ISSN:
1572-9222
Keywords:
semilinear control systems
;
bilinear control systems
;
linear control semigroups
;
control sets
;
chain control sets
;
control flows on vector bundles
;
93B05
;
93C10
;
58F25
;
58F12
;
34C35
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Linear control semigroupsL⊂=Gl(d,R) are associated with semilinear control systems of the form whereA:R m → gl(d,R) is continuous in some open set containingU. The semigroupL then corresponds to the solutions with piecewise constant controls, i.e., L acts in a natural way onR d {0}, on the sphereS d−1, and on the projective spaceP d−1. Under the assumption that the group generated byL in Gl(d,R) acts transitively onP d−1, we analyze the control structure of the action ofL onP d−1: We characterize the sets inP d−1, where the system is controllable (the control sets) using perturbation theory of eigenvalues and (generalized) eigenspaces of the matrices g εL For nonlinear control systems on finitedimensional manifoldsM, we study the linearization on the tangent bundleTM and the projective bundleP M via the theory of Morse decompositions, to obtain a characterization of the chain-recurrent components of the control flow onU×PM. These components correspond uniquely to the chain control sets onP M, and they induce a subbundle decomposition ofU×TM. These results are used to characterize the chain control sets ofL acting onP d−1 and to compare the control sets and chain control sets.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01053533
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