Optimal Control Of Surface Shape

Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Moreover, we provide error estimates for the state variable and adjoint state. Numerical results are shown to illustrate the minimizers and optimal controls on different domains

Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals

We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter $Q = Q(x)$. Equilibrium LC states correspond to $Q$ functions that (locally) minimize an LdG energy functional. Thus, we consider an $L^2$-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external ``force'' controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where $Q(x) = 0$) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure

A C-elegans stretch receptor neuron revealed by a mechanosensitive TRP channel homologue

The nematode Caenorhabditis elegans is commonly used as a genetic model organism for dissecting integration of the sensory and motor systems(1). Despite extensive genetic and behavioural analyses that have led to the identification of many genes and neural circuits involved in regulating C. elegans locomotion behaviour(1), it remains unclear whether and how somatosensory feedback modulates motor output during locomotion. In particular, no stretch receptors have been identified in C. elegans, raising the issue of whether stretch-receptor-mediated proprioception is used by C. elegans to regulate its locomotion behaviour. Here we have characterized TRP-4, the C. elegans homologue of the mechanosensitive TRPN channel. We show that trp-4 mutant worms bend their body abnormally, exhibiting a body posture distinct from that of wild-type worms during locomotion, suggesting that TRP-4 is involved in stretch-receptor-mediated proprioception. We show that TRP-4 acts in a single neuron, DVA, to mediate its function in proprioception, and that the activity of DVA can be stimulated by body stretch. DVA both positively and negatively modulates locomotion, providing a unique mechanism whereby a single neuron can fine-tune motor activity. Thus, DVA represents a stretch receptor neuron that regulates sensory - motor integration during C. elegans locomotion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62770/1/nature04538.pd

Life Beyond the Solar System: Remotely Detectable Biosignatures

For the first time in human history, we will soon be able to apply to the scientific method to the question "Are We Alone?" The rapid advance of exoplanet discovery, planetary systems science, and telescope technology will soon allow scientists to search for life beyond our Solar System through direct observation of extrasolar planets. This endeavor will occur alongside searches for habitable environments and signs of life within our Solar System. While these searches are thematically related and will inform each other, they will require separate observational techniques. The search for life on exoplanets holds potential through the great diversity of worlds to be explored beyond our Solar System. However, there are also unique challenges related to the relatively limited data this search will obtain on any individual world