Some aspects of control systems as dynamical systems |
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Authors: | Fritz Colonius Wolfgang Kliemann |
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Institution: | (1) Institut für Mathematik, Universität Augsburg, D-86159 Augsburg, FRG;(2) Department of Mathematics, Iowa State University, 50011 Ames, Iowa |
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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 L 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.Research supported in part by NSF Grant DMS 8813976 and DFG Grant Co 124/6-1. |
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Keywords: | semilinear control systems bilinear control systems linear control semigroups control sets chain control sets control flows on vector bundles |
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