## Journals

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DCDS-B

We study self-propelled stokesian robots composed of assemblies of
balls, in dimensions 2 and 3, and prove that they are able to
control their position and orientation. This is a result of

*controllability*, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [4] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [4] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.
CPAA

We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known
Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given

keywords:
stability
,
Lojasiewicz inequality
,
backward Euler scheme
,
convergence rates.
,
$\theta$-scheme

## Year of publication

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