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This paper proposes a topological framework for the analysis of the time shift on behaviors. It is shown that controllability is not a property of the time shift, while chain controllability is. This also leads to a global decomposition.
For control systems described by ordinary differential equations subject to almost periodic excitations the controllability properties depend on the specific excitation. Here these properties and, in particular, control sets and chain control sets are discussed for all excitations in the closure of all time shifts of a given almost periodic function. Then relations between heteroclinic orbits of an uncontrolled and unperturbed system and controllability for small control ranges and small perturbations are studied using Melnikov's method. Finally, a system with two-well potential is studied in detail.
The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.
For control systems described by ordinary differential equations subject to almost periodic excitations the controllability properties depend on the specific excitation. Here these properties and, in particular, control sets and chain control sets are discussed for all excitations in the closure of all time shifts of a given almost periodic function. Then relations between heteroclinic orbits of an uncontrolled and unperturbed system and controllability for small control ranges and small perturbations are studied using Melnikov's method. Finally, a system with two-well potential is studied in detail.
For continuous time control systems, this paper introduces invariance entropy as a measure for the amount of information necessary to achieve invariance of weakly invariant compact subsets of the state space. Upper and lower bounds are derived, in particular, finiteness is proven. For linear control systems with compact control range, the invariance entropy is given by the sum of the real parts of the unstable eigenvalues of the uncontrolled system. A characterization via covers and corresponding feedbacks is provided.
For continuous-time control systems with outputs, this paper analyzes invariance entropy as a measure for the information rate necessary to achieve invariance of compact subsets of the output space. For linear control systems with compact control range, relations to controllability and observability properties are studied. Furthermore, the notion of asymptotic invariance entropy is introduced and characterized for these systems.
For continuous-time linear control systems, a concept of entropy for controlled and almost controlled invariant subspaces is introduced. Upper bounds for the entropy in terms of the eigenvalues of the autonomous subsystem are derived.
Invariance entropy for the action of topological semigroups acting on metric spaces is introduced. It is shown that invariance entropy is invariant under conjugations and a lower bound and upper bounds of invariance entropy are obtained. The special case of control systems is discussed.
