DCDS-B
Optimally swimming stokesian robots
François Alouges Antonio DeSimone Luca Heltai Aline Lefebvre-Lepot Benoît Merlet
Discrete & Continuous Dynamical Systems - B 2013, 18(5): 1189-1215 doi: 10.3934/dcdsb.2013.18.1189
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.
keywords: movement and locomotion optimal control Biological and artificial micro-swimmers low-Reynolds-number (creeping) flow. propulsion efficiency
CPAA
Convergence to equilibrium for the backward Euler scheme and applications
Benoît Merlet Morgan Pierre
Communications on Pure & Applied Analysis 2010, 9(3): 685-702 doi: 10.3934/cpaa.2010.9.685
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

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