Section: New Results

Local controllability of magnetic micro-swimmers and more general classes of control systems

Participants : Laetitia Giraldi, Pierre Lissy [Univ. Paris Dauphine] , Clément Moreau, Jean-Baptiste Pomet.

As a part of Clément Moreau's PhD, we gave fine results on local controlability of magnetized micro swimmers actuated by an external magnetic field. We had shown that the “two-link” magnetic swimmer had some local controllability around its straight configuration but that it was not Small Time Locally Controllable” (STLC) in the classical sense that asks that poin ts close to the initial condition can be reached using “small” controls.

We derived in [30] some necessary conditions for STLC of affine control systems with two scalar controls, around an equlibrium where not only the drift vector field vanishes but one of the two control vector fields vanishes too; we state various necessary conditions (involving the value at the equilibrium of some iterated Lie brackets of the system vector fields), where the “smallness” of the controls is intended in the $L^{\infty }$ (classical) or $W^{1,\infty }$ (less classical, used in recent work by K. Beauchard and F. Marbach).

We also arrived to local controllability results in higher dimension than the “two-link” micro-robots, see [9]. This relies on the following remark: classical STLC does not hold, but STLC is concerned with small controls, hence with variations around the zero control... but, due to one of the control fields vanishing, the system also rests at the equilibrium for (infinitely many) nozero constant values of the control. It is proved that there is one nonzero value of the control such that STLC holds when considered around this constant control and not around the zero control. In other terms, classical STLC holds after a constant feedback transformation.