  This paper describes the optimal selection of a control policy to program the
steady state of controlled nonlinear systems with hyperbolic fixed points. This
work is motivated by the field of synthetic biology, in which saddle points are
common (along with limit cycles), and the aim is to program cells to perform
both digital and analog computation, though developing genetic digital
computation has been the main focus. We frame the analog computing challenge of
generating a steady state input-output function inside living cells. To program
the steady state, a data-driven approach is taken wherein an approximation of
the Koopman operator, identified via deep dynamic mode decomposition, is used
to describe the dynamics of the system linearly. The new representation of the
dynamics are then used to solve an optimization problem for the input which
maximizes a direction in state space. Some added structure on the Koopman
operator learning process for controlled systems is given for dynamics that are
separable in the state and input. Finally, the methods are demonstrated on
simulation examples of an incoherent feedforward loop and a combinatorial
promoter system, two common network architectures seen in the field of
synthetic biology.
