  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 introducing a consistent finite difference scheme. We proceed
investigating a reachability optimal control problem associated to our tentacle
model. We first focus on the stationary case, by 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.
