Soft-robotics is an emerging branch of robotics, focused on the design, construction and control of articulated manipulators (even with a large number of degrees of freedom) composed of elastic and soft materials, able to perform complex tasks and to adapt to the working environment. The main objective, compared to more classical rigid manipulators, is to ensure the safety of human-machine interactions, especially in the industrial, but also in the medical field, with futuristic applications to invasive surgery and rehabilitation of patients with limited mobility. In this seminar, I will present a mathematical model for a soft-robot inspired by an octopus tentacle, obtained using the theory of calculus of variations and studied through some tools of optimal control theory for partial differential equations. It is a two-dimensional model, which aims to capture the dynamics emerging from the biological structure of the tentacle. The reference theory is the Euler-Bernoulli beam, in particular its nonlinear formulation which includes an inextensibility constraint and a bending moment. To these properties a curvature constraint has been added, which prevents the tentacle from bending beyond a certain threshold, and also a distributed control term, which locally prescribes the curvature and models the voluntary contraction of the muscular system of the tentacle. I will also present the results obtained by numerically solving some typical optimal control problems in this field: reaching a point with the free-end of the tentacle, avoiding obstacles in the working environment, and grasping objects, guaranteeing stability at the contact points with respect to external perturbations. This is a joint work with A.C. Lai and P. Loreti at the Department of Basic and Applied Sciences for Engineering (SBAI), Sapienza University of Rome.