ISSN:
1433-0490

Keywords:
Nonlinear systems
;
controllability
;
reachable set
;
vector bundle

Source:
Springer Online Journal Archives 1860-2000

Topics:
Computer Science

Notes:
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.

Type of Medium:
Electronic Resource

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