Pattern selection in a biomechanical model for the growth of walled cells

In this paper, we analyse a model for the growth of three-dimensional walled cells. In this model the biomechanical expansion of the cell is coupled with the geometry of its wall. We consider that the density of building material depends on the curvature of the cell wall, thus yield-ing possible anisotropic growth. The dynamics of the axisymmetric cell wall is described by a system of nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with a Poisson equation. We develop the linear stability analysis of the spherical symmetric config-uration in expansion. We identify three critical parameters that play a role in the possible instability of the radially symmetric shape, namely the degree of nonlinearity of the coupling, the effective diffusion of the building material, and the Poisson's ratio of the cell wall. We also investigate numerically pattern selection in the nonlinear regime. All the results are also obtained for a simpler, but similar, two-dimensional model

Local Controllability of the Two-Link Magneto-Elastic Micro-Swimmer

A recent promising technique for robotic micro-swimmers is to endow them with a magnetization and apply an external magnetic field to provoke their deformation. In this note we consider a simple planar micro-swimmer model made of two magnetized segments connected by an elastic joint, controlled via a magnetic field. After recalling the analytical model, we establish a local controllability result around the straight position of the swimmer

Optimal Strokes for Driftless Swimmers: A General Geometric Approach

Swimming consists by definition in propelling through a fluid by means of bodily movements. Thus, from a mathematical point of view, swimming turns into a control problem for which the controls are the deformations of the swimmer. The aim of this paper is to present a unified geometric approach for the optimization of the body deformations of so-called driftless swimmers. The class of driftless swimmers includes, among other, swimmers in a 3D Stokes flow (case of micro-swimmers in viscous fluids) or swimmers in a 2D or 3D potential flow. A general framework is introduced, allowing the complete analysis of five usual nonlinear optimization problems to be carried out. The results are illustrated with examples coming from the literature and with an in-depth study of a swimmer in a 2D potential flow. Numerical tests are also provided

Addendum to "Local Controllability of the Two-Link Magneto-Elastic Micro-Swimmer"

In the above mentioned note (, , published in IEEE Trans. Autom. Cont., 2017), the first and fourth authors proved a local controllability result around the straight configuration for a class of magneto-elastic micro-swimmers.That result is weaker than the usual small-time local controllability (STLC), and the authors left the STLC question open. The present addendum closes it by showing that these systems cannot be STLC

Optimal Design for Purcell Three-link Swimmer

International audienceIn this paper we address the question of the optimal design for the Purcell 3-link swimmer. More precisely we investigate the best link length ratio which maximizes its displacement. The dynamics of the swimmer is expressed as an ODE, using the Resistive Force Theory. Among a set of optimal strategies of deformation (strokes), we provide an asymptotic estimate of the displacement for small deformations, from which we derive the optimal link ratio. Numerical simulations are in good agreement with this theoretical estimate, and also cover larger amplitudes of deformation. Compared with the classical design of the Purcell swimmer, we observe a gain in displacement of roughly 60%

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella.

This movie displays an animated version of the simulation presented in Fig.2

The Purcell Three-link swimmer: some geometric and numerical aspects related to periodic optimal controls

International audienceThe maximum principle combined with numerical methods is a powerful tool to compute solutions for optimal control problems. This approach turns out to be extremely useful in applications, including solving problems which require establishing periodic trajectories for Hamiltonian systems, optimizing the production of photobioreactors over a one-day period, finding the best periodic controls for locomotion models (e.g. walking, flying and swimming). In this article we investigate some geometric and numerical aspects related to optimal control problems for the so-called Purcell Three-link swimmer [20], in which the cost to minimize represents the energy consumed by the swimmer. More precisely, employing the maximum principle and shooting methods we derive optimal trajectories and controls, which have particular periodic features. Moreover, invoking a linearization procedure of the control system along a reference extremal, we estimate the conjugate points, which play a crucial role for the second order optimality conditions. We also show how, making use of techniques imported by the sub-Riemannian geometry, the nilpotent approximation of the system provides a model which is integrable, obtaining explicit expressions in terms of elliptic functions. This approximation allows to compute optimal periodic controls for small deformations of the body, allowing the swimmer to move minimizing its energy. Numerical simulations are presented using Hampath and Bocop codes

Hydrodynamics of elastic micro-filaments : model comparison and applications

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