free endpoint optimal control of inhomogeneous bilinear ensemble systems optimal control of bilinear systems has been a well studied subject in the areas of mathematical and computational optimal control however effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped these burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience in this work we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free endpoint conditions the central idea is to represent the bilinear ensemble system at each iteration as a time varying linear ensemble system and then solve it in an iterative manner we analyze convergence of the iterative procedure and discuss optimality of the convergent solutions the method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes we demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control

g2 structures and octonion bundles abstract we use a g 2 structure on a 7 dimensional riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non associative product we then define a metric compatible octonionic covariant derivative on this bundle that is compatible with the octonion product the torsion of the g 2 structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the riemann curvature that lies in the 7 dimensional representation of g 2 we also interpret the choice of a particular g 2 structure within the same metric class as a choice of gauge and show that under a change of this gauge the torsion does transform as an octonion valued connection 1 form finally we also show an explicit relationship between the octonion bundle and the spinor bundle define an octonionic dirac operator and explore an energy functional for octonion sections we then prove that critical points correspond to divergence free torsion which is shown to be an octonionic analog of the coulomb gauge

local geometric invariants of integrable evolution equations the integrable hierarchy of commuting vector fields for the localized induction equation of 3d hydrodynamics and its associated recursion operator are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept out surface

a pseudospectral method for the optimal control of constrained feedback linearizable systems we consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints in general a linearizing feedback control does not minimize the cost function such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls in this paper we consider a pseudospectral ps method to compute optimal controls we prove that a sequence of solutions to the ps discretized constrained problem converges to the optimal solution of the continuous time optimal control problem under mild and numerically verifiable conditions the spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution the proposed ideas are illustrated by several numerical examples

stability of nonlinearly elastic rods this paper describes previously unknown stabilities and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead load the governing equations are derived from variational principles including ones of isoperimetric type properties of stability are accordingly determined by study of the second variation stabilities to deformations both in the plane and out of the plane are considered among the newly discovered properties are secondary bifurcation from the first buckled mode marked differences between stability to two dimensional and to three dimensional variations and the stabilizing influence of resistance to twist in the isoperimetric examples the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints

lagrangian aspects of the kirchhoff elastic rod the relation between euler s planar elastic curves and vortex filaments evolving by the localized induction equation lie of hydrodynamics was discovered by hasimoto in 1971 basic facts about an integrable case of kirchhoff elastic rods are described here which amplify the connection between the variational problem for rods and the soliton equation lie in particular it is shown that the centerline of the kirchhoff rod is an equilibrium for a linear combination of the first three conserved hamiltonians in the lie hierarchy

technical note probability of success in the search for a moving target the probability of detection in the search for a randomly moving target is calculated for the case of a target whose motion is a diffusion process and known searcher path the probability of detection can be calculated by solving a backward diffusion equation corwin corwin t 1980 on multiplicative functionals on diffusion processes siam j appl math 38 112 119 gives a solution of the backward equation for a special case in general exact solutions do not exist and other methods are needed in this paper the backward equation is solved approximately by using a formal asymptotic method valid when the intensity of the random motion is small the general solution is illustrated for the case of spatially homogeneous drift and diffusion coefficients in this case the asymptotic solution can be evaluated analytically

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

a multiplicity of localised buckling modes for twisted rod equations the kirchhoff love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions the effects of shear and axial extension are not considered but the model does additionally allow for nonlinear constitutive laws under the assumption of infinite length the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths localizing solutions correspond to homoclinic orbits of the dynamical system under non dimensionalization the twisted rod equations are shown to depend on a single load parameter and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques first in the case of a rod with equal principal bending stiffnesses where the equilibrium equations are completely integrable a known one parameter family of localizing solutions is computed for a variety of subcritical loads load deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture the breaking of the geometrical circular symmetry destroys complete integrability and in particular breaks the non transverse intersection of the stable and unstable manifolds of the trivial steady state the resulting transverse intersection which is already known to lead to spatial chaos is explicitly demonstrated to imply a multitude of localized buckling modes a sample of primary and multi modal solutions are computed numerically aided by the reversibility of the differential equations finally parallels are drawn with the conceptually simpler problem of a strut resting on a nonlinear elastic foundation for which much more information is known about the global behaviour of localized buckling modes

