Controllability of general nonlinear systems

Abstract

Consider the nonlinear system

$$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t)), x(0) = x_0 \in M}$$

whereM is aC ^∞ realn-dimensional manifold,f, g ₁,⋯.,g _m areC ^∞ vector fields onM, andu ₁ ,..,u _m are real-valued controls. Ifm=n−1 andf, g ₁ ,⋯,g _m are linearly independent, then the system is called a hypersurface system, and necessary and sufficient conditions for controllability are known. For a generalm, 1 ≤m ≤n−1, and arbitraryC ^∞ vector fields,f, g ₁ ,⋯,g _m , assume that the Lie algebra generated byf, g ₁ ,⋯,g _m and by taking successive Lie brackets of these vector fields is a vector bundle with constant fiber (vector space) dimensionp onM. By Chow's Theorem there exists a maximalC ^∞ realp-dimensional submanifoldS ofM containingx ₀ with the generated bundle as its tangent bundle. It is known that the reachable set fromx ₀ must contain an open set inS. The largest open subsetU ofS which is reachable fromx ₀ is called the region of reachability fromx ₀. IfO is an open subset ofS which is reachable fromx ₀,S we find necessary conditions and sufficient conditions on the boundary ofO inS so thatO = U. Best results are obtained when it is assumed that the Lie algebra generated byg ₁,⋯,g _m and their Lie brackets is a vector bundle onM.