analysis of the energy based control for swinging up two pendulums this note studies the energy based control for swinging up two pendulums on a cart and it presents an original analysis of the convergence of the energy and the motion of the two pendulums for the two pendulums with the different natural frequencies irrespective of initial state of the two pendulums it is shown that the energy of each pendulum converges to either of two values and the motion of the two pendulums can be described by four invariant sets moreover the stability analysis of the motion of the two pendulums is performed via the concept of stability of an invariant set the results obtained in this note not only show theoretically the effectiveness of the energy based control for swinging up the two pendulums but also pave one way for analyzing and designing the energy based control for more complicated underactuated mechanical systems

fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions abstract let x be a banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict opial condition let c be a bounded closed convex subset of x kc x the family of all compact convex subsets of x and t a nonexpansive mapping from c into kc x with bounded range we prove that t has a fixed point the nonstrict opial condition can be removed if in addition t is an 1 contractive mapping

fixed point theorems for generalized metrically inward maps we prove fixed point results for multimaps satisfying a generalized metric or an metric inwardness condition our results extend generalize or improve several known results

nonlinear h control around periodic orbits abstract in this paper we study the nonlinear h control of systems with periodic orbits we develop the notion of an induced l 2 gain so called nonlinear h norm for systems where the no disturbance behavior of the system is a periodic orbit and provide conditions under which the induced l 2 gain of the system around the orbit can be made less than a specified value by state feedback this work is a natural extension of results on nonlinear h control of nonlinear systems in a neighborhood of a stable equilibrium point to the periodic orbit case synthesis of a nonlinear h state feedback controller is facilitated by the use of transverse coordinates and in particular the transverse linearization of the system

heat kernel expansion for fermionic billiards in an external magnetic field using seeley s heat kernel expansion the authors compute the asymptotic density of states of the dirac operator coupled to a magnetic field on a two dimensional manifold with boundary fermionic billiard local boundary conditions compatible with vector current conservation depend on a free parameter alpha it is shown that the perimeter correction identically vanishes for alpha 0 in that case the next order constant term is found to be proportional to the euler characteristic of the manifold these results are independent of the external magnetic field and of the shape of the billiard provided the boundary is sufficiently smooth for the flat circular billiard the constant term is found to be 1 2 in agreement with a numerical result of berry and mondragon

heat kernel coefficients for chiral bag boundary conditions we study the asymptotic expansion of the smeared l 2 trace of f e tp 2 where p is an operator of dirac type f is an auxiliary smooth smear ing function which is used to localize the problem and chiral bag boundary conditions are imposed special case calculations functorial methods and the theory of and invariants are used to obtain the boundary part of the heat kernel coefficientsa1 and a2

neutrino billiards time reversal symmetry breaking without magnetic fields a dirac hamiltonian describing massless spin half particles neutrinos moving in the plane r x y under the action of a 4 scalar not electric potential v r is in position representation h i h c v r z where x y and z are the pauli matrices acts on two component column spinor wavefunctions r 1 2 and has eigen values ck n does not possess time reversal symmetry t if v r describes a hard wall bounding a finite domain d billiards this is equivalent to a novel boundary condition for 2 1 t breaking is interpreted semiclassically as a difference of between the phases accumulated by waves travelling in opposite senses round closed geo desics in d with odd numbers of reflections the semiclassical large k asymptotics of the eigenvalue counting function spectral staircase n k are shown to have the weyl leading term ak 2 4 where a is the area of d but zero perimeter correction the dirac equation is transformed to an integral equation round the boundary of d and forms the basis of a numerical method for computing the k n when d is the unit disc geodesics are integrable and the eigenvalues which satisfy j l k n j l 1 k n are locally poisson distributed when d is an africa shape cubic conformal map of the unit disc the eigenvalues are locally distributed according to the statistics of the gaussian unitary ensemble of random matrix theory as predicted on the basis of t breaking and lack of geometric symmetry

inward contractions on metric spaces we study the existence of constrained fixed points of contractions in arbitrary complete metric spaces from a global and local point of view in particular we provide generalizations of results due to lim downing and kirk and others some aspects of the topological transversality in the spirit of frigon and granas of contractions under constraints are also considered

viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings let k be a nonempty closed and convex subset of a real banach space e let t k e be a nonexpansive weakly inward mapping with f t and f k k be a contraction then for t 0 1 there exists a sequence yt k satisfying yt 1 t f yt tt yt furthermore if e is a strictly convex real reflexive banach space having a uniformly g teaux differentiable norm then yt converges strongly to a fixed point p of t such that p is the unique solution in f t to a certain variational inequality moreover if ti i 1 2 r is a family of nonexpansive mappings then an explicit iteration process which converges strongly to a common fixed point of ti i 1 2 r and to a solution of a certain variational inequality is constructed under the above setting the family ti i 1 2 r need not satisfy the requirment that mathematical expression

feedback linearization of transverse dynamics for periodic orbits in r 3 with points of transverse controllability loss research supported in part by nsf under grant pyi ecs 9396296 by afosr under grant afosr 94nm006 and by a grant from hughes aircraft company

set valued dynamic systems and random iterations of set valued weaker contractions in uniform spaces set valued weaker contractions in uniform locally convex and metric spaces are defined and dynamic systems of such weaker contractions are studied conditions guaranteeing the convergence of generalized sequences of random iterations and iterations and the existence and uniqueness of endpoints of set valued weaker contractions are established our definitions and results are new for set valued maps in uniform locally convex and metric spaces and even for single valued maps

mixed boundary conditions in quantum field theory when quantum field theory is formulated in terms of transition amplitudes between initial and final configurations it is important to impose mathematically consistent boundary conditions on the fields the choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds although dirichlet and neumann boundary conditions are adequate for many purposes they are not always appropriate however a number of the remaining cases can be handled by imposing dirichlet conditions on certain field components and neumann conditions on the others quantum gravity and supergravity spinor field theory and various elliptic complexes all lend themselves to mixed boundary conditions of this sort the mathematical significance of mixed boundary conditions is investigated and a simple rule is obtained for determining which field components should obey which boundary conditions it will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary this result is used to define gauge invariant and brs invariant quantum amplitudes for boundary configurations the most general form for mixed boundary conditions on the dirac operator and the de rham complex will also be derived

stabilization of the inverted pendulum around its homoclinic orbit the inverted pendulum has been used as a benchmark for motivating the study of nonlinear control techniques we propose a simple controller for balancing the inverted pendulum and raise it to its upper equilibrium position while the cart displacement is brought to zero the control strategy is based on an energy approach of the cart and pendulum system

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

vss version of energy based control for swinging up a pendulum abstract a new algorithm ensuring global attractivity of the upright unstable equilibrium of a pendulum based on the variable structure system version of the energy speed gradient method is proposed it is shown that global attractivity cannot be obtained with continuous static state feedback a detailed global analysis of the transient behavior of the closed loop system is presented in addition it is shown that the global attractivity of the upright equilibrium can be achieved by applying a control of arbitrarily small magnitude

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

nonlinear control of a swinging pendulum we propose a nonlinear controller to regulate the swinging energy of the pendulum for a cart and pendulum system roughly speaking the controller is designed to regulate an output the swing energy while providing internal stability regulating the cart position it is difficult to apply many of the standard nonlinear control design techniques since the output zeroing manifold does not contain any equilibrium points and the relative degree of the system is not constant in contrast to controllers that use a command generator and possibly a time varying feedback our control law is a simple autonomous nonlinear controller we analyze the stability of the closed loop system using an lv small gain approach on a transverse linearization of the system about the desired periodic orbit one can easily extend this approach to analyze the robustness of the control system with respect to disturbances and parameter variations experimental results demonstrate the effectiveness of the nonlinear controller

feedback linearization of transverse dynamics for periodic orbits in this paper we give necessary and sufficient conditions for feedback linearization of the transverse dynamics tfl of a nonlinear affine single input system in a neighborhood of a periodic orbit the tfl procedure provides a means of finding coordinates that are tuned to the structure of the control system with respect to the periodic orbit an autonomous feedback control providing exponential stability of the periodic orbit is easily designed in the transverse coordinate system

converse lyapunov functions for exponentially stable periodic orbits abstract in this paper we present a technique for constructing a class of quadratic lyapunov functions for exponentially stable periodic orbits this construction is facilitated by the use of a special set of local coordinates that serve to highlight the tangential and transverse dynamics of the system