Markov–Dubins path via optimal control theory

Markov–Dubins path is the shortest planar curve joining two points with prescribed tangents, with a specified bound on its curvature. Its structure, as proved by Dubins in 1957, nearly 70 years after Markov posed the problem of finding it, is elegantly simple: a selection of at most three arcs are concatenated, each of which is either a circular arc of maximum (prescribed) curvature or a straight line. The Markov–Dubins problem and its variants have since been extensively studied in practical and theoretical settings. A reformulation of the Markov–Dubins problem as an optimal control problem was subsequently studied by various researchers using the Pontryagin maximum principle and additional techniques, to reproduce Dubins’ result. In the present paper, we study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins’ result again. We prove that abnormal control solutions do exist. We characterize these solutions, which were not studied adequately in the literature previously, as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem. Moreover, we prove that any feasible path of the types mentioned in Dubins’ result is a stationary solution, i.e., that it satisfies the Pontryagin maximum principle. We propose a numerical method for computing Markov–Dubins path. We illustrate the theory and the numerical approach by three qualitatively different examples.     

Kinetics and mechanism of myristic acid and isopropyl alcohol esterification reaction with homogeneous and heterogeneous catalysts

The reaction of myristic acid (MA) and isopropyl alcohol (IPA) was carried out by using both homogeneous and heterogeneous catalysts. For a homogeneously catalyzed system, the experimental data have been interpreted with a second order, using the power‐law kinetic model, and a good agreement between the experimental data and the model has been obtained. In this approach, it was assumed that a protonated carboxylic acid is a possible reaction intermediate. After a mathematical model was proposed, reaction rate constants were computed by the Polymath* program. For a heterogeneously catalyzed system, interestingly, no pore diffusion limitation was detected. The influences of initial molar ratios, catalyst loading and type, temperature, and water amount in the feed have been examined, as well as the effects of catalyst size for heterogeneous catalyst systems. Among used catalysts, p‐toluene sulfonic acid (p‐TSA) gave highest reaction rates. Kinetic parameters such as activation energy and frequency factor were determined from model fitting. Experimental K values were found to be 0.54 and 1.49 at 60°C and 80°C, respectively. Furthermore, activation energy and frequency factor at forward were calculated as 54.2 kJ mol^?1 and 1828 L mol^?1 s^?1, respectively. © 2008 Wiley Periodicals, Inc. 40: 136–144, 2008     

Dye-ligand immobilized IPNs membrane for removal heavy metal ions

We have developed a novel approach to obtain high metal sorption capacity utilizing a membrane containing chitosan and an immobilized reactive dye (i.e. Reactive Yellow-2). The composite membrane was characterized by SEM, FT-IR, swelling test, and elemental analysis. The membrane has uniform small pores distribution and the pore dimensions are between 5 and 10 μm, and the HEMA:chitosan ratio was 50:1. The reactive dye immobilized composite membrane was used in the removal of heavy metal ions [i.e., Pb(II), Hg(II) and Cd(II)] from aqueous medium containing different amounts of these ions (5-600 mg l⁻¹) and at different pH values (2.0-7.0). The maximum adsorption capacities of heavy metal ions onto the composite membrane under non-competitive conditions were 64.3 mmol m⁻² for Pb(II), 52.7 mmol m⁻² for Hg(II), 39.6 mmol m⁻² for Cd(II) and the affinity order was Pb(II) > Hg(II)>Cd(II).