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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.14683 |
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| _version_ | 1866909418723999744 |
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| author | Raginsky, Maxim |
| author_facet | Raginsky, Maxim |
| contents | We revisit the work of Roger Brockett on controllability of the Liouville equation, with a particular focus on the following problem: Given a smooth controlled dynamical system of the form $\dot{x} = f(x,u)$ and a state-space diffeomorphism $ψ$, design a feedback control $u(t,x)$ to steer an arbitrary initial state $x_0$ to $ψ(x_0)$ in finite time. This formulation of the problem makes contact with the theory of optimal transportation and with nonlinear controllability. For controllable linear systems, Brockett showed that this is possible under a fairly restrictive condition on $ψ$. We prove that controllability suffices for a much larger class of diffeomorphisms. For nonlinear systems defined on smooth manifolds, we review a recent result of Agrachev and Caponigro regarding controllability on the group of diffeomorphisms. A corollary of this result states that, for control-affine systems satisfying a bracket generating condition, any $ψ$ in a neighborhood of the identity can be implemented using a time-varying feedback control law that switches between finitely many time-invariant flows. We prove a quantitative version which allows us to describe the implementation complexity of the Agrachev-Caponigro construction in terms of a lower bound on the number of switchings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14683 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Remarks on Controllability of the Liouville Equation Raginsky, Maxim Optimization and Control We revisit the work of Roger Brockett on controllability of the Liouville equation, with a particular focus on the following problem: Given a smooth controlled dynamical system of the form $\dot{x} = f(x,u)$ and a state-space diffeomorphism $ψ$, design a feedback control $u(t,x)$ to steer an arbitrary initial state $x_0$ to $ψ(x_0)$ in finite time. This formulation of the problem makes contact with the theory of optimal transportation and with nonlinear controllability. For controllable linear systems, Brockett showed that this is possible under a fairly restrictive condition on $ψ$. We prove that controllability suffices for a much larger class of diffeomorphisms. For nonlinear systems defined on smooth manifolds, we review a recent result of Agrachev and Caponigro regarding controllability on the group of diffeomorphisms. A corollary of this result states that, for control-affine systems satisfying a bracket generating condition, any $ψ$ in a neighborhood of the identity can be implemented using a time-varying feedback control law that switches between finitely many time-invariant flows. We prove a quantitative version which allows us to describe the implementation complexity of the Agrachev-Caponigro construction in terms of a lower bound on the number of switchings. |
| title | Some Remarks on Controllability of the Liouville Equation |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2404.14683 |