A control theory on the configuration space and singular point assignment via the Conley index theory are proposed. To assign a global topological structure of control systems, a set of singular points is allocated on the null manifold of state equations. This topological structure of control systems can be described by homology Conley index braids and homology connection matrices. Generalized control systems are defined on configuration spaces. On the configuration space we discuss properties

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... f the uncontrolable invariant null manifold, the separating null manifold and projective dimensions of null manifolds for the input-output manifold.