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Abstract. Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence on the alphabets.

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with (p0, W)regulated cost, for some continuous, state-dependent function W and some constant p0 > 0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W (z)/p0. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W , called p0-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a p0-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with (p0, W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

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.

We investigate a reachability control problem for a soft manipulator inspired to an octopus arm. Cases modelling mechanical breakdowns of the actuators are treated in detail: we explicitly characterize the equilibria, and we provide numerical simulations of optimal control strategies. This is a pre-copy-editing, author-produced PDF of an article accepted for publication in following peer review. The definitive publisher-authenticated version 16th International Conference on Informatics in Control, Automation and Robotics (1) 82-90 (2019)