SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians.

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To want something now rather than later is a common attitude that reflects the brain's tendency to value the passage of time. Because the time taken to accomplish an action inevitably delays task achievement and reward acquisition, this idea was ported to neural movement control within the "cost of time" theory. This theory provides a normative framework to account for the underpinnings of movement time formation within the brain and the origin of a self-selected pace in human and animal motion. Then, how does the brain exactly value time in the control of action? To tackle this issue, we used an inverse optimal control approach and developed a general methodology allowing to squarely sample infinitesimal values of the time cost from experimental motion data. The cost of time underlying saccades was found to have a concave growth, thereby confirming previous results on hyperbolic reward discounting, yet without making any prior assumption about this hypothetical nature. For self-paced reaching, however, movement time was primarily valued according to a striking sigmoidal shape; its rate of change consistently presented a steep rise before a maximum was reached and a slower decay was observed. Theoretical properties of uniqueness and robustness of the inferred time cost were established for the class of problems under investigation, thus reinforcing the significance of the present findings. These results may offer a unique opportunity to uncover how the brain values the passage of time in healthy and pathological motor control and shed new light on the processes underlying action invigoration.

Abstract. In this paper we study the problem of the car with n trailers.It was proved in previous works ( 9], 12]) that when each trailer is perpendicular with the previous one the degree of nonholonomy i s Fn+3 (the (n + 3)-th term of the Fibonacci's sequence) and that when no two consecutive trailers are perpendicular this degree is n + 2 .W e compute here by induction the degree of non holonomy i n e v ery state and obtain a partition of the singular set by this degree of non-holonomy. W e g i v e also for each area a set of vector elds in the Lie Algebra of the control system wich makes a basis of the tangent space.

Abstract. When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -with respect to the Whitney topology -, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend results of [13,14]. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.1. Introduction. When addressing standard issues of control theory such as motion planning and stabilization, one may adopt an approach based on optimal control, e.g., Hamilton-Jacobi type methods and shooting algorithms. One is then immediately facing intrinsic difficulties due to the possible presence of singular trajectories. It is therefore important to characterize these trajectories, by studying in particular their existence, optimality status, and the related computational aspects. In this paper, we provide answers to the aforementioned questions for control-affine systems, under generic assumptions, and then investigate consequences in optimal control and its applications.Let M be a smooth (i.e. C ∞ ) manifold of dimension n. Consider the control-affine system

Nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability, such as nonholonomic mechanisms. However, conventional homogeneous approximations exhibit a drawback: in the neighborhood of singular points (where the system growth vector is not constant) the vector fields of the approximate dynamics do not vary continuously with the approximation point. The geometric counterpart of this situation is that the sub-Riemannian distance estimate provided by the classical Ball-Box Theorem is not uniform at singular points. With reference to a specific family of driftless systems, we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points. It is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance. Index Terms-Nilpotent approximations, nonholonomic systems, singularities, sub-Riemannian distance.A point x = (t), for t 2 [0; T ], is said to be accessible from x 0 .

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Optimal control is a prominent approach in robotics and movement neuroscience, among other fields of science. Methods for deriving optimal choices of action have been classically devised either in deterministic or stochastic settings. Here, we consider a setting in-between that retains the stochastic aspect of the controlled system but assumes deterministic open-loop control actions. The rationale stems from observations about the neural control of movement which highlighted that relatively stable behaviors can be achieved without feedback circuitry, via open-loop motor commands adequately tuning the mechanical impedance of the neuromusculoskeletal system. Yet, effective methods for deriving optimal open-loop controls for stochastic systems are lacking overall. This work presents a continuous-time approach for the efficient computation of optimal open-loop controls for a broad class of stochastic optimal control problems. We first consider simple synthetic examples showing that non-trivial departure from the optimal solutions of classical deterministic and stochastic approaches arises, and to stress the originality of the framework. We further exemplify its potential relevance to the planning of biological movement by showing that a well-known phenomenon in motor control, referred to as muscle co-contraction, occurs spontaneously. More generally, this stochastic optimal control framework may be suited to all fields where the design of optimal open-loop actions is relevant.2