This dissertation develops a variety of structure-preserving algorithms for mechanical systems with external forcing and also extends those methods to systems that evolve on non-Euclidean manifolds. The dissertation is focused on numerical schemes derived from variational principles – schemes that are general enough to apply to a large class of engineering problems. A theoretical framework that encapsulates variational integration for mechanical systems with external forcing and time-dependence and which supports the extension of these methods to systems that evolve on non-Euclidean manifolds is developed. An adaptive time step, energy-preserving variational integrator is developed for mechanical systems with external forcing. It is shown that these methods track the change in energy more accurately than their fixed time step counterparts. This approach is also extended to rigid body systems evolving on Lie groups where the resulting algorithms preserve the geometry of the configuration space in addition to being symplectic as well as energy and momentum-preserving. The advantages of structure-preservation in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching. / Doctor of Philosophy / Accurate numerical simulation of dynamical systems over long time horizons is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. In many of these applications, the governing physical equations derive from a variational principle and their solutions exhibit physically meaningful invariants such as momentum, energy, or vorticity. Unfortunately, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. In this dissertation, tools from geometric mechanics and computational methods are used to develop numerical integrators that respect the qualitative features of the physical system. The research presented here focuses on numerical schemes derived from variational principles– schemes that are general enough to apply to a large class of engineering problems. Energy-preserving algorithms are developed for mechanical systems by exploiting the underlying geometric properties. Numerical performance comparisons demonstrate that these algorithms provide almost exact energy preservation and lead to more accurate prediction. The advantages of these methods in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching.

The first and most imperative step when designing a biologically inspired robot is to identify the underlying mechanics of the system or animal of interest. It is most common, perhaps, that this process generates a set of coupled nonlinear ordinary or partial differential equations. For this class of systems, the models derived from morphology of the skeleton are usually very high dimensional, nonlinear, and complex. This is particularly true if joint and link flexibility are included in the model. In addition to complexities that arise from morphology of the animal, some of the external forces that influence the dynamics of animal motion are very hard to model. A very well-established example of these forces is the unsteady aerodynamic forces applied to the wings and the body of insects, birds, and bats. These forces result from the interaction of the flapping motion of the wing and the surround- ing air. These forces generate lift and drag during flapping flight regime. As a result, they play a significant role in the description of the physics that underlies such systems. In this research we focus on dynamic and kinematic models that govern the motion of ground based robots that emulate flapping flight. The restriction to ground based biologically inspired robotic systems is predicated on two observations. First, it has become increasingly popular to design and fabricate bio-inspired robots for wind tunnel studies. Second, by restricting the robotic systems to be anchored in an inertial frame, the robotic equations of motion are well understood, and we can focus attention on flapping wing aerodynamics for such nonlinear systems. We study nonlinear modeling, identification, and control problems that feature the above complexities. This document summarizes research progress and plans that focuses on two key aspects of modeling, identification, and control of nonlinear dynamics associated with flapping flight. / Ph. D. / In this work we focus on modeling flapping flight mechanics by focusing our attention in two aspects of modeling. We first model the behavior of aerodynamic forces in charge of keeping the flying animal airborn. We present a mathematical model for history dependent profile of these forces. Also, we propose a novel adaptive controller to compensate these unknown forces in the dynamic model of the system. We also propose an algorithm to derive dynamic equations of the animal motion by using video data. We expect the model derived by this novel method to emulate the animal motion closely.

Cette thèse est une contribution au calcul des variations et à la théorie du contrôle optimal dans les cadres discret, plus généralement time scale, et fractionnaire. Ces deux domaines ont récemment connu un développement considérable dû pour l’un à son application en informatique et pour l’autre à son essor dans des problèmes physiques de diffusion anormale. Que ce soit dans le cadre time scale ou dans le cadre fractionnaire, nos objectifs sont de : a) développer un calcul des variations et étendre quelques résultats classiques (voir plus bas); b) établir un principe du maximum de Pontryagin (PMP en abrégé) pour des problèmes de contrôle optimal. Dans ce but, nous généralisons plusieurs méthodes variationnelles usuelles, allant du simple calcul des variations au principe variationnel d’Ekeland (couplé avec la technique des variations-aiguilles), en passant par l’étude d’invariances variationnelles par des groupes de transformations. Les démonstrations des PMPs nous amènent également à employer des théorèmes de point fixe et à prendre en considération la technique des multiplicateurs de Lagrange ou encore une méthode basée sur un théorème d’inversion locale conique. Ce manuscrit est donc composé de deux parties : la Partie 1 traite de problèmes variationnels posés sur time scale et la Partie 2 est consacrée à leurs pendants fractionnaires. Dans chacune de ces deux parties, nous suivons l’organisation suivante : 1. détermination de l’équation d’Euler-Lagrange caractérisant les points critiques d’une fonctionnelle Lagrangienne ; 2. énoncé d’un théorème de type Noether assurant l’existence d’une constante de mouvement pour les équations d’Euler-Lagrange admettant une symétrie ; 3. énoncé d’un théorème de type Tonelli assurant l’existence d’un minimiseur pour une fonctionnelle Lagrangienne et donc, par la même occasion, d’une solution pour l’équation d’Euler-Lagrange associée (uniquement en Partie 2) ; 4. énoncé d’un PMP (version forte en Partie 1, version faible en Partie 2) donnant une condition nécessaire pour les trajectoires qui sont solutions de problèmes de contrôle optimal généraux non-linéaires ; 5. détermination d’une condition de type Helmholtz caractérisant les équations provenant d’un calcul des variations (uniquement en Partie 1 et uniquement dans les cas purement continu et purement discret). Des théorèmes de type Cauchy-Lipschitz nécessaires à l’étude de problèmes de contrôle optimal sont démontrés en Annexe. / This dissertation deals with the mathematical fields called calculus of variations and optimal control theory. More precisely, we develop some aspects of these two domains in discrete, more generally time scale, and fractional frameworks. Indeed, these two settings have recently experience a significant development due to its applications in computing for the first one and to its emergence in physical contexts of anomalous diffusion for the second one. In both frameworks, our goals are: a) to develop a calculus of variations and extend some classical results (see below); b) to state a Pontryagin maximum principle (denoted in short PMP) for optimal control problems. Towards these purposes, we generalize several classical variational methods, including the Ekeland’s variational principle (combined with needle-like variations) as well as variational invariances via the action of groups of transformations. Furthermore, the investigations for PMPs lead us to use fixed point theorems and to consider the Lagrange multiplier technique and a method based on a conic implicit function theorem. This manuscript is made up of two parts : Part A deals with variational problems on time scale and Part B is devoted to their fractional analogues. In each of these parts, we follow (with minor differences) the following organization: 1. obtaining of an Euler-Lagrange equation characterizing the critical points of a Lagrangian functional; 2. statement of a Noether-type theorem ensuring the existence of a constant of motion for Euler-Lagrange equations admitting a symmetry;3. statement of a Tonelli-type theorem ensuring the existence of a minimizer for a Lagrangian functional and, consequently, of a solution for the corresponding Euler-Lagrange equation (only in Part B); 4. statement of a PMP (strong version in Part A and weak version in Part B) giving a necessary condition for the solutions of general nonlinear optimal control problems; 5. obtaining of a Helmholtz condition characterizing the equations deriving from a calculus of variations (only in Part A and only in the purely continuous and purely discrete cases). Some Picard-Lindelöf type theorems necessary for the analysis of optimal control problems are obtained in Appendices.