Ivancevic_Applied-Diff-Geom

568 Applied Differential Geometry: A Modern Introduction{f, {g, h}}}+{g, {h, f}}+{h, {f, g}} = 0, and {fg, h} = {f, h}g +f{g, h})allows us to get a Hamiltonian vector–field X h with Hamiltonian h throughthe equalityL Xh f = {f, h}, for all f ∈ C ∞ (M),where L Xh f is the Lie derivative of f along X h . Note that the vector–fieldX h is well defined since the Poisson bracket verifies the Leibniz rule andtherefore defines a derivation on C ∞ (M) (see [Marsden and Ratiu (1999)]).Furthermore C ∞ (M) equipped with a Poisson bracket is a Lie algebra,called a Poisson algebra. Also, we say that the Poisson structure on M isnondegenerate if the {, }−associated map B # : T ∗ M → T M defined bydg(B # (x)(df)) = B(x)(df, dg),(where df denotes the exterior derivative of f) is an isomorphism for everyx ∈ M.An affine Hamiltonian control system Σ = (U, M, h) consists of asmooth manifold U (the input space), a Poisson manifold M with nondegeneratePoisson bracket (the state–space), and a smooth function H :M × U → R (the controlled Hamiltonian). Furthermore, H is locally ofthe form H = h 0 + h i u i (i = 1, ..., n), with h i locally defined smooth realvalued maps and u i local coordinates for U [Tabuada and Pappas (2001)].Using the controlled Hamiltonian and the Poisson structure on M wecan recover the familiar system map F : M × U → T M, locally given byF = X h0 + X hi u i ,and defines an affine distribution on M given byD M (x) = X h0 (x) + span{X h1 (x), X h2 (x), ..., X hn (x)}.This distribution captures all the possible directions of motion availableat a certain point x, and therefore describes a control system, up to aparametrization by control inputs. This affine distribution will is our mainobject of interest here, and we will assume that the rank of D M does notchange with x. Furthermore, we denote an affine distribution D M by X+∆,where X is a vector–field and ∆ a distribution. When this affine distributionis defined by a Hamiltonian control system we have X = X h0 and ∆ =span{X h1 (x), X h2 (x), ..., X hn (x)}. A similar reasoning is possible at thelevel of Hamiltonians. Locally, we can define the following affine space of