Generic Controllability of 3D Swimmers in a Perfect Fluid

We address the problem of controlling a dynamical system governing the motion of a 3D weighted shape changing body swimming in a perfect fluid. The rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow, the total impulse of the fluid-swimmer system being constant for all times. We prove the following tracking results: (i) Synchronized swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately follow any given trajectory while, in addition, undergoing approximately any given shape changes. In this statement, the control consists in arbitrarily small superimposed deformations; (ii) Freestyle swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately tracks any given trajectory by combining suitably at most five basic movements that can be generically chosen (no macro shape changes are prescribed in this statement).

💡 Research Summary

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The paper investigates the controllability of a three‑dimensional shape‑changing body (a “swimmer”) immersed in an ideal, incompressible, inviscid fluid. The authors first construct a rigorous mathematical model that couples the rigid motion of the swimmer with the potential flow of the surrounding fluid. Two reference frames are introduced: a fixed inertial frame E and a moving body‑attached frame e. The swimmer’s configuration at time t is described by a rotation matrix R(t)∈SO(3) and a translation vector r(t)∈ℝ³; their time derivatives define the linear velocity v and angular velocity Ω in the body frame. Shape changes are encoded by a diffeomorphism Θₜ = Id + ϑₜ, where ϑₜ belongs to the Banach space D¹₀(ℝ³) of C¹ vector fields that decay at infinity. The self‑propelled constraints (1.2) enforce conservation of the swimmer’s center of mass and angular momentum during deformation, reflecting the physical assumption that only internal forces generate shape changes.

The fluid dynamics are governed by the Euler equations. Because the flow is assumed irrotational at the initial time, it remains irrotational for all t, allowing the velocity field to be expressed as the gradient of a scalar potential. Six elementary potentials ψᵢ (i = 1,…,6) correspond to the six degrees of freedom of rigid motion; they are harmonic in the fluid domain Fₜ, vanish at infinity, and satisfy Neumann boundary conditions on the body surface Σₜ. An additional potential φₜ, associated with the shape deformation, solves a Neumann problem with boundary data given by the normal component of the deformation velocity wₜ = ∂ₜΘₜ∘Θₜ⁻¹.

Using these potentials the authors define the fluid‑added mass matrix Mᵣ(t) = Mᵣᵇ(t)+Mᵣᶠ(t) and a coupling vector N(t). Mᵣᵇ(t) encodes the swimmer’s intrinsic mass m and inertia tensor I(t), while Mᵣᶠ(t) captures the fluid‑induced added mass effects, both expressed as integrals of gradients of the elementary potentials over the fluid domain. N(t) represents the interaction between the rigid‑motion potentials and the deformation potential. The governing equations of motion reduce to the compact ODE system

  (Ω̇, v̇) = − Mᵣ⁻¹(t) N(t)  (1.6a),

supplemented by the kinematic equations for R(t) and r(t):

  Ṙ = R Ω̂,  ṙ = R v  (1.6b).

Here the control input is the time‑dependent deformation field ϑₜ; because the fluid domain and the potentials depend on ϑₜ, the system is highly nonlinear in the control.

The first analytical result (Proposition 1.1) establishes well‑posedness: for any absolutely continuous deformation ϑₜ and any initial rigid configuration, the ODE system admits a unique Carathéodory solution that depends continuously on the input. This provides a solid foundation for the subsequent controllability analysis.

The authors then introduce the notion of a “swimmer configuration” (SC). For a given integer n, an SC is a triple (ρ, ϑ, V) where ρ∈C⁰( (\bar B) )⁺ is the density, ϑ∈D¹₀(ℝ³) is a reference deformation, and V = (V₁,…,Vₙ) is a collection of C¹₀(ℝ³) vector fields satisfying two key properties: (i) the set {Vᵢ·eₖ | 1≤i≤n, k=1,2,3} is linearly independent, and (ii) each Vᵢ and each pair (Vᵢ,Vⱼ) satisfy the self‑propelled constraints (zero net mass and zero net angular momentum). The space C(n) of all such configurations is shown to be an infinite‑dimensional analytic submanifold of a Banach space.

The controllability problem is recast in the language of geometric control theory. The dynamics (1.6) are expressed as an affine control system on the state space SO(3)×ℝ³ with control vector fields derived from the partial derivatives of the mass matrix and coupling vector with respect to the deformation parameters. The authors prove that the mass matrix Mᵣ and the coupling N depend analytically on the SC parameters. Consequently, the Lie algebra generated by the control vector fields is an analytic function of the configuration.

A crucial step is to exhibit at least one specific SC for which the Lie algebra spans the entire tangent space of the state manifold. By explicit computation (using a carefully chosen set of five basic deformations V₁,…,V₅), the authors verify that the Lie brackets up to a finite order are linearly independent, satisfying the Lie‑algebra rank condition. Analyticity then implies that the set of configurations for which the rank condition fails is a proper analytic subset, i.e., of measure zero in C(n). Hence, “generic” swimmers are controllable.

Two main theorems follow from this analysis:

1. Theorem 1.2 (Synchronized Swimming). Given any target density (\barρ), a C¹ reference deformation (\barϑ(t)), and a desired rigid trajectory ((\bar R(t),\bar r(t))), for any ε>0 there exists an actual density ρ and a deformation ϑ(t) (arbitrarily close to the target in C⁰ and C¹ norms) such that the solution of (1.6) with initial condition ((\bar R(0),\bar r(0))) stays within ε of the target trajectory for all t∈