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On local linearization of control systems. (English) Zbl 1203.93044
Summary: We consider the problem of topological linearization of smooth ($$C^{\infty }$$ or $$C^{\omega }$$) control systems, i.e., of their local equivalence to a linear controllable system via point-wise transformations on the state and the control (static feedback transformations) that are topological but not necessarily differentiable. We prove that the local topological linearization implies the local smooth linearization, at generic points. At arbitrary points, it implies the local conjugation to a linear system via a homeomorphism that induces a smooth diffeomorphism on the state variables, and, except at “strongly” singular points, this homeomorphism can be chosen to be a smooth mapping (the inverse map needs not be smooth). Deciding whether the same is true at “strongly” singular points is tantamount to solve an intriguing open question in differential topology.
##### MSC:
 93B18 Linearizations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 37C10 Dynamics induced by flows and semiflows
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