Controllability and nonlinearity

A usual problem in control theory is the problem of controllability: given two states, is it possible to go from the first one to the second one by means of suitable control? Even in finite dimension, find a necessary and sufficient condition for controllability is out of reach. One can restrict our goal to the study of local controllability. In this case the two states are close to some given equilibrium. A major method to study the local controllability around an equilibrium is to look at the controllability of the linearized control system around this equilibrium. Indeed, using the inverse mapping theorem, the controllability of this linearized control system implies the local controllability of the nonlinear control system, in any cases in finite dimension and in many cases in infinite dimension. In infinite dimension the situation can be more complicated due to some problems of “loss of derivatives” as we shall see on examples. However, classical iterative schemes, as we shall see for hyperbolic systems [7], and the Nash-Moser method as introduced by Karine Beauchard in [1] for Schrödinger (see also [2]) can allow to handle some of these cases. When the linearized control system around the equilibrium is not controllable, the situation is more complicated. However, for finite-dimensional systems, one knows powerful tools to handle this situation. These tools rely on iterated Lie brackets. They lead to many sufficient or necessary conditions for local controllability of a nonlinear control system. We shall recall some of these conditions. In infinite dimension, iterated Lie brackets give some interesting results. However we shall see that these iterated Lie brackets do not work so well in many interesting cases. We present here three methods to get in some cases controllability results for some control systems modeled by partial differential equations even if the linearized control system around the equilibrium is not controllable. These methods are