On local linearization of control systems
Résumé
We consider the problem of topological linearization of smooth (meaning either infinitely differentiable or real analytic) control systems, i.e. of their local equivalence to a linear controllable system via pointwise transformations on the state and the control that are topological but not necessarily differentiable. In the language of control theory, such transformations reduce to (topological) static feedback transformations. We prove that topological linearization implies smooth (with the same meaning as above) linearization, away from singularities. At "regular" singular points, we show that topological linearizability implies local conjugation to a linear system via a smooth mapping, although this time the inverse map needs not be smooth. Finally, deciding whether the same is true at "strongly" singular points is tantamount to solve an intriguing open question in differential topology. Our results entail that control systems sharply differ from ordinary differential equations, in that a naive analog to the Grobman-Hartman theorem (i.e. topological linearizability at generic fixed points) cannot hold.