Optimal Motion of a Scallop: Some Case Studies

In this letter, we analyze two optimal control problems for the scallop: a two-link swimmer that is able to self-propel changing dynamics between two fluids regimes. We address and solve explicitly the minimum time problem and the minimum quadratic one, computing the cost needed to move the swimmer between two fixed positions using a periodic control. We focus on the case of only one switching in the dynamics and exploiting the structure of the equation of motion we are able to split the problem into simpler ones. We solve explicitly each sub-problem obtaining a discontinuous global solution. Then we approximate it through a suitable sequence of continuous functions.