Linear control semigroups acting on projective space

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.