Some aspects of control systems as dynamical systems

Abstract

Consider the smooth control system $$\dot x = X_0 (x) + \sum\limits_{i = 1}^m {u_i X_i (x)}$$ (C) on a manifold M with admissible controlsu eU={u:R →U, locally integrable} and compact control spaceU ⊂Rm. Associated with (C) is a dynamical system whereθt is the shift byt eR to the right onU, andϕ(t, x, u) is the solution of (C) at timet eR with initial condition ϑ(0,x, u) = x, under the control action of ueU We discuss some connections between control properties of (C) and basic notions for dynamical systems, such as topological mixing, chain recurrence, recurrence, invariant (ergodic) measures, and their support. It turns out that these concepts for (D) are related to the control sets and chain control sets of (C): A setD ⊂M is a control set of (C) iff the liftD=cl{(u, x) eU ×M,ϕ(t, x, u) e D for allt eR} toU×M is a maximal topologically mixing (transitive) component ofφ, similarly for the lifts of chain control sets and the components of the chain recurrent set ofφ. Furthermore, ifμ is an ergodic, invariant measure ofφ, thenπM(suppμ) ⊂=D for some control setD ⊂M, and the pointsx e M that are contained in control sets, are the projections ontoM ofφ-recurrent points.