volumes of vector fields on spheres in this paper we study the problem what is the unit vector field of smallest volume on an odd dimensional sphere we exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four we conjecture that this volume is the infimum for volumes of bona fide unit vector fields and is only achieved by the singular vector field we generalize the construction of the singular vector field to give a family of cycles in stiefel manifolds each of which is a smooth manifold except for one singular point except for some low dimensional cases the tangent cones at these singular points are volume minimizing and half of the cones are nonorientable thus we obtain a new family of nonorientable volume minimizing cones

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

vibration of thin pre twisted helical beams abstract a helical beam has non zero curvature and tortuosity when there is a pre twist the frenet triad is not necessary coincident with the principal triad when an initially straight rod undergoing a large deformation it will be curved and twisted during each deformation sequence the purpose of the paper is to establish the governing equations and the associated natural boundary conditions for a pre twisted helical beam by variational principles and differential geometry although circular helix is taken as example the formulation is valid for inhomogeneous and non uniform parameters along the centerline that variable curvature and tortuosity can be treated without difficulties the method can be used to treat helical anisotropy if the pre twist rate of a ring is 0 5 we show a thick mobius ring for the first time the solution of the natural vibration problem is through chebyshev discretization and clenshaw curtis integration by means of the galerkin formulation

the intricate global structure of buckled states of compressible columns singularity theory and global multiparameter bifurcation theory are combined to show that nonbifurcating buckled states of compressible nonlinearly elastic columns lie on branches having detailed qualitative properties these properties distinguish the branches globally together with a delicate asymptotic analysis they are used to show that the global disposition of solution branches both bifurcating and not is strikingly different from that for incompressible columns

on the dynamics of rods in the theory of kirchhoff and clebsch we discuss here the dynamical equations of a theory of elastic rods that is due to kmcm ioff 1 2 and cl bsc 3 4 this properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small although rotations may be large it is constructed to be a first order theory i e a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness curvature twist and extension in a first order theory of thin rods one can treat the rod as inextensible and we do so here at the outset thus at each time t the arc length parameter s for the axial curve t is employed as a material coordinate i e a parameter whose value at a material point is constant in time and not only the resultant of the shearing forces on a cross section but also the tension in the rod are reactive quantities not given by constitutive equations consider for a moment a rod that is naturally prismatic and dynamically symmetric i e a rod that in an undistorted stress free configuration is a cylinder whose directrix although not necessarily a circle bounds a figure with equal principal moments of inertia a motion of the rod is said to be planar and twist flee i e a motion of pure flexure if it is such that t lies at all times in a fixed plane which contains a principal axis of inertia of each cross section for such a motion we employ a fixed cartesian coordinate system on and for consistency with a discussion of more general motions to be given later in this paper we call the abscissa z and the ordinate x we may write fz s t fx s t for the z and x components of the resultant force f at time t on the cross section with arc length coordinate s the motion of the rod may be described by giving the z x coordinates of the points on t as functions of s and t with o s t the counterclockwise angle from the z axis to the tangent of t at s we have

consistent approximations for optimal control problems based on runge kutta integration this paper explores the use of runge kutta integration methods in the construction of families of finite dimensional consistent approximations to nonsmooth control and state constrained optimal control problems consistency is defined in terms of epiconvergence of the approximating problems and hypoconvergence of their optimality functions a significant consequence of this concept of consistency is that stationary points and global solutions of the approximating discrete time optimal control problems can only converge to stationary points and global solutions of the original optimal control problem the construction of consistent approximations requires the introduction of appropriate finite dimensional subspaces of the space of controls and the extension of the standard runge kutta methods to piecewise continuous functions it is shown that in solving discrete time optimal control problems that result from runge kutta integration a non euclidean inner product and norm must be used on the control space to avoid potentially serious ill conditioning effects

