We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols S n δ = n j=0 F δ−j . We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H 2 diff (vect(1), D ν,µ ) where D ν,µ is the space of differential operators from F ν to F µ . Necessary second-order integrability conditions of any infinitesimal deformations of S n δ are given. We describe completely the formal deformations for some spaces S n δ and we give concrete examples of non trivial deformations.
We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.
We investigate the first cohomology space associated with the embedding of the Lie Orthosymplectic superalgebra osp(n|2) on the (1,n)-dimensional superspace R 1|n in the Lie superalgebra S DO(n) of superpseudodifferential operators with smooth coefficients, where n = 0, 1, 2. Following Ovsienko and Roger, we give erxplicit expressions of the basis cocycles.
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