Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Complexes of 2,2′,2″-Nirtilotriphenol. Part IV. Cage Compounds with Phosphorus(III) and Phosphorus(V)

The ligand 2,2′,2″-nitrilotriphenol reacts with P(III) and P(V) compounds to form corresponding phosphorus complexes. Syntheses and NMR data of 2,2′,2″-nitrilotriphenyl phosphite ( II ), 2,2′,2″-nitrilotriphenyl phosphate ( III ) and of a hydrolysis product of II , 2,2′-[N-(2-hydroxyphenyl)imino]diphenly phosphonate ( IV ), are reported, as well as crystal structures of II and IV . Phosphite II shows a bicycloundecane framework; no N?Pinteraction is present. The phosphonate IV shows two coordinated and one dangling phenol group; the N-atom does not interact with the P-atom. Strong acids protonate II as well as III to form cations: in these, NMR evidence indicates coordination of the N-atom to the P-atom.