Optimal control of bifurcation structures
Abstract
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. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain or a parameter in the equation. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving an optimization problem constrained by an augmented system of equations that characterize the location of the branch points. The flexibility and robustness of the method also allow us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency. We will apply this technique on systems arising from biology, fluid dynamics, and engineering, such as the FitzHugh-Nagumo model, Navier-Stokes, and hyperelasticity equations.