Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

An elastoplastic theory of dislocations as a physical field theory with torsion

We consider a static theory of dislocations with moment stress in an anisotropic or isotropic elastoplastical material as a T(3)-gauge theory. We obtain Yang-Mills type field equations which express the force and the moment equilibrium. Additionally, we discuss several constitutive laws between the dislocation density and the moment stress. For a straight screw dislocation, we find the stress field which is modified near the dislocation core due to the appearance of moment stress. For the first time, we calculate the localized moment stress, the Nye tensor, the elastoplastic energy and the modified Peach-Koehler force of a screw dislocation in this framework. Moreover, we discuss the straightforward analogy between a screw dislocation and a magnetic vortex. The dislocation theory in solids is also considered as a three-dimensional effective theory of gravity.Comment: 38 pages, 6 figures, RevTe

Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor

We derive general relations between grain boundaries, rotational deformations, and stress-free states for the mesoscale continuum Nye dislocation density tensor. Dislocations generally are associated with long-range stress fields. We provide the general form for dislocation density fields whose stress fields vanish. We explain that a grain boundary (a dislocation wall satisfying Frank's formula) has vanishing stress in the continuum limit. We show that the general stress-free state can be written explicitly as a (perhaps continuous) superposition of flat Frank walls. We show that the stress-free states are also naturally interpreted as configurations generated by a general spatially-dependent rotational deformation. Finally, we propose a least-squares definition for the spatially-dependent rotation field of a general (stressful) dislocation density field.Comment: 9 pages, 3 figure

Measuring rhizosphere hydraulic properties: impact of root mucilage on soil hydraulic conductivity and water retention curve

Roots are hypothesized to alter rhizosphere hydraulic properties by release of mucilage. This mechanism is expected to have strong implications for root water uptake under drought conditions. Direct measurement of rhizosphere hydraulic properties is hindered by the dynamic nature of the components involved; root hydraulics change with ontology; mucilage production, composition and diffusion are not constant; soil water content changes. An experimental approach was developed which enables to simultaneously measure hydraulic conductivity and apparent water retention curve in a radial flow setup, mimicking the flow geometry around roots. The method consists of extracting water at constant suction via a suction cup, which is centrally placed in a soil filled cylinder and recording water outflow and soil matric potential. In the past, the setup was tested for homogeneous distribution of a model substance (calcium-polygalacturonic acid) frequently used to mimic the properties of root mucilage. Now the system has been applied to investigate the impact of plant root mucilage collected from white lupine. As the system allows a local placement of mucilage treated soil around the suction cup to simulate a ‘rhizosphere’ between bulk soil and suction cup, it can be set up with the limited quantity of natural plant root mucilage available from direct collection. Quartz sand has been treated with lupine root mucilage by mixing liquid mucilage with dry sand at a concentration of 2 mg mucilage per gram soil. Treated sand has been placed as a circular layer with 3.75 mm thickness around the suction cup, which has a radius of 1.25 mm. All around this layer, the device has been filled up with untreated sand. The radius of the whole device was 25 mm. To determine soil hydraulic conductivity we inversely fitted the outflow curves and soil matric potential by solving the Richards’ equation in radial coordinates. Water outflow curves show a significant impact of lupine mucilage on water flow rate – it slows water flow from bulk soil to suction cup. Currently modelling is in process to determine soil hydraulic conductivity and water retention curves. Decreasing hydraulic conductivities and increasing water retention due to lupine mucilage treatment are expected

Semiclassical back reaction around a cosmic dislocation

The energy-momentum vacuum average of a conformally coupled massless scalar field vibrating around a cosmic dislocation (a cosmic string with a dislocation along its axis) is taken as source of the linearized semiclassical Einstein equations. The solution up to first order in the Planck constant is derived. Motion of a test particle is then discussed, showing that under certain circumstances a helical-like dragging effect, with no classical analogue around the cosmic dislocation, is induced by back reaction.Comment: Published version, 4 pages, no figures, REVTeX4 fil

Volume elements and torsion

We reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In Riemann-Cartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, $T_\mu=\partial_\mu\theta$, and that the modified volume element $\tau_\theta = e^\theta \sqrt{g} dx^1\wedge...\wedge dx^n$ be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects.Comment: Revtex, 8 pages, 1 figure. v2 includes a discussion on $\lambda$-symmetr

A gauge theoretic approach to elasticity with microrotations

We formulate elasticity theory with microrotations using the framework of gauge theories, which has been developed and successfully applied in various areas of gravitation and cosmology. Following this approach, we demonstrate the existence of particle-like solutions. Mathematically this is due to the fact that our equations of motion are of Sine-Gordon type and thus have soliton type solutions. Similar to Skyrmions and Kinks in classical field theory, we can show explicitly that these solutions have a topological origin.Comment: 15 pages, 1 figure; revised and extended version, one extra page; revised and extended versio

Diffuse retro-reflective imaging for improved mosquito tracking around human baited bednets

Robust imaging techniques for tracking insects have been essential tools in numerous laboratory and field studies on pests, beneficial insects and model systems. Recent innovations in optical imaging systems and associated signal processing have enabled detailed characterisation of nocturnal mosquito behaviour around bednets and improvements in bednet design, a global essential for protecting populations against malaria. Nonetheless, there remain challenges around ease of use for large scale in situ recordings and extracting data reliably in the critical areas of the bednet where the optical signal is attenuated. Here we introduce a retro-reflective screen at the back of the measurement volume, which can simultaneously provide diffuse illumination, and remove optical alignment issues whilst requiring only one-sided access to the measurement space. The illumination becomes significantly more uniform, although, noise removal algorithms are needed to reduce the effects of shot noise particularly across low intensity bednet regions. By systematically introducing mosquitoes in front and behind the bednet in lab experiments we are able to demonstrate robust tracking in these challenging areas. Overall, the retro-reflective imaging setup delivers mosquito segmentation rates in excess of 90% compared to less than 70% with back-lit systems

Mesoscale theory of grains and cells: crystal plasticity and coarsening

Solids with spatial variations in the crystalline axes naturally evolve into cells or grains separated by sharp walls. Such variations are mathematically described using the Nye dislocation density tensor. At high temperatures, polycrystalline grains form from the melt and coarsen with time: the dislocations can both climb and glide. At low temperatures under shear the dislocations (which allow only glide) form into cell structures. While both the microscopic laws of dislocation motion and the macroscopic laws of coarsening and plastic deformation are well studied, we hitherto have had no simple, continuum explanation for the evolution of dislocations into sharp walls. We present here a mesoscale theory of dislocation motion. It provides a quantitative description of deformation and rotation, grounded in a microscopic order parameter field exhibiting the topologically conserved quantities. The topological current of the Nye dislocation density tensor is derived from a microscopic theory of glide driven by Peach-Koehler forces between dislocations using a simple closure approximation. The resulting theory is shown to form sharp dislocation walls in finite time, both with and without dislocation climb.Comment: 5 pages, 3 figure

Aharonov-Bohm Effect and Disclinations in an Elastic Medium

In this work we investigate quasiparticles in the background of defects in solids using the geometric theory of defects. We use the parallel transport matrix to study the Aharonov-Bohm effect in this background. For quasiparticles moving in this effective medium we demonstrate an effect similar to the gravitational Aharonov- Bohm effect. We analyze this effect in an elastic medium with one and $N$ defects.Comment: 6 pages, Revtex