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

Steady state programming of controlled nonlinear systems via deep dynamic mode decomposition