Linear control semigroups acting on projective space

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 eL 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.