Optimal undulatory swimming with constrained deformation and actuation intervals

In nature, many unicellular organisms are able to swim with the help of beating filaments, where local energy input leads to cooperative undulatory beating motion. Here we investigate, by employing reinforcement learning, how undulatory microswimmers modeled as a discretized bead-bend-spring filament are actuated by torques which are constrained locally. We show that the competition between actively applied torques and intrinsic bending stiffness leads to various optimal beating patterns characterized by distinct frequencies, amplitudes, and wavelengths. Interestingly, the optimum solutions depend on the action interval, i.e., the timescale of how fast the microswimmer can change the applied torques based on its internal state. We show that optimized stiffness- and action-interval-dependent beating is realized by bang-bang solutions of the applied torques with distinct optimum time-periodicity and phase shift between consecutive joints, which we analyze in detail by a systematic study of possible bang-bang wave solution patterns of applied torques. Our work not only sheds light on how efficient beating patterns of biological microswimmers can emerge based on internal and local constraints, but also offers actuation policies for potential artificial elastic microswimmers.