Structure of norbornene-fused 3,1-oxazine double cycloadducts

Pentacyclic isoxazolines were obtained by the cycloaddition of benzonitrile oxide to norbornene-azetidinone-fused 3,1-oxazines. The constitutions of two of the isomers obtained, and the configurations and conformations of all products, were determined by means of ¹H and ¹³C NMR spectroscopy and DNOE experiments.     

Phasing of three vircators simulated in terms of coupled van der Pol oscillators

Phase synchronization in a system of three virtual-cathode microwave oscillators (vircators) simulated by coupled van der Pol oscillators is studied. The phasing dynamics of the vircators is visualized with the phase portraits of the system in the triangular coordinates. Different phasing conditions are found.     

Fundamental aspects of initiated oxidation of tridecane and polyethylene in solutions

Specific features of the initiated oxidation of polyethylene and its low-molecular-weight analogue tridecane, associated with the occurrence of the process in the short-chain mode, were studied.     

Optimality problem for infinite dimensional bilinear systems

For any finite dimensional control system with arbitrary cost, Pontryagin's Maximum Principle (PMP) [N. Bensalem, Localisation des courbes anormales et problème d'accessibilité sur un groupe de Lie hilbertien nilpotent de degré 2, Thèse de doctorat, Université de Savoie, 1998. [6]] gives necessary conditions for optimality of trajectories. In the infinite dimensional case, it is well known that these conditions are no more true in general. The purpose of this paper is to establish an “approached” version of PMP for infinite dimensional bilinear systems, with fixed final time and without constraints on the final state. Moreover, if the set of control is contained in a closed bounded convex subset with operators defining its dynamics are compact, or if it is contained in a finite dimensional space, we get an “exact” version of PMP. We also give two applications of these results. The first one deals with sub-Riemannian geometry on nilpotent Hilbertian Lie groups for which we can define a sub-Riemannian distance. The second one deals with heat equation for which we analyse the necessary conditions to give the optimal controls.