  Many problems in engineering can be understood as controlling the bifurcation
structure of a given device. For example, one may wish to delay the onset of
instability, or bring forward a bifurcation to enable rapid switching between
states. We propose a numerical technique for controlling the bifurcation
diagram of a nonlinear partial differential equation by varying the shape of
the domain. Specifically, we are able to delay or advance a given branch point
to a target parameter value. The algorithm consists of solving a shape
optimization problem constrained by an augmented system of equations, the
Moore--Spence system, that characterize the location of the branch points.
Numerical experiments on the Allen--Cahn, Navier--Stokes, and hyperelasticity
equations demonstrate the effectiveness of this technique in a wide range of
settings.
