{\mathbb R}^d)$ and $L^r(T^*{\mathbb T}^d,{\mathbb R})$
  5. A system in ${\mathbb R}^d$
  6. A system in ${\mathbb T}^d$
  7. Proof strategy
  8. Some additional literature
  9. Proof of Theorem \ref{['thm:closure-orbit']}
  10. Proof of the inclusion $\overline{\mathcal{O}(\rho_0)}\subset\mathcal{L}(\rho_0)$
  11. Proof of the inclusion $\mathcal{L}(\rho_0)\subset\overline{\mathcal{O}(\rho_0)}$
  12. Approximation with a permutation of the mesh
  13. Approximation by a Hamiltonian diffeomorphism
  14. Some properties of vector fields
  15. Some properties of the approximately reachable sets
  16. ...and 12 more sections

Key Result

Theorem 3

Let $M$ be ${\mathbb T}^d={\mathbb R}^d/2\pi{\mathbb Z}^d$ or ${\mathbb R}^d$ and $r\in[1,\infty)$. Then, for any $\rho_0\in L^r(T^*M)$, where the closure is taken in the $L^r$-topology.

Figures (7)

  • Figure 1: The permutation $F_h$
  • Figure 2: The Hamiltonian transformation $e^{\overrightarrow{f}}$
  • Figure 3: The Hamiltonian transformation $e^{\overrightarrow{g}}$
  • Figure 4: Localized vertical translations
  • Figure 5: Localized Hamiltonian rotation of angle $\frac{\pi}{2}$
  • ...and 2 more figures

Theorems & Definitions (56)

  • Definition 1: Small-time reachable densities
  • Definition 2: Small-time approximately reachable densities
  • Theorem 3: $L^r$-closure of the orbits
  • Definition 4: Small-time approximate controllability
  • Definition 5: Small-time approximately reachable diffeomorphisms and controllability
  • Lemma 6
  • Theorem 7
  • Corollary 8
  • Theorem 9
  • Remark 10
  • ...and 46 more