on melnikov s method in the study of chaotic motions of gyrostat the homoclinic solutions of the attitude motion of a gyrostat with wheels along its principal axes are formulated using the procedure developed by wittenburg the generic melnikov function for the attitude motion of a gyrostat subject to small non linear damping torques and small periodic torques is derived the derivation is based on the chaotic theory for a one and half degrees of freedom system established by wiggins and shaw the conditions for the physical parameters at the onset of the homoclinic solutions are given in detail the chaotic criteria are investigated using melnikov s function in particular the conditions of the physical parameters moments of inertia damping coefficients amplitudes and frequencies of external excitation moments of momentum of wheels that ensure the convergence of melnikov s integral are also worked out in order to acquire some knowledge of long term behaviours of the chaotic attitude motion the fourth order runge kutta numerical algorithms are utilized to sim

search for a randomly moving object after a brief discussion of the operational origin of search problems the mathematical problem is formulated the mathematical quantity of interest is the joint density for location of the object sought and unsuccessful search when the object moves according to a diffusion process this joint density satisfies a parabolic equation after the introduction of scaled variables the search equation can be approximately solved by the ray method the interpretation of the terms in the approximate solution is discussed the case of constant diffusion and drift parameters and piecewise linear searching paths arises often in operational situations this case is considered in detail

spatial chaos of 3 d elastica with the kirchhoff gyrostat analogy using melnikov integrals summary the kirchhoff kinetic analogy from which the similarity between the governing equations for the static spatial equilibrium of a 3 d elastica and those for the temporal dynamics of a rigid body is constituted is revisited the melnikov integrals for detecting chaos cannot be easily formed for an elastica we shall modify the previous procedure of using the melnikov integrals for detecting temporal chaos for a gyrostat to solve the spatial chaos problem of an elastica one way to find the disturbed hamiltonian equations of the equilibrium equations of the elastica are by means of deprit s variables using the melnikov integrals the spatially chaotic deformation patterns can be resulted from the homoclinic transversal intersections of the stable and unstable manifolds at a saddle point in the poincare map when the elastica is disturbed by constant stress resultants the melnikov integrals are integrated for detecting homoclinic intersections the equations governing the evolution of the stress couples and the stress resultants are numerically integrated using the fourth runge kutta algorithms to crosscheck the analytical results the phase portraits of the poincare sections are created on a number of hyper planes to demonstrate the chaotic patterns of the stationary spatial deformations along the centreline of the elastica under weightless conditions the bounded non periodic solutions to the equilibrium of the elastica clearly show the existence of spatially chaotic deformation patterns the findings in this paper are useful in many structures in which the elastica are important examples are the mathematical modelling of the dna super coiling submarine cables and the design of the tethered satellites copyright 2004 john wiley sons ltd

consistent approximation of a nonlinear optimal control problem with uncertain parameters this paper focuses on a non standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain the research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non evading moving targets in this paper we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem the framework is proved to produce a zeroth order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem in addition we demonstrate the convergence of the corresponding adjoint variables the accumulation points of a sequence of optimal state adjoint pairs for the approximate problem satisfy a necessary condition of pontryagin minimum principle type which facilitates assessment of the optimality of numerical solutions

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

an adaptive differential quadrature element method for large deformation contact problems involving curved beams with a finite number of contact points abstract contact problems involving large deformation of curved beams are difficult to analyze due to uncertainty of contact positions and strong nonlinearity a nonlinear large deformation model of curved beams is formulated in arc length coordinates a new adaptive differential quadrature element method adqem is proposed to predict contact positions of a curved beam with a finite number of contact points where a dragging method and continuity conditions are combined to determine the contact positions simulation results show that the adqem greatly improves efficiency and accuracy of the large deformation contact problem of the curved beam the number of iterations in the present method does not greatly increase with the number of contact points