In this work, we focus on the controllability and its cost for some linear or nonlinar partial differential equations coming from physics. The first part of the thesis deals with the null controllability of the three-dimensional Navier-Stokes equations with Dirichlet boundary conditions and internal control distributed on a subdomain acting only one of the three equations. The proof is based on the return method as well as an original method of algebraic solvability of differential systems inspired by the works of Gromov. The second part of the thesis concerns the cost of control in small time or in the vanishing viscosity limit for linear unidimensional equations. At first, we show that we can in some cases make a link between these two issues, notably that it is possible to obtain results of uniform controllability for the transport-diffusion equation with constant coefficients controlled on the left side from known results concerning the heat equation. In a second step, we look at the cost of boundary control in small time for some equations for which the associated spatial operator is self-adjoint or skew-adjoint with compact resolvent and having eigenvalues that behaves asympotically as a polynomial, using the method of moments. We deduce results for linearized Korteweg-de-Vries equation, fractional diffusion equation and fractional Schrödinger equation.
