ISSN:
1572-9222
Keywords:
Control systems
;
dynamical systems
;
control sets
;
topological mixing
;
chain recurrence
;
recurrence
;
invariant measures
;
93C10
;
93B05
;
58F11
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Consider the smooth control system (C) $$\dot x = X_0 (x) + \sum\limits_{i = 1}^m {u_i X_i (x)}$$ on a manifold M with admissible controlsu εU={u:R →U, locally integrable} and compact control spaceU ⊂R m. Associated with (C) is a dynamical system whereθ t is the shift byt ε R to the right onU, andϕ(t, x, u) is the solution of (C) at timet ε R with initial condition ϑ(0,x, u) = x, under the control action of uεU 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) εU ×M,ϕ(t, x, u) ε D for allt ε R} 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, thenπ M(suppμ) ⊂=D for some control setD ⊂M, and the pointsx ε M that are contained in control sets, are the projections ontoM ofφ-recurrent points.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01053532
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