Note. Continuous-time systems only fit into action framework when using piecewise constant controls. If we want the action manifold to be second countable, the we must further restrict piecewise constant controls to take values in a countable dense subset of control space. To develop a general framework that encompasses both discrete-time and continuous-time systems (with arbitrary admissible controls) we need to move to pseudo semi-groups of local diffeomorphisms (PSG).

Consider a set Λ formed as a disjoint union Λ = ⋃ w∈ W (0, infinity) x { w }. Note that O has the structure of a one-dimensional differentiable manifold, but if O has an uncountable number of points, then the set Λ is not second countable.

Let M be a Cr manifold of dimension m with 2 ≤ r ≤ infinity or r = w , and let O be a separable metric space. Assume that manifolds are second countable and Hausdorff, hence metrizable. In many useful situations the normal accessibility property has been shown to be equivalent to the accessibility property. This equivalence was established first for systems of vector fields (that is, control systems with piecewise constant controls), and later for control systems with arbitrary measurable controls by K. Grasse and H. Sussmann. The results of this thesis further extend the equivalence of the accessibility and normal accessibility properties to actions, and to an even more general formulation of control systems involving pseudo semi-groups of local diffeomorphisms.

Using the formalism of pseudo semi-groups we provide a different prove of a known result that for a system of vector fields the accessibility property implies the normal accessibility property.

Theorem. Let M and Λ be finite dimensional C1 differentiable manifolds with Λ being second countable. If the action Gamma : M x Λ → M of Λ on M has the accessibility property, then it has the normal accessibility property.

Definition. A Cr action of Λ on M is a Cr map Gamma : M x Λ → M such that dom(Gamma) is an open subset of M x Λ and lambda ∈ Λ ⇒ Gammalambda ∈ Diff1loc (M), where Gammalambda is the partial map, Gammalambda(x) = Gamma(x, lambda).

Our interest in actions arises from the possibility of dealing with both continuous and discrete-time control systems under the framework of a single theory. Embracing both these types of control systems, Sontag's formalism of actions offers an effective tool of research in geometric control theory.

Theorem. Let S be a parametrized PSG containing the PSG generated by an action Gamma with Λ being second countable. If for any x0 ∈ M certain imbedded submanifolds have certain stability property and if the action Gamma is dense relative to the given parametrization, then the accessibility property of S implies the normal accessibility property of Gamma.

Let M and Λ be Cr differentiable manifolds with dim(M) = m ∈ IN and dim(Λ ) = l ∈ IN ∪ {0}.