Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

Study of Effective 2+1 Dimensional Gravity in Ferrofluid-Based Hyperbolic Metamaterials

Recent theoretical and experimental work demonstrated that nonlinear optics of ferrofluid-based hyperbolic metamaterials exhibit very unusual spatiotemporal dynamics. Here a detailed theoretical and experimental study of mutual interactions of individual self-focused optical filaments inside this metamaterial is reported. In agreement with theoretical expectations, the observed mutual interactions of individual filaments exhibit strong similarities with general relativity in 2+1 dimensions, which predicts that these interactions must have predominantly non-Newtonian topological character. This observation is important since 2+1-dimensional gravity is an exactly solvable theory even in the quantum gravity limit.     

Characterization of the products of aniline peroxydisulfate oligo/polymerization in media with different pH by resonance Raman spectroscopy at 413.1 and 1064 nm excitation wavelengths

The pH–structure correlation of the products of aniline peroxydisulfate reaction was mainly investigated by resonance Raman spectroscopy. The reactions of aniline and ammonium peroxydisulfate were carried out in aqueous solutions of initial pH ranging from 4.9 to 13.2 and monomer/oxidant molar ratio of 4/1. For an initial pH of 4.9, the spectroscopic techniques showed that the emeraldine salt form of polyaniline (PANI–ES) is the main product, corroborating that the usual head‐to‐tail coupling mechanism is taking place. The resonance Raman spectra at 1064 nm exciting wavelength were useful to detect the emeraldine salt as a minor product for reactions at an initial pH of 5.3–11.5. The Raman spectra of the main product of the reaction at initial pH of 13.2 excited at 1064 and 413.1 nm showed new spectral features consistent with 1,4‐Michael‐type adducts of aniline monomers and 1,4‐benzoquinone‐monoimine unit. These compounds and their products of hydrolysis/oxidation are the predominant species for the reaction media of initial pH from 5.3 to 13.2. In order to get PANI with different nanoscale morphologies, a pH value of more than 0 or 1 was used in the aniline polymerization. The spectroscopic data obtained in this work reveal that head‐to‐tail coupling does not occur when aniline reacts at media pH higher than about 5. It is suggested that chemical structures of the products of aniline oxidation by an unusual mechanism are the driving force for the development of assorted morphologies. Copyright © 2011 John Wiley & Sons, Ltd.     

Calculus of variations on time scales: weak local piecewise Crd solutions with variable endpoints

A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C¹_prd). For this problem, the Euler-Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.     

New Biomimetic Catalysts for the Electrochemical Processes on the Basis of Redox-Active Macrocyclic Frame Structures

Abstract

Efficient catalytic systems mimicking the action of metalloenzymes based on cyclic frame structures containing phosphorus, nitrogen, and sulfur coordinating atoms for hydrogen oxidation or reduction under mild conditions are proposed.