# Optimal motion of a scallop: some case studies

## TL;DR

This paper investigates the optimal control strategies for a two-link swimmer, called scallop, moving between two positions in fluid environments with switching dynamics, providing explicit solutions and numerical insights.

## Contribution

It explicitly solves two optimal control problems for a scallop swimmer with switching fluid dynamics, revealing that fewer switches are optimal for both minimum time and quadratic cost.

## Key findings

- Fewer switches in control are optimal for both cost functions.
- Explicit solutions are obtained for the control problems with one switch.
- Numerical simulations support the strategy of minimizing switches.

## Abstract

In this paper we focus on a two-link swimmer called scallop which moves changing dynamics between two fluids regimes. We address and solve explicitly two optimal control problems, the minimum time one and the minimum quadratic cost needed to move the swimmer between two fixed positions using a periodic control. Considering 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. Finally, we show numerical simulations suggesting that to switch less times is the best strategy for both costs.