Modeling and optimal control of an octopus tentacle

We present a control model for an octopus tentacle based on the dynamics of an inextensible string with curvature constraints and curvature controls. We derive the equations of motion together with an appropriate set of boundary conditions, and we characterize the corresponding equilibria. The model results in a system of fourth-order evolutive nonlinear controlled PDEs, generalizing the classic Euler's dynamic elastica equation, that we approximate and solve numerically by introducing a finite difference scheme. We proceed by investigating a reachability optimal control problem associated to our tentacle model. We first focus on the stationary case, establishing a relation with the celebrated Dubins car problem. Moreover, we propose an augmented Lagrangian method for its numerical solution. Finally, we address the evolutive case obtaining first order optimality conditions, then we numerically solve the optimality system by means of an adjoint-based gradient descent method.