This thesis is divided in two parts. The main tool of this work is time optimal control. We first consider the Pontryagin maximum principle for control system of finite dimension. After that, we give an application of this principle for the Brockett integrator with state constraints. Then, we study an extension of the Pontryagin maximum principle in the case of infinite dimensional systems. More precisely, this extension concerns the case of exactly controllable systems in any time. For instance, this can be the Schrödinger equation with internal control. Especially under some condition of approximate controllability, we can show the existence of a bang-bang control defined on a time set of positive measure. In the second part, we study the problem of swimming at low Reynolds number. A convenient physical model allows us to formulate it under the form of a control problem. We then get a controllability result on this problem. More precisely, we will show that whatever the shape of the swimmer is, the swimmer can slightly modify its shape in order to steer any prescribed trajectory. To complete this part, we consider the case of an axi-symmetric swimmer. The results of the first part allow us to find an optimal time control.
