Pontryagin Maximum Principle and Stokes Theorem

We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived in a new insightful way. It also suggests generalizations in diverse directions of such famous principle.Comment: 21 pages, 7 figures; we corrected a few minor misprints, added a couple of references and inserted a new section (Sect. 7); to appear in Journal of Geometry and Physic

Finite Mechanical Proxies for a Class of Reducible Continuum Systems

We present the exact finite reduction of a class of nonlinearly perturbed wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A-C-Z and a discrete mechanical model, a well definite finite spring-mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.Comment: 15 pages, 3 figure

Swim-like motion of bodies immersed in an ideal fluid

The connection between swimming and control theory is attracting increasing attention in the recent literature. Starting from an idea of Alberto Bressan [A. Bressan, Discrete Contin. Dyn. Syst. 20 (2008) 1\u201335]. we study the system of a planar body whose position and shape are described by a finite number of parameters, and is immersed in a 2-dimensional ideal and incompressible fluid in terms of gauge field on the space of shapes. We focus on a class of deformations measure preserving which are diffeomeorphisms whose existence is ensured by the Riemann Mapping Theorem. After making the first order expansion for small deformations, we face a crucial problem: the presence of possible non vanishing initial impulse. If the body starts with zero initial impulse we recover the results present in literature (Marsden, Munnier and oths). If instead the body starts with an initial impulse different from zero, the swimmer can self-propel in almost any direction if it can undergo shape changes without any bound on their velocity. This interesting observation, together with the analysis of the controllability of this system, seems innovative. Mathematics Subject Classification. 74F10, 74L15, 76B99, 76Z10. Received June 14, 2016. Accepted March 18, 2017. 1. Introduction In this work we are interested in studying the self-propulsion of a deformable body in a fluid. This kind of systems is attracting an increasing interest in recent literature. Many authors focus on two different type of fluids. Some of them consider swimming at micro scale in a Stokes fluid [2,4\u20136,27,35,40], because in this regime the inertial terms can be neglected and the hydrodynamic equations are linear. Others are interested in bodies immersed in an ideal incompressible fluid [8,18,23,30,33] and also in this case the hydrodynamic equations turn out to be linear. We deal with the last case, in particular we study a deformable body -typically a swimmer or a fish- immersed in an ideal and irrotational fluid. This special case has an interesting geometric nature and there is an attractive mathematical framework for it. We exploit this intrinsically geometrical structure of the problem inspired by [32,39,40], in which they interpret the system in terms of gauge field on the space of shapes. The choice of taking into account the inertia can apparently lead to a more complex system, but neglecting the viscosity the hydrodynamic equations are still linear, and this fact makes the system more manageable. The same fluid regime and existence of solutions of these hydrodynamic equations has been studied in [18] regarding the motion of rigid bodies

Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces

The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $\Lambda$ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on $\Lambda$ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold $\Lambda$, under the stronger assumption that the dynamics on $\Lambda$ is strongly chain transitive.Comment: 26 pages, 2 figure

A Basic Geometric Framework for Quasi-Static Mechanical Manipulation

In this work, we propose a geometric framework for analyzing mechanical manipulation, for example, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. We first review and show that the natural geometric setting is represented by the cotangent bundle and its Lagrangian submanifolds. These are standard concepts in geometric mechanics but usually presented within dynamical frameworks. We review the basic definitions from a static mechanics perspective and show how Lagrangian submanifolds are naturally derived from a first order analysis. Then, via a second order analysis, we derive the Hessian of total energy. As this is not necessarily positive-definite from a control perspective, we propose the use of the squared-Hessian for optimality measures, motivated by insights {derived from both mechanics (Gauss's Principle) and biology (Separation Principle)}. We conclude by showing how such methods can be applied, for example, to the simple case of an elastically driven pendulum. The example is simple enough to allow for analytical solution. However, an extension is further derived and numerically solved, which is more realistically connected with actual robotic manipulation problems.Comment: 28 pages, 6 figures, 1 tabl

Commuting Hamiltonians and multi-time Hamilton-Jacobi equations

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting