In this talk, I will discuss two independent nonlinear control
problems. First, I will consider the damped cubic focusing wave
equation. I will present both positive and negative stabilization
results below the ground state energy and discuss some extensions above
it, including results derived from the analysis of the Duffing equation.
Second, I will examine small-time local controllability near the ground
state for a bilinear Schrödinger equation with Neumann boundary
conditions. I will present a new result showing that, if the linearized
system is not controllable, a Lie bracket–type condition ensures that
either the nonlinear system exhibits a quadratic obstruction or,
remarkably, recovers controllability at the quadratic order. This second
part is a joint work with Karine Beauchard and Frédéric Marbach.