A control system is a dynamical system on which one can act by means of suitable controls. The are two important issues for these systems, namely the controllability problem and the stabilization problem.

The controllability problem is the following one. Given two states, is it possible to steer the control system from the first state to the second one.

The stabilization problem is the following one. We have an equilibrium which is unstable without the use of the control. Let us give a concrete example. One has a stick that is placed vertically on one of his fingers. In principle, if the stick is exactly vertical with a vanishing speed, it should remain vertical. But, due to various small errors (the stick is not exactly vertical, for example), in practice, the stick falls down. In order to avoid this, one moves the finger in a suitable way, depending on the position and speed of the stick; we use  feedback laws  (or closed-loop controls) which stabilize the equilibrium.The problem of stabilization is the existence and construction of such stabilizing feedback laws for a given control system.

We first study these two problems when the control system is modeled by means of ordinary differential equations. We present applications to simple physical systems. The emphasis is put on cases where the nonlinearity plays a crucial role both for the controllability and stabilization issues.

Then we move to the case of control systems modeled by means of partial differential equations. We start by presenting various methods for the controllability of linear systems and then study cases where again the nonlinear terms are crucial. Applications are presented to equations coming from fluid and quantum mechanics.

Jean-Michel Coron, Control and nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence, 2007.
Jean-Michel Coron, Control of partial differential equations,  Scholarpedia.
Alberto Isidori, Nonlinear control systems, third ed., Communications and Control Engineering Series, Springer-Verlag, Berlin, 1995.
Jacques-Louis Lions, Contrôlabilite exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées, vol. 8, Masson, Paris, 1988.
Henk Nijmeijer et Arjan van der Schaft, Nonlinear dynamical control systems, Springer-Verlag, New York, 1990.