1,898 research outputs found
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
A Homotopy-Based Method for Optimization of Hybrid High-Low Thrust Trajectories
Space missions require increasingly more efficient trajectories to provide payload transport and mission goals by means of lowest fuel consumption, a strategic mission design key-point. Recent works demonstrated that the combined (or hybrid) use of chemical and electrical propulsion can give important advantages in terms of fuel consumption, without losing the ability to reach other mission objectives: as an example the Hohmann Spiral Transfer, applied in the case of a transfer to GEO orbit, demonstrated a fuel mass saving between 5-10% of the spacecraft wet mass, whilst satisfying a pre-set boundary constraint for the time of flight. Nevertheless, methods specifically developed for optimizing space trajectories considering the use of hybrid high-low thrust propulsion systems have not been extensively developed, basically because of the intrinsic complexity in the solution of optimal problem equations with existent numerical methods. The study undertaken and presented in this paper develops a numerical strategy for the optimization of hybrid high-low thrust space trajectories. An indirect optimization method has been developed, which makes use of a homotopic approach for numerical convergence improvement. The adoption of a homotopic approach provides a relaxation to the optimal problem, transforming it into a simplest problem to solve in which the optimal problem presents smoother equations and the shooting function acquires an increased convergence radius: the original optimal problem is then reached through a homotopy parameter continuation. Moreover, the use of homotopy can make possible to include a high thrust impulse (treated as velocity discontinuity) to the low thrust optimal control obtained from the indirect method. The impulse magnitude, location and direction are obtained following from a numerical continuation in order to minimize the problem cost function. The initial study carried out in this paper is finally correlated with particular test cases, in order to validate the work developed and to start investigating in which cases the effectiveness of hybrid-thrust propulsion subsists
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
Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux
State-dependent time-impulsive perturbations to a two-dimensional autonomous
flow with stable and unstable manifolds are analysed by posing in terms of an
integral equation which is valid in both forwards- and backwards-time. The
impulses destroy the smooth invariant manifolds, necessitating new definitions
for stable and unstable pseudo-manifolds. Their time-evolution is characterised
by solving a Volterra integral equation of the second kind with discontinuous
inhomogeniety. A criteria for heteroclinic trajectory persistence in this
impulsive context is developed, as is a quantification of an instantaneous flux
across broken heteroclinic manifolds. Several examples, including a kicked
Duffing oscillator and an underwater explosion in the vicinity of an eddy, are
used to illustrate the theory
-Minimization for Mechanical Systems
Second order systems whose drift is defined by the gradient of a given
potential are considered, and minimization of the -norm of the control is
addressed. An analysis of the extremal flow emphasizes the role of singular
trajectories of order two [25,29]; the case of the two-body potential is
treated in detail. In -minimization, regular extremals are associated with
controls whose norm is bang-bang; in order to assess their optimality
properties, sufficient conditions are given for broken extremals and related to
the no-fold conditions of [20]. An example of numerical verification of these
conditions is proposed on a problem coming from space mechanics
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Reformulations for control systems and optimization problems with impulses
This dissertation studies two different techniques for analyzing control systems whose dynamics include impulses, or more specifically, are measure-driven. In such systems, the state trajectories will have discontinuities corresponding to the atoms of the Borel measure driving the dynamics, and these discontinuities require further definition in order for the control system to be treated with the broad range of results available to non-impulsive systems. Both techniques considered involve a reparameterization of the system variables including state, time, and controls. The first method is that of the graph completion, which provides an explicit reparameterization of the time and state variables. The reparameterization is continuous, which allows for the analysis of the system within classical control theory, yet it retains enough information about the discontinuous, or impulsive, trajectories that the results of such analyses may be interpreted for the original impulsive system. We utilize this reparameterization to formulate equivalent solution concepts between impulsive differential inclusions and impulsive differential equations. We also demonstrate that the graph completion is generally equivalent to a solution concept established for a neural spiking model, and make use of a specific such model as a numerical example. The second method considered is similar to the graph completion but differs in that it utilizes implicit reparameterizations of all variables considered as families of functions which meet continuity and other requirements. This is particularly beneficial to optimal control problems as the choices of controls, impulsive and non-impulsive, may be varied within the optimization problem and analysis thereof. Necessary conditions for optimal control problems of Mayer form with fixed end time have been established under this reparameterization technique, and we extend these necessary conditions in a general context to a Mayer problem with free end time. Corollary to this, we deduce necessary conditions for a Bolza problem and a minimum time problem for impulsive control systems. Much of these results are obtained through reformulation techniques
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