Michail Zhitomirskii

Feedback classification of nonlinear control systems on 3-manifolds

We consider nonlinear control-affine systems with two inputs evolving on three-dimensional manifolds. We study their local classification under static state feedback. Under the assumption that the control vector fields are independent we give complete classification of generic systems. We prove that out of a “singular” smooth curve a generic control system is either structurally stable and thus equivalent to one of six canonical forms (models) or finitely determined and thus equivalent to one of two canonical forms with real parameters. Moreover, we show that at points of the “singular” curve the system is not finitely determined and we give normal forms containing functional moduli. We also study geometry of singularities, i.e., we describe surfaces, curves, and isolated points where the system admits its canonical forms.

Simple germs of corank one affine distributions

1.1. Affine distributions. All objects are assumed to be smooth (of class C∞). An affine distribution on R of rank m (or corank n − m) is a family A = {Ap}p∈Rn of m-dimensional affine subspaces Ap ⊂ TpR. If Ap is a subspace, i.e., Ap contains the zero tangent vector, then p is called an equilibrium point of A. Two germs A and A of corank one affine distributions, at points p and p respectively, are equivalent if there exists a local diffeomorphism Φ sending p to p such that Φ∗(Ax) = AΦ(x) for each x close to p.

SINGULARITIES AND BIFURCATIONS OF 3-DIMENSIONAL POISSON STRUCTURES

We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable 1-forms on ℝ3.

Odd-dimensional pfaffian equations: Reduction to the hypersurface of singular points

Abstract Let ω be a local nonvanishing differential 1-form on a (2k+1)-dimensional manifold with structurally smooth hypersurface S of singular points (the points at which us A (dw)k vanishes). We prove that in the holomorphic, real-analytic, and C ∞ categories, the Pfaffian equation (ω) is determined, up to a diffeomorphism, by its restriction to S , a canonical connection, and (in the real-analytic and the C ∞ cases) a canonical orientation. On the other hand, if we exclude certain degenerations of infinite codimension, then the restriction determines the connection and the orientation. Then (ω) is determined, up to a diffeomorphism, by its restriction to S .

Modules of vector fields, differential forms and degenerations of differential systems

AbstractWe deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X1,..., Xn−1) (codimension 1 differential systems) defined locally on ℝn or ℂn, and extend the standard duality(X1,..., Xn−1)↦(ω), ω=Ω(X1,...,Xn−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsXi are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and

Completely symmetric centers

(\tilde X_1 ,...,\tilde X_{n - 1} )

Points and curves in the Monster tower

are equivalent if and only if so are the corresponding Pfaffian equations (ω) and