  In various biomedical applications, precise focusing of nonlinear ultrasonic
waves is crucial for efficiency and safety of the involved procedures. This
work analyzes a class of shape optimization problems constrained by general
quasi-linear acoustic wave equations that arise in high-intensity focused
ultrasound (HIFU) applications.
  Within our theoretical framework, the Westervelt and Kuznetsov equations of
nonlinear acoustics are obtained as particular cases. The quadratic gradient
nonlinearity, specific to the Kuznetsov equation, requires special attention
throughout. To prove the existence of the Eulerian shape derivative, we
successively study the local well-posedness and regularity of the forward
problem, uniformly with respect to shape variations, and prove that it does not
degenerate under the hypothesis of small initial and boundary data.
Additionally, we prove H\"older-continuity of the acoustic potential with
respect to domain deformations. We then derive and analyze the corresponding
adjoint problems for several different cost functionals of practical interest
and conclude with the expressions of well-defined shape derivatives.
